A Prototype Quantum Computer Using Nuclear Spins in Liquid Solution

A PROTOTYPE QUANTUM COMPUTER USING NUCLEAR SPINS IN LIQUID SOLUTION

a dissertation submitted to the department of electrical engineering and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy

Matthias Steﬀen June 2003 c Copyright by Matthias Steﬀen 2003

ii I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

James S. Harris (Principal adviser)

I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

Isaac L. Chuang (Co-adviser)

I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

Yoshihisa Yamamoto

Approved for the University Committee on Graduate Studies.

iii iv Abstract

Quantum computers can potentially solve real and relevant mathematical and physical problems that are intractable using classical machines. However, the experimental realization of quantum computers represents a signiﬁcant challenge, because several opposing experimental requirements must be met. A set of coupled quantum bits must be manipulated and measured while coherently retaining entangled quantum states. Yet the manipulation and measurement processes almost inevitably lead to the decay of these fragile states. This thesis work takes signiﬁcant steps towards building a practical quantum computer using nuclear spins and liquid state nuclear magnetic resonance (NMR) techniques. I present experimental results for proof of principle of quantum computing in a series of small implementations of quantum algorithms, culminating in the implementation of an adiabatic quantum optimization algorithm. The performance of adiabatic algorithms compared with classical optimization methods is unknown, but small quantum computers could provide crucial insight into answering this question. Furthermore, adiabatic algorithms also shed new light on the usefulness of quantum resources for computational tasks and hence they represent an important class of algorithms. The second part of this thesis presents several methods developed for improving the control over NMR quantum computing experiments. Even though liquid state NMR quantum computers have well known and accepted scaling limitations, the developed tools are of general use. I will show how several tools are directly transferable to two other implementations of quantum computers; one using optical methods and the other using loops of superconducting material. This work provides insight into what is needed to build a practical quantum computer, and delivers several tools and techniques that will be useful in future large-scale implementations of quantum computers.

v Acknowledgements

Equipped with a background in Physics from Emory University, I came to Stanford Univer- sity, looking for an exciting research project to join. Before I knew it, I found myself interested in a wide selection of research groups including Jim Harris’ group, or more specifically the quantum computing subgroup. He referred me to Lieven Vandersypen who patiently explained what quantum computing is all about and how he was involved in it. This topic truly fascinated me not just because of its enormous future applications but also because of the significant experimental work involved. Soon after, I met Isaac Chuang who initiated the entire NMR Quantum Computing project at IBM. He offered me to join the group, and this was the beginning of an exciting 4.5 year journey that started at IBM Almaden in San Jose and ended up at MIT in Boston. Jim Harris (aka coach) was my advisor at Stanford and introduced me to the right people that made my work a success. I deeply appreciate his encouragement and support over the years and his outlook on life in general. Isaac Chuang, my co-advisor guided and inspired me always at the right times. He taught me how to put my work into perspective, to think outside the box and helped me continue to develop important presentation and writing skills. After he chose to accept a position at MIT, I decided to follow him and I appreciate having learned about the research setting in industry (at IBM) and academia (at MIT). Lieven Vandersypen has been a tremendous mentor during my 2.5 years at IBM. I started as his apprentice but we soon grew together to form a very productive team. I look back at this time with great pride, and I am truly fortunate to have with him during this period. Since we worked so closely together over the years, there is substantial overlap between his and my thesis. It’s hard for any one of us to fully take credit for an experiment that we worked on together. Most of our ideas and work has been a combined effort, but there are different areas of concentrations. Lieven worked a lot on the early simulation code, the

vi decoherence model, did a significant portion of the writing of the papers, and worked out some of the quantum circuits used in chapter 6. I subsequently simplified several of these circuits, and focused on the software reference frame used for the seven spin framework (shown in the Appendix) as well as the solutions to several artifacts that arose throughout our experiments (shown in chapter 5). Plenty of thanks to the entire IBM crew. Nino Yannoni (Nino) found most of the molecules we ended up using - very spectacular molecules I might add. I enjoyed his words of wisdom which he was very willing to share with us during the most enjoyable lunch hour. Mark Sherwood always offered helpful hints and comments pertaining to NMR techniques and molecule choices, while ensuring adequate coffee supplies. Greg Breyta synthesized all of the molecules that Nino discovered during his free time - a commitment he did not have to make. My other colleagues, Anne Verhulst and Oskar Liivak, contributed to a very pleasant working atmosphere and provided numerous useful discussions. Of course, I also wish to thank the new group that was formed at MIT. Andrew Houck was the first student that joined our group. He continuously found ways to bring humor into the laboratory (Maxwell’s Demon). Steve Huang, our electronics guru, followed soon after and I had some great discussions with him. Many thanks also to (and this is in no particular order) Teri, Aram, Francois, Andrew (Cross), Josh, Joshua, and Zilong for being such spectacular group members. Murali Kota, a fine enthusiastic student, started as my apprentice at MIT, but unfortunately I was not able to convince him to pursue NMR quantum computation “full time”. Nonetheless, he had several very interesting experimental proposals, which are now part of chapters 4, 6 and 7 of my thesis. Bin is an extraordinary undergraduate student who worked with us for his undergraduate thesis. He learned the spectrometer details at an amazing pace, and started running experiments soon after he joined the group. Him, Murali and I together worked on the connection between the EIT effect and quantum computing as outlined in section 7.5. Gail Chun Creech was extremely helpful in coordinating a lot of the complex adminis- trative tasks resulting from my move to Boston. The same goes for the rest of the admin- istrational staff of Stanford and MIT (Maureen, Mike, and the Ex’s Murray and Claire). Thanks to everyone else who enriched my life throughout my graduate school career. The Neil Gershenfeld gang (Jason, Yael, Ben), Wim van Dam and Tad Hogg from HP, who

vii helped me tremendously on one of my later projects. To my friends and “the boys” who supported me when needed. My girlfriend Jennifer Padgett was always there for me when I had concerns or just whenever I needed support. Last but certainly not least, thank you to my parents, brother, uncle and aunt who have always believed in me and my abilities, and taught me how to succeed in life.

viii Contents

Abstract v

Acknowledgements vi

1 Introduction 1 1.1 Historical background ...... 1 1.2 Related ﬁelds and additional literature ...... 5 1.3 Goals of my work ...... 6 1.4 Organization of the dissertation ...... 7

2 Theory of quantum computing 10 2.1 Quantum computing vs. classical computing ...... 10 2.1.1 Quantum bits vs. classical bits ...... 10 2.1.2 Quantum computing subsumes classical computing ...... 15 2.1.3 Quantum parallelism ...... 19 2.1.4 Complexity theory ...... 22 2.1.5 Quantum error correction ...... 23 2.2 Quantum gates ...... 25 2.2.1 Universal quantum gates ...... 26 2.2.2 Single qubit gates ...... 26 2.2.3 Two-qubit gates ...... 29 2.2.4 Remarks on unitary gates ...... 31 2.2.5 Multiple (n > 2) qubit gates ...... 32 2.3 Quantum algorithms ...... 34 2.3.1 The Deutsch-Jozsa algorithm ...... 35

ix 2.3.2 Grover’s algorithm ...... 38 2.3.3 Order-ﬁnding and Shor’s algorithm ...... 42 2.3.4 Adiabatic quantum algorithms ...... 50 2.3.5 Quantum simulations ...... 52 2.3.6 Perspectives ...... 53 2.4 Decoherence ...... 54 2.4.1 Energy dissipation ...... 54 2.4.2 Phase randomization ...... 56 2.4.3 Remarks on amplitude and phase damping ...... 58 2.4.4 Other sources of decoherence ...... 58 2.5 Summary ...... 59

3 Implementation of quantum computers 61 3.1 Requirements ...... 61 3.1.1 System of qubits ...... 63 3.1.2 Quantum gates ...... 64 3.1.3 Initialization ...... 67 3.1.4 Measurement ...... 70 3.1.5 Coherence ...... 72 3.2 State of the art ...... 73 3.3 Summary ...... 74

4 Liquid-state NMR quantum computing 75 4.1 System of qubits ...... 75 4.1.1 Single-spin Hamiltonian ...... 76 4.1.2 Distinguishing the qubits ...... 77 4.2 Single-qubit quantum gates ...... 79 4.2.1 Rotations in thex ˆ-ˆy plane ...... 79 4.2.2 Rotations about thez ˆ axis ...... 82 4.2.3 Selective excitation using shaped RF pulses ...... 83 4.3 Two-qubit quantum gates ...... 87 4.3.1 Interaction Hamiltonian ...... 87 4.3.2 The controlled NOT gate ...... 90 4.3.3 Refocusing J-couplings ...... 91

x 4.4 Initialization ...... 95 4.4.1 Thermal equilibrium ...... 96 4.4.2 Schulman-Vazirani cooling ...... 96 4.4.3 Eﬀective pure states ...... 98 4.4.4 Spatial labeling ...... 99 4.4.5 Logical labeling ...... 101 4.4.6 Temporal labeling ...... 102 4.5 Measurement ...... 106 4.5.1 Interpretation of NMR spectra ...... 106 4.5.2 Signal-to-noise ratio ...... 110 4.5.3 Quantum state tomography ...... 111 4.6 Decoherence ...... 112 4.6.1 Causes and eﬀects of decoherence ...... 113 4.6.2 Measurement of decoherence times ...... 116 4.7 Higher-order spins ...... 118 4.7.1 System of qubits ...... 119 4.7.2 Single and multiple-qubit gates ...... 119 4.7.3 Initialization ...... 121 4.7.4 Measurement ...... 122 4.7.5 Coherence times ...... 122 4.8 Molecule design ...... 123 4.9 The ”quantumness” of NMRQC ...... 124 4.10 Summary ...... 125

5 Tools of liquid-state NMR quantum computing 126 5.1 Excitation of spins oﬀ-resonance ...... 126 5.2 Transient Bloch-Siegert shifts ...... 128 5.2.1 Calculation of transient Bloch-Siegert shifts ...... 128 5.2.2 Correction of the transient Bloch-Siegert shift ...... 129 5.3 Simultaneous selective pulses - artifacts and solutions ...... 130 5.4 Unwinding J-coupling during selective pulses ...... 133 5.4.1 Two qubit model ...... 134 5.4.2 Multiple qubit model ...... 136

xi 5.4.3 Extension to multiple single qubit rotations ...... 139 5.5 Pulse sequence design and simpliﬁcation ...... 141 5.6 Pulse sequence frameworks ...... 143 5.7 Other advances ...... 144 5.8 Summary ...... 144

6 Experimental realization of NMR quantum computers 146 6.1 The NMR spectrometer ...... 146 6.1.1 Sample ...... 147 6.1.2 Magnet ...... 149 6.1.3 Probe ...... 153 6.1.4 Transmitter ...... 156 6.1.5 Receiver ...... 159 6.1.6 Console (Workstation) ...... 159 6.2 Overview of NMRQC experiments ...... 160 6.3 Grover search algorithm (3 qubits) ...... 165 6.3.1 Problem statement ...... 165 6.3.2 Experimental set-up ...... 166 6.3.3 Experimental results ...... 166 6.3.4 Discussion ...... 167 6.4 Schulman-Vazirani cooling (3 qubits) ...... 169 6.4.1 Problem statement ...... 169 6.4.2 Experimental set-up ...... 170 6.4.3 Experimental results ...... 172 6.4.4 Discussion ...... 173 6.5 Order-ﬁnding (5 qubits) ...... 175 6.5.1 Problem statement ...... 175 6.5.2 Experimental set-up ...... 176 6.5.3 Experimental results ...... 179 6.5.4 Discussion ...... 179 6.6 Shor’s factoring algorithm (7 qubits) ...... 180 6.6.1 Problem statement ...... 180 6.6.2 Experimental set-up ...... 185

xii 6.6.3 Experimental results ...... 190 6.6.4 Discussion ...... 196 6.7 Grover search using higher-order spins (2 qubits) ...... 196 6.7.1 Problem statement ...... 196 6.7.2 Experimental set-up ...... 199 6.7.3 Experimental results ...... 200 6.7.4 Discussion ...... 207 6.8 Adiabatic quantum optimization algorithm (3 qubits) ...... 207 6.8.1 Problem statement ...... 207 6.8.2 Experimental set-up ...... 208 6.8.3 Experimental results ...... 213 6.8.4 Discussion ...... 213 6.9 Summary ...... 215

7 Applications of NMRQC 217 7.1 Superconducting qubits ...... 217 7.1.1 The Cooper pair box ...... 219 7.1.2 Flux qubits ...... 228 7.1.3 Josephson phase qubits ...... 236 7.1.4 Decoherence and recent experiments ...... 239 7.2 Ion trap quantum computers ...... 240 7.2.1 The qubit states and single-qubit rotations ...... 241 7.2.2 Measurement and initialization ...... 241 7.2.3 Trapping and motion of the ions ...... 242 7.2.4 Two-qubit gates ...... 245 7.2.5 Decoherence, summary, and perspectives ...... 247 7.3 Design of atomic physics pulse sequences ...... 248 7.4 Design of pulses for superconducting loop quantum computers ...... 251 7.4.1 Problem statement ...... 251 7.4.2 Results: Pulse shaping ...... 254 7.4.3 Results: Composite pulses ...... 257 7.4.4 Eﬀects of tunneling ...... 262 7.4.5 Summary ...... 268

xiii 7.5 Quantum computing and quantum optical phenomena ...... 268 7.5.1 The EIT eﬀect ...... 269 7.5.2 Simulation of the EIT eﬀect (3 energy levels) ...... 276 7.6 Summary ...... 287

8 Conclusions 289

A Numerical Models 292 A.1 Set-up Hamiltonians and Pauli matrices ...... 293 A.2 Ideal single-qubit rotations ...... 294 A.3 Time evolution under the natural Hamiltonian ...... 295 A.4 Action of a unitary matrix on a density matrix ...... 296 A.5 Generalized operator ...... 296 A.6 Kronecker product function ...... 297 A.7 Simulated single-qubit rotations ...... 297 A.8 Plots excitation proﬁle of a shaped RF pulse ...... 299 A.9 Calculation of the transient Bloch-Siegert shift ...... 301 A.10 Generalized amplitude damping model for 7 spins ...... 303 A.11 Phase damping model for 7 spins ...... 305 A.12 Sample code simulating decoherence eﬀects ...... 307

B The three-spin Grover pulse sequence 309

C The 7-spin framework 312 C.1 Used variables and functions ...... 313 C.2 The 7-spin pseudo-code ...... 314

Bibliography 317

xiv List of Tables

2.1 Example of the FFT for N = 8...... 43 2.2 Example of the phase shifting property of the FFT for N = 8, and r = 4 . . 43

3.1 Crude estimates for decoherence lifetimes (τc), operation times (τop), and

maximum number of operations nop...... 73

4.1 Larmor frequencies [MHz] of several species of nuclei, at 11.74 Tesla . . . . 77

5.1 Table listing the optimized values of α1 for ﬂip angles of 90◦ and 45◦ and a variety of diﬀerent pulse shapes...... 138

5.2 Table listing the optimized values of α2 for ﬂip angles of 180◦ and 90◦ and a variety of diﬀerent pulse shapes...... 140

6.1 The table lists f(x) = ax mod 15 for all valid values of a. From f(x) we can determine the period r for each value of a which is used to determine ar/2 1. ± Finally, we calculate the greatest common denominator of ar/2 1 with 15 ± to determine the prime factors of N = 15...... 182 6.2 Summary of the 36 temporal labeling experiments on the spins, grouped into

four sets. Time goes from left to right; Cij denotes a cnot operation ﬂipping

spin j if spin i is in the state 1 ; Ni is a NOT and simply ﬂips spin i. . . . 187 | i 6.3 Summary of the density matrix reconstruction procedure. The shown entries only reﬂect which part of the observable is used for the density matrix reconstruction. The entry n/a means that we do not use this measurement for our density matrix reconstruction. From these measurements, all terms in the density matrix can be reconstructed...... 200

xv List of Figures

2.1 (a) Bloch sphere representation of an arbitrary quantum state ψ for a single | i qubit. (b) Representation of several important quantum states, ignoring the normalization factor...... 12

2.2 Truth tables for (a) the traditional irreversible and gate, (b) the traditional and gate but retaining the ﬁrst of the inputs, and (c) the reversible and gate, also referred to as toffoli gate...... 17

2.3 Quantum circuit representation of (a) an arbitrary single qubit rotation U, and (b) the not gate...... 29

2.4 Truth tables for (a) the cnot12 gate, and (b) the cnot21 gate...... 29

2.5 Quantum circuit representation of (a) a controlled-not gate, (b) a zero- controlled-not, (c) a swap gate, and (d) a controlled-U gate. The symbol • denotes the control qubit - the controlled operation is only executed if the qubit is in the 1 state. The symbol denotes the zero-controlled qubit, i.e. | i ◦ the operation is only executed if the control qubit is in the state 0 . . . . . 31 | i 2.6 Quantum circuit representation of a toffoli gate, and its decomposition into two-qubit gates...... 33

2.7 Quantum circuit representation of a fredkin gate, and its decomposition into toffoli gates. This decomposition can be further simpliﬁed, as shown here, and explained in the text...... 33

2.8 Quantum circuit outline for the Deutsch-Jozsa algorithm...... 36

xvi 2.9 Illustration of Grover’s algorithm for the case N = 8 (n = 3) and x0 = 110. The diagrams show the amplitude of the eight terms 000 through 111 . | i | i The algorithm begins with an equal superposition of all 8 terms. Next, the amplitude of the 110 is ﬂipped (oracle function call), followed by an | i inversion about the average of all amplitudes. One Grover iteration consists of an oracle function call and the inversion about average. For N = 8, the

amplitude of the x0 element reaches almost unity after just 2 iterations. . 40 | i

2.10 Quantum circuit outline for the 3-qubit QFT. The most signiﬁcant input bit is qubit 3 whereas the most signiﬁcant output bit is qubit 1. We can explicitly reverse the ordering of the qubits by applying a series of swap gates, or implicitly by simply keeping this reverse order in mind when applying additional operations following the QFT...... 44

2.11 Illustration of an M = 4 order-ﬁnding problem. In this example, the order r of the permutation π is r = 1 for y = 2, and r = 3 for y = 0 , 1 , or 3. . . . . 45

2.12 Illustration of an M = 4 order-ﬁnding circuit...... 47

2.13 Illustration of the evolution of H(t) for the adiabatic algorithm with H(t) =

(t/T )Hp + (1 t/T )Hb (solid line). The vertical axis represents the strength − of the terms in H(t), and the horizontal axis represents time. This continuous evolution can be approximated by discretizing the Hamiltonian and applying a zeroth order interpolation (dashed line)...... 52

3.1 Three coupling network architectures for ﬁve qubits. (a) All qubits are coupled with each other. (b) The qubits are only coupled via nearest neighbor interaction. (c) Coupling of the qubits via a bus...... 66

4.1 Zeeman splitting for a spin-1/2 particle subject to a static magnetic ﬁeld. . 76

4.2 Precession of a spin-1/2 particle about the axis of a static magnetic ﬁeld with frequency ω. The precession can be visualized on the Bloch sphere by assuming that the magnetic ﬁeld is along thez ˆ direction...... 77

xvii 4.3 Structure of (a) tetrafluoroethylene. This molecule has four equivalent fluorine nuclei and hence they all have the same chemical shift. (b) Bromotriflu- oroethane has three inequivalent fluorine nuclei. The two fluorines on the left are not free to rotate about the double bond between the two carbon nuclei and hence they each have a different chemical environment with respect to the bromine. The third fluorine has an even different chemical environment compared to the previous two. If the bond between the two carbons was only a single bond, the two fluorines on the left would be free to rotate about the axis of the bond and hence their chemical shift would be the same due to molecular tumbling...... 78

4.4 Dynamic evolution of a spin subject to an on-resonance RF ﬁeld (a) in the rotating frame and (b) in the laboratory frame...... 80

4.5 Time proﬁle (left) and excitation proﬁle (right) for several pulse shapes all of

the same length and calibrated to result in a 180◦ rotation when applying the RF ﬁeld on resonance. The excitation proﬁle shows thez ˆ, andx ˆyˆ-component of the spin magnetization, starting from thez ˆ-axis...... 85

4.6 Visualization of the cnot12 operation of two spins using the Bloch-sphere representation. Spin 2 is shown on the Bloch-sphere which rotates aboutz ˆ at ω2/2π. Spin 2 is left untouched if spin 1 is 0 (solid line), and ﬂipped if 0 | i spin 1 is 1 (dashed line)...... 90 | i xviii 4.7 Bloch-sphere representation of a refocusing scheme with n = 3 spins, with time going left to right, and the spins (1,2, and 3) are ordered from top to bottom. All spins start out in thex ˆyˆ-plane. The refocusing scheme begins 2 with a time evolution UJ (t), followed by X3 (a 180◦ rotation on spin 3 about

xˆ), and finally another time evolution UJ (t). The first row indicates what would happen to spin 1 when it points along yˆ and the remaining spins are − along zˆ. The four possible states for the remaining two spins 2 and 3 are ± also drawn for clarification. Similarly, row 2 indicates what would happen to spin 2 if it was pointed along yˆ and the remaining spins along zˆ, and − ± so forth. The couplings J13 and J12 are refocused while the coupling J12 is turned on. Hence spin 3 ends up alongy ˆ. To flip spin 3 back to yˆ and − 2 complete the refocusing scheme, we need to apply a final X3 pulse. Spin 1 ends up along xˆ, depending only on the state of spin 2, but not 3. Similarly, ± spin 2 also ends up along xˆ depending only on the state of spin 1, but not 3. 92 ±

4.8 Refocusing scheme for a 7 spin system in which all spins are coupled to each other. A + or indicates whether the spin points in its original direction or − whether it is ﬂipped. The black rectangles denote 180◦ instantaneous spin rotations, ﬂipping the spins (+ to or vice versa). All couplings except for − J12 are refocused. A coupling acts “forward” in time when both spins point in the same direction (++ or ), and “backwards” in time when both spins −− point in the opposite direction (+ or +)...... 93 − −

4.9 Simpliﬁed refocusing for a 7 spin system, useful when spins 3 through 7

are along zˆ. The coupling J12 is active, and the couplings J1i and J2i are ± refocused. All remaining couplings have no measurable eﬀect on the quantum state of the system...... 94

4.10 Thermal equilibrium spectrum of a 5 spin molecule, used in one of our experiments (section 6.5). The frequencies are in units of Hz with respect to the Larmor frequency of that spin. The signal strength is in arbitrary

units. Knowledge about the couplings (J12 < 0, J13, J14, J15 > 0 and

J12 > J13 > J15 > J14) allows us to assign each line with a particular | | | | | | | state of the remaining spins...... 109

xix 4.11 (a) Schematic energy level diagram for an I = 3/2 system with quadrupolar splitting. The energy levels correspond to the spin states I = 3/2, m = 3/2 , | i I = 3/2, m = 1/2 , I = 3/2, m = 1/2 , and I = 3/2, m = 3/2 , and can | i | − i | − i be assigned the logical labels 00 , 01 , 10 , and 11 . (b) Thermal spectrum | i | i | i | i of 133Cs displaying three of the allowed seven transitions, forming a spin-3/2 subsystem...... 120

4.12 Illustration how to implement a universal set of quantum gates in an I = 3/2 system. Time goes from left to right. The ﬁrst and third column show qubit operations with the horizontal lines denoting the qubits, and the subscript denoting the rotation angle in units of π/2. Columns two and four show the corresponding transition selective operations for the I = 3/2 system,

where the ﬁrst line denotes the transition at frequency ω12, and so forth. Single qubit Y -rotations are implemented similar to the shown X-rotations. From these, single qubit Z-rotations can be implemented. Together with the

controlled Zα-gate, arbitrary 4x4 unitary gates can be implemented. . . . . 121

5.1 (a) Plot of ωBS during the application of a Hermite 180 shaped pulse with

ωrf ω0 = 5kHz and a pulse width of 2000µs. (b) Plot of φBS during the − same pulse. The total acquired phase reaches 27◦ - a substantial amount that must be corrected...... 130

5.2 (a) Simulation of the amplitude of thez ˆ andz ˆyˆ-component of a spin as a function of its frequency. The spin starts out along +ˆz and is subject to two simultaneous Hermitian-shaped pulses with carrier frequencies at 0 and 3273 Hz (vertical dashed lines), with a calibrated pulse length of 2650µs (ideally

180◦). (b) Same as (a) but with the frequency shift correction...... 131

xx 5.3 The ﬁgure shows four pairs of spectra experimentally measured spectra (absolute value). For each pair, the spectrum of spin a is on the left and the

spectrum of spin b is on the right. The ﬁrst pair was taken after a 90◦ Her-

mitian pulse on a, and the second pair was taken after a Hermitian 180◦ pulse on a. These serve as reference spectra. The third pair was taken after

simultaneous Hermitian 180◦ pulses on the two spins without the correction scheme. The last pair is the same but with the correction scheme. The experimental parameters are the same as those used in the simulations of the previous figure...... 133 5.4 Gate fidelity of soft pulses as a function of β, over a range typically encountered in NMR. The solid line shows the rapid decay of the gate fidelity for stronger couplings without any correction scheme. The dashed line shows improvement when adding a ’negative’ time delay either before or after the single qubit rotation. The gate fidelity stays almost constant for the symmetric compensation by introducing ’negative’ time delays before and after the pulse...... 139

6.1 Schematic overview of a NMR spectrometer...... 147 6.2 A typical NMR sample: A sample tube containing the dissolved quantum computer molecules is held in place by a sample holder...... 148 6.3 An Oxford Instruments 500MHz narrow-bore NMR magnet (the IBM magnet was wide-bore). The fill ports for liquid nitrogen and helium are at the top of the magnet. The cabinet next to one of the magnet legs (visible on left side of the picture) contains transmit/receive switches, preamplifiers and mixers. The probe is inserted at the bottom of the magnet, while the sample is inserted from the top...... 150 6.4 A quench of the IBM 500 MHz magnet during its installation. The superconducting solenoid suddenly dissipates all of its energy by boiling off the liquid helium...... 151 6.5 A Nalorac HFX probe. The RF coils are located near the top of the probe, connected by the BNC connectors at the bottom. The cooling air inlet and connectors for the gradient coil are also located near the bottom. The tune/match capacitor rods stick out at the bottom of the probe...... 154

xxi 6.6 An open 200 MHz Bruker probe showing the RF coils (indicated by the white arrow)...... 154

6.7 The electronics cabinet of our spectrometer. The four PTS frequency synthesizers are visible on the left. The waveform generators, attenuators, and high power ampliﬁers are on the right...... 157

6.8 Overview of quantum computing experiments performed to date. The complexity of the experiment depends on the number of quantum gates involved, the demands on the coupling network, and the number of qubits. The numbers are the year published. The dotted ovals indicate experiments I was directly involved with, and are the focus of this chapter...... 164

6.9 Experimental deviation density matrices ρexp for x0 = 1 0 1 , shown in | i | i| i| i magnitude with the sign of the real part (all imaginary components were small), after 2 (a) and 28 (b) Grover iterations. The diagonal elements give the population difference with respect to the average. The off-diagonal elements represent coherences between the basis states. Inset: The corresponding 13C spectra (13C was the least significant qubit). The receiver phase and read-out pulse are set such that the spectrum be absorptive and positive for a spin in 0 ...... 167 | i

6.10 (a). Experimental (error bars) and ideal (circles) amplitude of dx0 , with ﬁts (dotted) to guide the eye. Dashed line: the signal decay for 13C due to 13 intrinsic phase randomization or decoherence (for C, T2 0.65 s). Solid ≈ line: the signal strength retained after applying a continuous RF pulse of the same cumulative duration per Grover iteration as the pulses in the Grover sequence (averaged over the three spins; measured up to 4 iterations and

then extrapolated). (b). The relative error r...... 168

6.11 A quantum circuit that implements the boosting procedure for Schulman- Vazirani’s algorithm. The controlled-swap operation has been replaced by an equivalent set of gates: two cnot’s and a toffoli...... 170

6.12 Pulse sequence to implement the boosting procedure. This pulse sequence is

designed for molecules with Jab < 0 and Jac,Jbc > 0...... 172

xxii 6.13 Experimentally measured spectra of spin a (Left), spin b (Center) and spin c (Right), after a readout pulse on the corresponding spin, for the spin system in thermal equilibrium (Top) and after applying the boosting procedure (Bot- tom). The real part of the spectra is shown, and the spectra were rescaled in order to obtain unit amplitude for the thermal equilibrium spectra. Fre- quencies are in Hz with respect to the Larmor frequency of the respective spins...... 173 6.14 Pictorial representation of the theoretical (left) and experimentally measured (right) density matrices, shown in magnitude with the sign of the real part (all imaginary components were very small)...... 174 6.15 (Left) The probabilities that the measurement result m is 0, 1,..., or 7, given r (for an ideal single quantum computer). (Right) The optimal probabilities

with which to make a guess r0 for r, given m...... 176 6.16 Structure of the pentaﬂuorobutadienyl cyclopentadienyldicarbonyliron complex, with a table of the relative chemical shifts of the 19F spins at 11.7 T [Hz], and the J-couplings [Hz]. A total of 76 out of the 80 lines in the 5 spectra are resolved...... 177 6.17 (a) The spectrum of spin 1 in equilibrium. The 16 lines are due to shifts in

the transition frequency ω1 by J1j/2, depending on whether spin j is in 0 ± | i or 1 . In equilibrium, all the 32 states are nearly equally populated, hence | i the 16 lines in each spectrum have virtually the same intensity. Taking into

account the sign and magnitude of the J1,j, the 16 lines in the spectrum of spin 1 can be labeled as shown. (b) The same spectrum when the spins are in an effective pure state. Only the line labeled 0000 is present. All spectra shown here and in the following figure display the real part of the spectrum in the same arbitrary units, and were obtained without phase cycling or signal- averaging. A 0.1 Hz filter was applied. Frequencies are in units of Hz with

respect to ω1...... 179 6.18 Spectra of spin 1 acquired after executing the order-ﬁnding algorithm. The respective permutations are shown in inset, with the input element highlighted. The 16 marks on top of each spectrum indicate the position of the 16 lines in the thermal equilibrium spectrum. The spectra obtained for the r = 3 case was averaged 16 times to obtain better signal-to-noise...... 180

xxiii 6.19 Outline of the quantum circuit to factor the number N = 15. The first register consists of n = 8 qubits, initialized to 0 . The second register | i consists of m = 4 and is initialized to 1 . We then apply Hadamard gates | i to all qubits in the first register, followed by the multiplication of ax mod 15. Next the Quantum Fourier Transform is applied, and finally the first register is measured...... 181 6.20 Simplified quantum circuit for factorizing the number N = 15...... 183 6.21 (a) Detailed quantum circuit for the case N = 15 and a = 11. (b) Detailed quantum circuit for the case N = 15 and a = 7. The gates shown in dotted lines can be left out and the gates shown in dashed lines can be replaced by simpler gates (see following section)...... 183

6.22 The seven spin molecule, along with its J-coupling constants, T1 and T2 relaxation times (in seconds), and chemical shifts (in Hertz) at 11.7 Tesla. . 185 6.23 Diagram of the synthesis of the 7 qubit molecule...... 186 6.24 Pulse sequence for implementing Shor’s algorithm to factor N = 15 with

a = 7, using seven qubits. The tall lines represent 90◦ pulses selectively acting on one of the seven qubits (horizontal lines) about positivex ˆ (no cross),

negativex ˆ (lower cross) and positivey ˆ (top cross). Note how single 90◦ pulses correspond to Hadamard gates and pairs of such pulses separated by delay

times correspond to two-qubit gates. The smaller lines denote 180◦ selective pulses used for refocusing, about positive (darker shade) and negativex ˆ (lighter shade). Rotations aboutz ˆ are denoted by even smaller and thicker lines and were implemented with frame-rotations. Time delays are not drawn to scale. The vertical dashed black lines visually separate the steps of the algorithm; step (0) shows one of the 36 temporal averaging sequences. . . . 189 6.25 Experimentally obtained thermal equilibrium spectra of the ﬁve ﬂuorine spins. The real part of the spectrum is shown, in arbitrary units. The

frequencies are with respect to ωi/2π, in Hertz...... 190 6.26 Experimentally obtained thermal equilibrium spectra of the two carbon spins. The real part of the spectrum is shown, in arbitrary units. The frequencies

are with respect to ωi/2π, in Hertz...... 191 6.27 Experimentally measured spectra of the ﬁrst register, similar to the thermal equilibrium spectra, after putting the seven spins into an eﬀective-pure state. 191

xxiv 6.28 Experimentally measured spectra of the spins in the first register (Bottom) and the ideally expected spectra (Top), after the completion of the “easy” case of Shor’s algorithm (a = 11). Positive and negative integrals of the spectra denote the qubit state 0 and 1 respectively...... 192 | i | i 6.29 Similar to the previous figure but for the “difficult” case (a = 7)...... 192 6.30 Comparison between our experimentally obtained spectra (Bottom) and the simulated spectra based on the decoherence model (Top), for the “easy” case of Shor’s algorithm (a = 11)...... 195 6.31 Similar to the previous figure but for the “difficult” case (a = 7)...... 196 6.32 Procedure to obtain the pulse sequence for a spin-3/2 system implementing a two-qubit Grover algorithm. Step A initializes the system into an equal superposition. Step B implements the oracle function call. Step C implements the inversion about average where f(0) flips the phase of the 00 state. The | i letters in the boxes denote the axis of rotation, and the subscript denotes the angle of rotation in units of π/2. (a) Two-qubit quantum circuit diagram. (b) Outline of the pulse sequence for a spin-3/2 system, implementing the circuit shown in part (a). (c) Detailed pulse sequence implementing the hadamard gate...... 201 6.33 Plot of the experimentally obtained absolute value of the deviation density

matrices for x0 = 3, x0 = 2, x0 = 1 and x0 = 0. For visual clarity, each plot has been adjusted such that the minimum diagonal value equals to zero. . . 202 6.34 Plot of the expected absolute value of the deviation density matrices for

x0 = 3, x0 = 2, x0 = 1 and x0 = 0, without any compensation of the Bloch-Siegert shifts...... 203 6.35 Plot of the experimentally obtained absolute value of the deviation density

matrices for x0 = 3, x0 = 2, x0 = 1 and x0 = 0, without any compensation of the Bloch-Siegert shifts. For visual clarity, each plot has been adjusted such that the minimum diagonal value equals to zero...... 204 6.36 Plot of the signal strength as a function of the phase of the second pulse when

applying two 180◦ pulses on one transition of a spin-7/2 system. Assuming the usual spin dynamics, we would expect the plot to be a horizontal line through the 0 point...... 206

xxv 6.37 Illustration of Eq. (6.8). The shaded and clear boxes denote the strength

and duration of the Hamiltonians Hb and Hp respectively...... 209

6.38 Illustration of a graph consisting of 3 nodes and 3 edges. The edges carry

weights w12, w13, and w23. When min(wij) = w23 as indicated by the length of the edges, the maxcut corresponds to the drawn cut. The solution is therefore s = 100 and also s = 011 due to symmetry. This symmetry can be

broken by assigning the weights w1, w2, and w3 to the nodes...... 210

6.39 Refocusing scheme to effectively change Jij into wij. The delay segments are of length α, β, γ, and δ. When all segments are of equal length, all of the couplings are effectively turned off...... 212

6.40 Illustration of the pulse sequence which implements our discrete adiabatic optimization algorithm with a total of four steps. The horizontal lines indicate the three spins, and time goes from left to right. The empty rectangles

denote the Hamiltonian Hb[m]/2 while the black rectangles denote the 180◦ refocusing pulses. The arrows indicate the delay segments during which we go oﬀ-resonance with respect of the Larmor frequencies of the spins. The ﬁrst set of rectangles denote the initialization into the highest excited state

of Hb...... 213

6.41 Plot of the absolute value of the deviation density matrix for M = 100 (T = 374 ms), M = 60 (T = 226 ms), M = 30 (T = 115 ms), and M = 15 (T = 59.2 ms), adjusted by an identity portion such that the minimum diagonal value equals zero. The scale is arbitrary...... 214

6.42 Experimental performance of the adiabatic algorithm. Plot of the error as a function of M (Left). The error measure is the trace distance D(ρ, σ) = ρ σ /2 where σ is the traceless deviation density matrix for M = 400, | − | approximating M , and ρ equals the ideal expected (circles), the exper- → ∞ imentally obtained (crosses), or the ideal expected traceless deviation density matrix with decoherence eﬀects (diamonds). The minimum error occurs at about M = 60 indicating an optimal run-time of the algorithm. A similar observation can be made when plotting the element 101 101 of the traceless | ih | density matrices from the previous ﬁgure as a function of M (Right). . . . . 215

xxvi 7.1 Circuit representation of a Cooper pair box. The Josephson Junction with

Josephson energy EJ and a capacitance of CJ is connected to a superconduct-

ing island which in turn is connected to a gate voltage Vg with a capacitance

Cg...... 219

7.2 The energy eigenvalues of a Cooper pair box for EC = 10EJ and working

only with the lowest 10 charge states. If EJ was equal to zero, there would 1 exits degeneracies at ng = 2 mod 1. These are lifted even for small values of EJ , and hence the lowest two energy levels at such a degenerate point eﬀectively reduce to a 2-state quantum system...... 221

7.3 Circuit representation of a split Cooper pair box. The two Josephson Junc-

tions both have Josephson energies EJ /2 and EJ /2 and capacitances CJ1

and CJ2. The junctions are connected to a superconducting island which in

turn is connected to a gate voltage Vg with a capacitance Cg. The ﬂux Φx

induces a phase diﬀerence Φ0δ across the two junctions...... 223

7.4 A register of many charge qubits coupled by the oscillator modes in the LC- circuit...... 226

7.5 The circuit consisting of the qubit and the SET used as measurement device. 228

7.6 Two ﬂux qubit designs. (a) The rf-SQUID, consisting of a superconducting loop and one Josephson junction, form the simplest ﬂux qubit. (b) An im-

proved design in which the ﬂux Φ˜x controls the eﬀective Josephson coupling constant of the rf-SQUID...... 229

7.7 Plot of the potential energy of a ﬂux qubit for two values of Φx. The lowest two energy levels of the wells form the basis of the qubit...... 230

7.8 Sketch of the energy eigenvalues as a function of Φx. Far away from the avoided level crossing, the eigenstates correspond to current ﬂowing either clockwise or anti-clockwise. At the avoided level crossing, the eigenstates are equal superpositions of the two current ﬂows...... 230

7.9 Two 3-junction ﬂux qubit designs. (a) The 3-junction design by Mooij reduces the inductance of the system and hence lowers the coupling of the circuit to the environment. (b) A multi-junction design by Orlando where

EJ is controlled by the application of an external ﬂux Φ...... 232 f e xxvii 7.10 Contour plot of the potential energy U(φ1, φ2) for Φ/Φ0 = 0.5 and E˜J /EJ = 0.7. The nearly circular shapes enclose maxima while the hourglass-shaped contours enclose two minima. The solid line indicated the direction along which the potential is double-well-shaped. The dashed line indicates the direction from one minimum to the next nearest-neighbor minimum in a diﬀerent unit cell...... 233

7.11 Coupling ﬂux qubit in two diﬀerent ways. The qubits can be inductively coupled via an external superconducting loop (dashed line) or the coupling is provided by an external LC-circuit (solid line), similar to the coupling for charge qubits...... 234

7.12 Set-up for measuring a ﬂux qubit. The qubit (on the left) is inductively coupled to the meter (on the right)...... 235

7.13 Schematic representation of a Josephson junction (left), and circuit representation (right)...... 237

7.14 (a) Schematic drawing of the energy levels of the particle in the tilted wash-

board potential U(δ) with I < I0. When the anharmonic term is suﬃciently large only 3 energy levels exist whose energy separations diﬀer by a few percent. In this case, the bottom two levels can be used as the qubit states 0 | i and 1 . (b) Energy level diagram of the three-level system...... 238 | i 7.15 Schematic of the energy levels of an ion isolated from the rest of the energy levels for its use as a qubit...... 242

7.16 A linear ion trap consisting of four parallel rods. An oscillating electric field is generated by applying an rf voltage to the electrodes, providing radial confinement. The axial confinement is achieved by applying different dc voltages to different segments of the rods...... 244

7.17 Schematic of the energy levels of an ion isolated from the rest of the energy levels for its use as a qubit...... 245

7.18 Schematic diagram of the magnetic sublevels of a cesium atom...... 248

xxviii 7.19 Outline of the derivation of the quantum circuit to implement the desired

rotation, given the Hamiltonians Ha and Hb. The two wires correspond to the two qubits. The goal is to implement a zero-controlled operation with qubit 1 being the control, and qubit 2 the target. This corresponds to a rotation in the desired Hilbert space. The individual steps are explained in the text...... 250

7.20 Estimate of the error as a function of normalized pulse width tpw = Tpwδω/2π =

2ατδω/2π via Fourier analysis using the untruncated and truncated gaussian shape, compared with the exact calculation...... 255

7.21 Plot of the error as a function of the normalized pulse width tpw for three diﬀerent pulse shapes and several diﬀerent levels of truncation...... 256

7.22 Plot of the error as a function of normalized pulse width tpw for several traditional NMR pulse shapes. The Gaussian shape here corresponds to α = 3.257

7.23 Plot of the error (maximum occupation probability) as a function of time

normalized with respect to the total pulse length tpw during a hard and gaussian shaped pulse and the second composite pulse. A Gaussian-shaped 8 pulse with a pulse width of tpw = 4 has a very small error (about 10− ), but 2 during the pulse the error can be as high as 5 x 10− . The error during the pulse can be reduced by applying the pulse for a longer duration since the

maximum error scales as approximately 1/tpw, based on Fourier analysis of the truncated pulses...... 258

7.24 Plot of the error as a function of normalized time tpw for the two composite pulses (n = 2), compared with a hard and gaussian shaped pulse. The ﬁrst composite method outperforms hard pulses whenever the pulses are multiples

of 2π/δω even though the limit of g δ does not apply here. For the second composite pulse, when the applied pulses are integer multiples of 2π/δω, the error is minimized as expected...... 262

7.25 Plot of Pt as a function of normalized pulse width for a hard, gaussian and

composite pulse taking Γ2 δω/2π (ω10/2π)/100. Clearly, the estimate ≈ ≈ of the tunneling probability is several orders of magnitude higher than , indicating that tunneling eﬀects are important...... 264

xxix 7.26 Plot of the error as a function of time when including tunneling eﬀects with

Γ2 δω (ω10/2π)/100. The error is given by the maximum of 2 2 + ≈ ≈ | ih | A A over all possible input superpositions a 0 + b 1 . Now, the hard pulse | ih | | i | i shows the best performance, but still with a large error...... 266

7.27 Plot of the error as a function of time when including tunneling eﬀects in

a four level system. The tunneling rate Γ2 is (δω/2π)/1000, and Γ3 is 1000 times higher. The error is given by 2 2 + 3 3 + A A after maximizing | ih | | ih | | ih | over all possible input superpositions a 0 + b 1 . In this case, the composite | i | i pulse can perform about as well as a gaussian pulse...... 267

7.28 (a) Energy level diagram of Λ-like spin states of a three-level atomic system.

(b) Schematic energy level diagram for the Iz = 1/2, 1/2, 3/2 levels a multi- − level system with quadrupole splitting...... 271

7.29 Experimental data verifying the transparency behavior of an NMR EIT system in the strong control field limit. Here, we observe the signal corresponding to the 1 2 element of the density matrix. The intensity of the probe field | ih | was set to a strength equivalent to a π pulse. The dash-dot line represents the 1/b trend of Rabi oscillations in the 1 2 element. The dotted line is the | ih | simulation result, which includes the RF inhomogeneity of the control field and the Bloch-Siegert shifts. The points are experimental results obtained by averaging over five experiments. The height of the error bars were obtained

from standard deviations normalized by Nexperiments 1, where Nexperiments − is the number of repeated experiments...... 277

7.30 Experimental data verifying the transparency behavior of an NMR EIT system in the strong control ﬁeld limit. Here, the intensity of the probe ﬁeld was set to a strength equivalent to 3π/2 units...... 278

7.31 Experimentally reconstructed deviation density matrix of the “dark state” 1 3 − (a) before applying the EIT Hamiltonian and (b) after applying the | i√2| i EIT Hamiltonian for a duration equivalent to 3π/2 pulses. Note that the density matrices presented here show only the absolute values of the elements for the sake of visual clarity...... 279

xxx 7.32 (a) Energy level diagram of Λ-like spin states of a three-level atomic system coupled to an additional reference level R . (b) Schematic energy level di- | i agram for the Iz = ( 3/2, 1/2, 1/2, 3/2) levels a multi-level system with − − quadrupole splitting. Here, 11 = R , 10 = 1 , 01 = 2 , and 00 = 3 . 280 | i | i | i | i | i | i | i | i 7.33 Simulations and experimental results of visibility as a function of b/a in the strong control field limit. The solid line indicates the ideal expectation of visibility, the dashed line indicates the simulated behavior including decoherence effects and pulse imperfections, and the points indicate the experimental results. The experimental results are in close agreement with simulations. . 284 7.34 The expectation of the reference state R as a function of θ. The solid | i line indicates the ideal value in the expectation of R , the dashed-line the | i simulated expectation of R including decoherence effects and the squares | i indicate experimental results. They show the characteristic cos2 θ behavior derived...... 285 7.35 Visibility of of the expectation value of R state as a function of b/a for | i EIT with a coherent dark state. The solid line is the ideal case scenario, the dashed-line indicates simulated expectation with decoherence effects and the 1 3 points indicate experimental results. Here, the dark state is − , while | i√2| i the ratio b/a is varied with a fixed probe field and varying control field. . . 287

xxxi xxxii Chapter 1

Introduction

1.1 Historical background

“There is plenty of room at the bottom”

For over 40 years and following Moore’s Law, the semiconductor industry has been rapidly shrinking circuit and device dimensions. If this rapid trend continues, device dimensions will approach the length scale of a single atom within about a decade. At these length scales however, the devices no longer obey classical rules but instead the rules of quantum mechanics, leading to unexpected behavior and even failure of these devices. Should we therefore consider quantum mechanical effects as a foe and avoid them at all costs? The answer to this question is a resounding NO and it is now clear that we can use quantum mechanical effects to our advantage! In fact, already in 1959, Richard Feynman gave a talk entitled “There is plenty of room at the bottom” at the annual meeting of the American Physical Society [Fey60], challenging scientists to use quantum mechanical effects and design devices at a very small scale: using the discretized energy levels of atoms, wires of 10 to 100 atoms in diameter, or circuits with a few tens of nanometers across.

Feynman’s daring idea to build very small computers inspired researchers to investigate the fundamental physical laws that govern computation: How much energy, time, memory, and space are required to compute or solve a problem? This question was just the beginning of a journey towards building real, practical quantum computers.

1 2 CHAPTER 1. INTRODUCTION

Energy dissipation and computation

The ﬁrst insight came when Rolf Landauer studied the relationship between energy consumption and computation [Lan61]. He showed that a ﬁxed amount of energy is released into the environment when one bit of information is erased. This means when irreversible logic gates such as the nand gate (used in most computers today) are executed, the computer must dissipate energy because information is erased (it is impossible to recover the input bits of an irreversible gate just based on the output bits). It turns out however that computation does not fundamentally require energy! Lecerf [Lec63] and Bennett [Ben73] have shown that universal computing can be performed without energy dissipation by applying reversible logic gates. This is an important concept because as we will discover, closed quantum systems can only perform reversible operations.

Time, memory, space and computation

The study of how much time and space are required for computation falls under the category of complexity theory. Specifically, complexity theory analyzes how the resources (time, memory, and space) scale with the problem size. The core concept of this field is Alan Turing’s introduction of a Turing machine [Tur36] and the formulation of the strong Church-Turing thesis [Chu36]. It can be summarized by saying that all universal computers are polynomially equivalent. This means that if a computation for a certain problem on one universal computer requires N computational steps, another universal computer requires no more than N x steps where x is a positive real number that is fixed and independent of the problem. Therefore, the simplest abacus dating back to 500 B.C. is polynomially equivalent to the fastest supercomputer, like IBM’s Deep Blue supercomputer for example1. While polynomial differences can be huge, the fundamental laws that govern classical computation have not changed in over 50 years.

The idea of a quantum computer is born

But what if we let go of the classical picture? Could a quantum mechanical system do more than a classical Turing machine? This question has been a motivating hypothesis for studying quantum computing. Paul Benioﬀ was the ﬁrst who showed in the early

1This comparison is actually not completely accurate because Turing machines are assumed to have inﬁnite memory, which is not true for an abacus. This comparison however gives us an idea of what is meant by the Church-Turing thesis. 1.1. HISTORICAL BACKGROUND 3

eighties that a quantum mechanical Hamiltonian can represent a universal Turing machine [Ben80]. This is possible because classical computation can be performed reversibly in an eﬃcient way, as we described above. Manin and independently Feynman then proposed that quantum systems might be able to simulate each other eﬃciently (i.e. using only a polynomial number of resources), a task that is impossible on classical computers [Man80, Fey82]. David Deutsch then took these ideas, fully developed the notion of a quantum Turing machine, and introduced the idea of quantum parallelism to illustrate how quantum computers might require less computation time [Deu85b].

The usefulness of a quantum computer

The major breakthrough that jump-started the field quantum computing came in 1994 when Peter Shor showed that a quantum computer could factor integer numbers in only polynomial time [Sho94a]. This quantum algorithm offers an exponential speed-up compared with the best known classical algorithms. Besides being able to break most secure encryption schemes used today, this result appears to directly violate the Church-Turing thesis! Thus, if we are able to succeed in building a quantum computer and prove that classical factoring takes exponential effort, then the Church-Turing thesis will have to be rewritten. However, if we fail in our attempt, we stand to learn a fundamental new physical concept. In either case, we will learn new physics through the study of quantum computation. Three years later, in 1997, Lov Grover proved that a quantum computer could search an unsorted database in only the square root of the time that would be required on classical computers [Gro97]. Meanwhile, Seth Lloyd proved Richard Feynman’s conjecture on efficiently simulating quantum systems [Llo96]. Quantum computers thus offer potentially dramatic algorithmic speed-ups.

Quantum computing in the presence of errors?

These discoveries are all truly remarkable, but can we actually build a practical quantum computer? To answer this question, we need to ﬁrst ask ourselves if a quantum computer can still achieve the described algorithmic feats in the presence of errors and decoherence because any real system will undergo errors. At ﬁrst, this problem resulted in much skepticism about the practicality of quantum computers. In an amazing discovery however, Peter Shor [Sho95] and Andrew Steane [Ste96] showed that quantum error correction is possible, similar to classical error correction. This then lead to the development of fault tolerant constructions, 4 CHAPTER 1. INTRODUCTION

based on von Neumann’s work on fault tolerance in the 1950’s: Not just are the algorithmic speed-ups on a quantum computer still attainable even in the presence of errors, but in fact it is possible to built a fault-tolerant quantum computer from faulty components provided the error rate per gate is below some threshold [Got97, ABO97, ABO99, Kit97a, Kit97b]. Another large and interesting class of quantum-error correcting codes, called noiseless quantum codes and decoherence-free subspaces, have been invented. The result of these codes is that quantum states can be encoded in such a way that they are completely invulnerable to decoherence eﬀects [ZR98, LCW98, BKLW99, KLV99].

Experimental realizations of quantum computers

Given the fact that we can build perfect quantum computers in theory, does this mean we can also do so in practice? As it turns out, building real quantum computers capable of solving problems that are beyond the reach of classical computers appears to be a grand challenge, much like Feynman’s original challenge. There have been numerous proposals including a silicon-based nuclear spin quantum computer [Kan98], trapped ions [CZ95b], cavity quantum electrodynamics [THL+95], nuclear spins [DiV95], electron spins in quantum dots [LD98], superconducting loops [MOL+], as well electrons ﬂoating on liquid helium [PD99], among others. The technological state of the art of each of these respective ﬁelds may only permit modest quantum algorithm implementations within the next few years.

Nuclear magnetic resonance and quantum computing

This grim outlook on experimental quantum computers changed dramatically in 1997 when Gershenfeld and Chuang [GC97], and independently Cory, Havel and Fahmy [CFH97] showed how liquid state nuclear magnetic resonance (NMR) techniques can be used to implement small prototype quantum computers. NMR technologies have been developing for over 50 years, and can be adapted for the purpose of quantum computing in a straightforward manner. In 1998 the ﬁrst quantum algorithm was experimentally demonstrated in Isaac Chuang’s laboratory, using NMR techniques [CVZ+98] and nearly simultaneously by Jones’s group [JM98]. Their experiments clearly demonstrated a quantum algorithm and suggested that much larger implementations using NMR techniques are accessible. This is the setting in which I commenced my work in quantum computing. It turned out to be a tremendously exciting time to join the experimental ﬁeld of NMR quantum computing. 1.2. RELATED FIELDS AND ADDITIONAL LITERATURE 5

In fact, the time frame from about 1998 to 2002 has been labeled by some as the “golden age” for NMR quantum computing because of the tremendous advancements made. The size of quantum computers increased from just two to seven qubits, which is also currently the largest quantum computer ever built. Despite the enormous progress however, NMR quantum computing has some well known limitations which may prohibit straightforward implementations beyond ten or twenty qubits. Nonetheless, I believe the end of interesting NMR quantum computations is not yet in sight as there is still a lot we can learn from NMR quantum computing.

1.2 Related ﬁelds and additional literature

Besides quantum computing, the field of quantum information processing developed in parallel. It is the quantum analogue to classical information theory (see for example [CT91]). Quantum information processing deals with the description of quantum sources and quantum channels including how much information can be transported on a quantum channel, and studies techniques for compressing quantum information. This field has produced some marvelous results including provably secure quantum communication [BBE92, BB84], quantum teleportation [BBC+93], and superdense coding [BW92]. Quantum teleportation has already been achieved experimentally [BPM+97, BBM+98], and quantum cryptography has been demonstrated through optical fibers tens of kilometers long [MZG96, SGG+02]. In fact, some companies already sell quantum cryp- tographic equipment for commercial applications (“id Quantique” and “MagiQ Technolo- gies”). Clearly, quantum cryptography is at a much more mature stage than quantum computing because of its vastly simpler physical implementation. References to relevant work are included throughout. However, there are a few books and articles that I would like to mention in particular. Nielsen and Chuang’s book “Quantum Computation and Quantum Information” [NC00] has been an invaluable resource, not just for chapters 2 and 3 of this thesis, but also throughout my research. Bennett and DiVincenzo wrote a relatively recent review article [BD00]. Another good review has been written by Barenco [Bar96] and Steane [Ste98]. A great introduction to quantum computing, especially Shor’s algorithm, is given by Ekert and Jozsa [EJ96]. There are also several books that were very helpful for learning the basics of NMR 6 CHAPTER 1. INTRODUCTION

and some advanced concepts. Freeman’s book, “Spin Choreography”, gives a spectacular overview of high-resolution NMR techniques [Fre97]. Another great book which helped me write about the decoherence mechanisms in liquid state NMR (chapter 4) is by Levitt [Lev01]. No textbook on NMR quantum computing exists yet, but several good and detailed introductory papers exist. Jones [Jon01] as well as Vandersypen, Yannoni and Chuang [VYC01] wrote excellent introductory review articles.

1.3 Goals of my work

The goal of my work has been to explore experimental quantum computing via well known liquid state NMR techniques [GC97, CFH97] by answering a series of questions:

1. Can we experimentally demonstrate quantum computing?

A theory is only a theory until proven by experiment. Since quantum computation had been solely a theoretical fancy for years, it is of signiﬁcant fundamental interest to experimentally demonstrate quantum computation. This work unambiguously demonstrates proof of principle of quantum computation.

2. Can experimental quantum computation stimulate fundamental theoretical discussions?

No scientific field can grow without both theoretical and experimental advances. In fact, both fields reinforce each other continuously. The experimental results in this work do not just demonstrate quantum computation, but also spark theoretical discussions about the fundamentals of quantum computing or unexpected experimental observations.

3. Besides quantum computation, what else can our methods be applied to?

Throughout any experiment, new tools and techniques are continuously developed. But often such techniques are rather speciﬁc to the particular experiment and hence are not widely applicable. This work develops tools that, in addition to advancing NMR techniques, are also directly applicable to other proposals for building scalable quantum computers. This work also begins to explore the connections between the 1.4. ORGANIZATION OF THE DISSERTATION 7

apparently unrelated ﬁelds of atomic, molecular, and optical (AMO) physics and quantum computation.

These three questions have served as a guiding light throughout my work and the answers to them embody my contributions to the scientiﬁc community.

1.4 Organization of the dissertation

This thesis answers those three questions though not chronologically. The goal is to reach a definite answer to question 1 first because of its fundamental value, but along the way questions 2 and 3 are partially answered. I then combine answers to all three questions in the last chapter of the thesis. Specifically, following the brief historical background of quantum computation of this chapter, chapter 2 describes the principles of quantum computing, introduces the language of quantum circuits, outlines several quantum algorithms, and provides a short mathematical description of decoherence. Chapter 3 describes the five fundamental practical requirements for building a quantum computer. Chapter 4 then details how these requirements are ideally met by liquid state solution NMR techniques. The implementation of these ideas often results in experimental artifacts such as off-resonance effects and phase shifts, among others. These artifacts can be a dominant source of errors, even more so than natural decoherence effects. In chapter 5, I develop several tools and techniques to overcome these errors. In chapter 6, I explore quantum computation by describing a series of experiments. This includes an experimental demonstration of Grover’s algorithm, Shor’s algorithm and an adiabatic optimization algorithm, among others. These experiments clearly demonstrate the proof-of-principle of quantum computing. Finally, in chapter 7, I extend our work beyond NMR quantum computation by showing how it applies to other implementations of quantum computers. For example, I apply NMR techniques to improve qubit control in Josephson phase qubits. I also begin to explore some of the connections between AMO physics and quantum computation to demonstrate the electromagnetically induced transparency effect via NMR. The topics in chapter 2 as well as the examples are chosen in light of the experiments of chapter 6. Similarly, several of the tools and techniques described in chapters 4 and 5 are invented in the context of the experiments in chapter 6. 8 CHAPTER 1. INTRODUCTION

The contents of chapters 1, 2, and 3 are not my own contributions to the ﬁeld, and have been discussed at length in the literature before I even began my research. Most of the ideas of chapter 4 have similarly been discussed, though some of my contributions begin to surface here. The bulk of my contributions are in chapters 5, 6, and 7, sometimes in conjunction with the work of Lieven Vandersypen as indicated by footnotes. My contributions detailed in this thesis have been or are in the process of being published: An accessible introduction for engineers was written and published in IEEE Micro.

M. Steﬀen, L.M.K. Vandersypen, and I.L. Chuang, Towards Quantum Computing: A • Five-Qubit Quantum Processor, IEEE Micro, 2001 [SVC01].

Each of the experiments described in sections 6.3 through 6.7 have been published in refereed journals (or are in preparation).

6.3: L.M.K. Vandersypen, M. Steﬀen, M.H. Sherwood, C.S. Yannoni, G. Breyta, and • I.L. Chuang, Implementation of a three-quantum-bit search algorithm, Appl. Phys. Lett., 2000 [VSS+00]

6.4: D.E. Chang, L.M.K. Vandersypen, and M. Steﬀen, NMR Implementation of a • Building Block for Scalable Quantum Computing, Chem. Phys. Lett., 2001 [CVS01]

6.5: L.M.K. Vandersypen, M. Steﬀen, G. Breyta, C.S. Yannoni, R. Cleve, and I.L. • Chuang, Experimental Realization of an Order-Finding Algorithm with an NMR Quantum Computer, Phys. Rev. Lett., 2000 [VSB+00]

6.6: L.M.K. Vandersypen, M. Steﬀen, G. Breyta, C.S. Yannoni, M. Sherwood, and • I.L. Chuang, Experimental Realization of Shor’s Algorithm using Nuclear Magnetic Resonance, Nature, 2001 [VSB+01]

6.7: M. Steﬀen, M. Kota, P. Judeinstein, and I.L. Chuang, Experimental Implemen- • tation of a Quantum Search Algorithm using Higher Order Spins, in preparation, 2003

6.8: M. Steﬀen, W. van Dam, T. Hogg, G. Breyta, and I.L. Chuang, Experimental • Implementation of an Adiabatic Quantum Optimization Algorithm, Phys. Rev. Lett., 2003 [SvDH+03] 1.4. ORGANIZATION OF THE DISSERTATION 9

Many of these experimental papers (especially [VSS+00],[CVS01],[VSB+00], and [VSB+01]) already include parts of chapter 5, but the techniques from sections 5.3 and 7.4 have been published separately. The ideas connecting AMO physics and quantum computing together with an experimental demonstration is also in preparation.

5.3: M. Steﬀen, L.M.K. Vandersypen, and I.L. Chuang, Simultaneous Soft Pulses • Applied at Nearby Frequencies, J. Magn. Reson., 2000 [SVC00]

5.4: M. Steﬀen, L.M.K. Vandersypen, G. Breyta, C.S. Yannoni, M.H. Sherwood, I.L. • Chuang, in preparation, 2003

7.4: M. Steﬀen, J. Martinis, and I.L. Chuang, Reliable Control of Josephson Phase • Qubits, in preparation, 2003

7.5: M. Kota, H. Son, M. Steﬀen, P. Judeinstein, and I.L. Chuang, Electromagnetically • induced Transparency by NMR, in preparation, 2003 Chapter 2

Theory of quantum computing

This chapter outlines the basic theory of quantum computing from a practical viewpoint, instead of a purely mathematical one. The chosen topics form a basis for understanding quantum computing, and also provide a good platform to explain NMR quantum computation. The discussion begins by comparing classical bits to quantum bits (section 2.1). After introducing the proper quantum language, we proceed to describe some quantum circuits (section 2.2), and move into a short description of the known quantum algorithms (section 2.3). Finally, we provide a mathematical description of decoherence (section 2.4), which we use in our experiments of chapter 6 to predict the impact of decoherence.

2.1 Quantum computing vs. classical computing

This section deals with the similarities and diﬀerence between classical and quantum bits. We describe how quantum computation subsumes classical computation. Then we introduce some ideas from complexity theory to form the basis for comparing the performance of a quantum computer and classical machines. We show how quantum bits are useful for speeding up certain computations using quantum algorithms.

2.1.1 Quantum bits vs. classical bits

The classical bit

The classical bit can take on the logical binary values 0 or 1, and can be represented by any physical system that has at least two distinguishable states. For example, the information

10 2.1. QUANTUM COMPUTING VS. CLASSICAL COMPUTING 11

stored on a hard drive is represented by the orientation of tiny magnetic particles. Similarly, the information in your PC processor is represented by voltages at speciﬁc nodes of the logic circuit. A small optical computer implementing a toffoli gate (see section 2.1.2) has also been built [TMH96]. In fact, even billiard balls can be used to represent information where the presence or absence of a ball represents a logical 0 or 1.

The quantum bit

A two-level quantum system such as a spin-1/2 or polarized photon can also be used to represent information. The two distinguishable computational basis states 0 and 1 can | i | i then encode the logical value 0 and 1 respectively1. Similarly, the 0 and 1 states for a | i | i spin-1/2 system subject to a static magnetic ﬁeld are represented by the spin up ( ) or | ↑i spin down ( ) states. | ↓i But how do quantum bits (qubits) diﬀer from classical bits? Qubits are not just limited to being in the state 0 or 1 , but they can also be in some superposition of these two | i | i states:

ψ = c0 0 + c1 1 (2.1) | i | i | i 2 2 where c0 and c1 are complex numbers that satisfy the normalization condition c0 + c1 = | | | | 1. The state ψ can in this case be written as a column vector consisting of two entries, | i c0 and c1. When either c0 = 1 or c1 = 1, the quantum system is in a classical state. | | | | But when this is not the case, then ψ is in a superposition. A natural question arising | i here is the physical interpretation of superpositions. Being in a superposition of two states can be verbally expressed by saying that we are both in 0 and 1 at the same time. This | i | i sounds completely paradoxical, but it is true and has been shown to be true via numerous experimental demonstrations via interference eﬀects. Young’s double slit experiment or its variation with electrons, for example, are the most clear demonstrations of how “objects” can be in “two locations at the same time”. We can multiply an arbitrary ψ by any phase factor without loss of generality because | i this overall phase is unmeasurable, so we can rewrite ψ as: | i

iφ ψ = cos(θ/2) 0 + e− sin(θ/2) 1 (2.2) | i | i | i

Visually, this state can be represented on the Bloch sphere as shown in Fig. 2.1. This picture

1The symbol denotes quantum mechanical state of the system. |i 12 CHAPTER 2. THEORY OF QUANTUM COMPUTING

will help us understand the dynamics of nuclear spins in the context of NMR quantum computing. Even though this sketch is very useful for visualizing single qubit quantum

0 0

ψ 0 − 1 θ y 0 +i 1 φ x 0 + 1

1 1 (a) (b)

Figure 2.1: (a) Bloch sphere representation of an arbitrary quantum state ψ for a single qubit. (b) Representation of several important quantum states, ignoring the| normalizationi factor.

states, it unfortunately also portrays the incorrect picture that ψ is classical with two | i variables θ and φ. For multiple qubits, this picture may not even hold in general, as we will see next.

Multiple qubits

Suppose we now have two qubits. The overall quantum state ψ can now be written as: | i

ψ = c00 00 + c01 01 + c10 10 + c11 11 (2.3) | i | i | i | i | i where 00 is the short hand notation for 0 0 , is the tensor product or Kronecker | i | i ⊗ | i ⊗ product symbol, and the coeﬃcients c00, c01, c10, c11 satisfy the normalization condition. Two qubits can exist in four states at the same time, and ψ is given by a column vector | i of four entries. 2.1. QUANTUM COMPUTING VS. CLASSICAL COMPUTING 13

For n qubits, a general quantum state ψ can be described as: | i 2n 1 − ψ = ci i (2.4) | i X | i i=0 where i is the decimal instead of binary representation, and the ci satisfy the normalization 2n 1 2 n condition − ci = 1. Describing the state of n pure qubits thus requires 2 complex Pi=0 | | numbers. This is in contrast to the description of n classical bits which only requires a number of complex numbers linear in n. This is the ﬁrst hint at why quantum computers could solve some problems faster than classical computers. In general, a state ψ of multiple qubits falls into two distinct categories: The state can | i either be entangled or unentangled. The distinction between the two categories is the focus of the next subsection.

Entanglement

The n-qubit state ψ can sometimes be written as a Kronecker product of several individual | i qubit states: n ψ = ψi (2.5) | i M | i i=1 n where ψi is the state of the i-th qubit and is of the form of Eq. 2.2 and represents | i Li=1 the n Kronecker products of the states ψi . If this equation holds, then each qubit can | i be represented on the Bloch sphere, similar to Fig. 2.1. Thus we only require a number of complex numbers linear in n to describe the n pure qubits2, similar to n classical bits. Such a quantum state is therefore also known as a separable state. For example, the two-qubit state ψ = ( 00 + 01 + 10 + 11 )/2 = [( 0 + 1 )/√2] [( 0 + 1 )/√2] is separable. | i | i | i | i | i | i | i ⊗ | i | i If ψ cannot be written into the product form of Eq. 2.5, then the two qubits are | i said to be entangled because the qubits are correlated. Entanglement is a very important concept and is believed to be a necessary resource for speeding up quantum algorithms3. For example, the important two-qubit Bell state or EPR pair ψ = ( 00 + 11 )/√2 is | i | i | i 2We will explain the details of what a pure qubit is shortly. For the time being, it suffices to say that a pure qubit is in a known state. 3Entanglement has been intensely debated in the context of NMR quantum computing, as we will describe later, and sparked a debate over what is the cause for algorithmic speed-ups in quantum computers. Entan- glement itself is not sufficient to speed up quantum algorithms. Instead, the sufficient condition appears to be how entanglement is used. 14 CHAPTER 2. THEORY OF QUANTUM COMPUTING

an entangled state. The two qubits are correlated because the measurement of one qubit directly aﬀects the state of the other qubit (we will discuss the measurement process in more detail shortly). This correlation is what Einstein referred to as spooky action at a distance, and often it is referred to as a non-local eﬀect. As mentioned above, unentangled qubits may each be represented on a Bloch sphere. Entanglement however cannot easily be visually represented. Instead, the state is mathematically described in Hilbert space, a 2n dimensional complex vector space. Any quantum state ψ (entangled or not) corresponds to a vector of unit length in Hilbert space. The | i concept of Hilbert space will be important when we start describing the dynamic evolution of qubits.

Mixed states and the density operator

Thus far, we have only discussed the case where ψ is in a pure state, which is to say | i the quantum state is well known. However, this may not always be true. There can be cases where all we know about the quantum state is that it can be in the state ψi with | i probability pi. The state is then in a statistical mixture of diﬀerent states, or simply a mixed state. We can conveniently describe a mixed state by introducing the density operator ρ:

T r(ρ) = 1 (2.7)

because T r(ρ) = piTr( ψi ψi ) = pi. This is also known as the trace condition. Pi | ih | Pi Furthermore, the density operator is a positive operator because φ ρ φ 0 where φ h | | i ≥ | i is any other quantum state. Therefore, the density operator must have some spectral decomposition into a set of orthogonal states:

ρ = λk k k (2.8) X | ih | k 2.1. QUANTUM COMPUTING VS. CLASSICAL COMPUTING 15

where λk are the real-valued eigenvalues of ρ and k are orthogonal (the ψi above need not | i | i be orthogonal). Since a pure state has only one eigenvalue, necessarily equal to unity, and a mixed state has several eigenvalues, we can derive a very useful condition which allows us to distinguish between mixed and pure states4:

T r(ρ2) = 1 when ρ is pure → T r(ρ2) < 1 when ρ is mixed →

A quantum system can only be in one single quantum state when measured, however. What then is the physical representation of mixed states? A mixed state is a manifestation of the lack of knowledge about the quantum system or ensemble of quantum systems. For example, all NMR quantum computers at present consist of several (on the order of 1018) quantum computers acting in parallel, yet not all of them start out in the same state (see section 4), requiring a description using mixed states. The distinction between mixed and pure states has some other important implications which we discuss later.

Qubits and their applications

As we discovered, multiple qubits can encode an exponential number of complex numbers, which is what led Feynman to believe that quantum computers could exponentially outperform classical computers. In the next section, we complement the so far static description of qubits by analyzing their dynamic behavior. This provides an adequate basis to verify that qubits can be used for classical computation as well. This then opens the door to a discussion of quantum parallelism, and quantum algorithms.

2.1.2 Quantum computing subsumes classical computing

Unitary Evolution

One of the core postulates of quantum mechanics is Schr¨odinger’sEquation. It describes how a quantum state ψ(t) of a closed quantum system5 evolves as a function of time. It | i is written as: d ψ(t) i~ | i = (t) ψ(t) (2.9) dt H | i 4For a much more in-depth discussion about this topic, we refer the reader to [NC00] 5A closed quantum system does not have any interaction with the environment, i.e. the rest of the universe. We will discuss the description of open quantum systems later. 16 CHAPTER 2. THEORY OF QUANTUM COMPUTING

where ~ is Planck’s constant, and is the Hamiltonian, a total energy operator of the H quantum system. When the Hamiltonian is time-independent, this equation has an easy solution: i t ψ(t) = exp − H ψ(0) (2.10) | i ~ | i When the Hamiltonian is time-dependent, then the solution often has no easy solution but can be approximated by a concatenation of time-independent evolutions6. The operator that evolves ψ(0) to ψ(t) is called a unitary operator U, regardless of whether the Hamiltonian | i | i is time dependent and time independent. For time independent Hamiltonians U can be written in closed form: i t U = exp (2.11) −~H For the one qubit case, U is a 2x2 matrix that operates on the two-entry column vector ψ(0) . In general for n qubits, U is a 2n x 2n matrix. The operator U also satisﬁes a | i few other important conditions. Since the Hamiltonian that generates U is Hermitian, i.e. it is its own Hermitian conjugate ( = where symbolizes the Hermitian conjugate H H† † operation), the time evolution operator U is unitary:

UU † = U †U = I (2.12) where I is the identity matrix. This means that any evolution U can be reversed by the application of U †. Therefore, any evolution of a closed quantum system must be reversible. We can similarly describe the evolution of the density operator ρ using the unitary operator U:

ρ(t) = pi ψi(t) ψi(t) = piU ψi(0) ψi(0) U † = Uρ(0)U † (2.13) X | ih | X | ih | i i

The time evolution U can be visually represented by a rotation of the vector ψ in the | i Bloch sphere (if applicable), or more general, in Hilbert space. Since the length of the vector ψ(t) has to be unity, we know that the application of U cannot change the length | i of that vector. We also know that there must exist another operator that can rotate the vector back to where it started from. Single qubit unitary operations can be visualized as rotations on the Bloch sphere. This visual picture is heavily used in chapters 4 and 5, and

6We use such an approximation often later. 2.1. QUANTUM COMPUTING VS. CLASSICAL COMPUTING 17

partially in chapter 7. Clearly, any evolution of a closed quantum system has to be fundamentally reversible. But since most classical computations are done reversibly, is it possible to perform computation on a quantum computer at all? We already claimed it is possible, and in the next subsection we describe how this is exactly done.

Reversible vs. irreversible computing

In chapter 1, we already stated that one can compute reversibly, but how is that possible? Almost none of the computing devices today operate reversibly. Note for example that our computers produce heat and require energy - a sign that information is irreversibly erased. Most machines today use the and gate, which forms a universal building block for arbitrary logic circuits, together with the not gate. The truth table for the and gate is given in Fig. 2.2a. This gate only has one output bit, compared with two input bits. There is no procedure to determine the input bits to this gate based on just the output bit. Therefore, information must get erased during the execution of this gate, which translates to heat being generated. Even if we were to retain one of the input bits (Fig. 2.2b), it would still be impossible to determine the input bits. A natural question that arises is whether

Input Output 0 0 0 0 0 0 Input Output Input Output 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 1 1 0 1 (a) (b) 1 1 1 1 1 0

(c)

Figure 2.2: Truth tables for (a) the traditional irreversible and gate, (b) the traditional and gate but retaining the ﬁrst of the inputs, and (c) the reversible and gate, also referred to as toffoli gate. there exist a construction for reversible computation. Rolf Landauer and Charles Bennett showed that it is possible to compute reversibly without energy dissipation [Ben73, Lan61]. We next outline a general procedure to achieve this. 18 CHAPTER 2. THEORY OF QUANTUM COMPUTING

Suppose there exists some logic function f(x) (the and gate for example). Also suppose there exists an additional bit string y of the same length of f(x), and suppose we add f(x) to this bit string: (x, y) (x, y f(x)) (2.14) 7→ ⊕ where denotes the bitwise addition modulo two. If we simply set all bits of y to 0, then ⊕ we get: (x, 0) (x, f(x)) (2.15) 7→ Thus, by using one additional input bit, we can construct a reversible scheme for the and gate, as shown in Fig. 2.2c. Incidentally, this gate is also known as a toffoli gate [Tof80]. We can convert the reversible and gate to a nand gate, which is universal on its own by either applying a reversible not gate, or simply by letting y = 1 in Fig. 2.2c. Since we can construct any logic circuit solely from nand gates, we can also construct any logic circuit from only toffoli gates. The only diﬀerence is that the construction from nand gates has to be irreversible while the ones from toffoli gates has to be reversible. Assume for now it is possible to eﬃciently construct Hamiltonians such that the resulting unitary evolution operator corresponds to the transformation of bits in the classical truth table of a toffoli gate7. Then, it becomes clear that we can use a quantum computer to perform any classical logic gate! If we use the computational basis states 0 and 1 for the | i | i individual qubits, we can achieve any transformation

x 0 x f(x) (2.16) | i| i 7→ | i| i and such a construction can be done eﬃciently. In other words:

Quantum computation subsumes classical computation.

Provided a toffoli gate can be implemented eﬃciently on a quantum computer, any classical computation can be eﬃciently performed on a quantum computer. The description we delivered so far assumes the qubits to be in a classical state (for example either 0 or 1 but not in some superposition). Suppose we relax this assumption | i | i and allow superposition states as inputs to logic functions. We next discuss what happens as a result of this.

7Such a transformation exists provided a special set of elementary operations is allowed. We return to the discussion of quantum gates in section 2.2 and their implementation in chapters 4 and 7. 2.1. QUANTUM COMPUTING VS. CLASSICAL COMPUTING 19

2.1.3 Quantum parallelism

Quantum parallelism

8 Suppose we apply some classical reversible gate f(x) when x = c0 0 + c1 1 . Since | i | i | i quantum mechanics is fundamentally linear, we obtain the following transformation:

c0 0 + c1 1 c0 f(0) + c1 f(1) (2.17) | i | i 7→ | i | i

Even though we applied the gate f(x) only once, it has been computed for two values, x = 0 and x = 1, at the same time! Now consider a classical reversible gate with two input and output bits. If we prepare the input state into the superposition

c00 00 + c01 01 + c10 10 + c11 11 (2.18) | i | i | i | i then the execution of f(x) results in the output state:

c00 f(00) + c01 f(01) + c10 f(10) + c11 f(11) (2.19) | i | i | i | i

Now, f has been been evaluated four times in parallel. In general, for every added input qubit, we double the number of parallel computations. Thus, a function with n possible input qubits, can be evaluated for all 2n input values at the same time:

2n 1 2n 1 − − cx x cx f(x) (2.20) X | i 7→ X | i x=0 x=0 where x is an integer encoded by n qubits. While the number of parallel computations on a classical computer can only grow linearly with their size,

a quantum computer can perform an exponential number of computations in parallel.

This spectacular feature was ﬁrst introduced by David Deutsch and was coined quantum parallelism [Deu85a]. As a consequence of quantum parallelism, computers which rely on qubits in a superposition appear to be exponentially more powerful than classical computers. But this would

8If f(x) itself is not reversible we can make it reversible using techniques from section 2.1.2. 20 CHAPTER 2. THEORY OF QUANTUM COMPUTING

only be so if we can also read-out this information. Otherwise, the computation is meaningless. It is therefore important to discuss the measurement process of an arbitrary quantum state.

Quantum measurement

The measurement process of a quantum system has been at the center of discussion for many decades. The following however is an accepted mathematical model for quantum measurements. It postulates that a measurement can be understood by applying a set of measurement operators Pm to the quantum system. Suppose the state of a quantum system is described by ψ . When we measure this state, we obtain the outcome m with probability: | i

p(m) = ψ Pm ψ (2.21) h | | i where we require the measurement operators Pm to be Hermitian and that they satisfy

P = I (2.22) X m m as well as P P = δ P (2.23) m m0 mm0 m where δ = 1 only if m = m , otherwise δ = 0. A measurement transforms the state mm0 0 mm0 of the quantum system into: Pm ψ ψ | i (2.24) | i 7→ ψ Pm ψ ph | | i We can associate an orthonormal set of basis states m with any measurement operator | i Pm, so that Pm = m m . The state after applying the measurement operator Pm is then | ih | simply m . | i For example, consider the case where we are given a single qubit in a superposition

ψ = c0 0 + c1 1 where 0 and 1 form a basis set. When we measure this system | i | i | i | i | i in the same basis set, the measurement operators are P0 = 0 0 and P1 = 1 1 . We | ih | | ih | 2 measure outcome m = 0 with probability p(0) = c0 and outcome m = 1 with probability | | 2 p(1) = c1 . If we measure outcome m = 0, then the state of the quantum system collapses | | to 0 , and similarly if we measure outcome m = 1, the post-measurement state is 1 . | i | i If we measure m = 0, we actually still do not know anything about c0, or about c1. 2.1. QUANTUM COMPUTING VS. CLASSICAL COMPUTING 21

It thus appears that we cannot extract information about c0 and c1 using only one measurement. What if we were to perform the same measurement again? Since we are already either in the state 0 or 1 due to the previous measurement, the following measurement | i | i gives us the same answer again. How about we measure again but in a diﬀerent basis? The state of the system after the ﬁrst measurement has already lost all information about c0

and c1, and hence we are no longer able to extract that information. In fact, there exists

no single measurement that can fully reveal the state of an unknown quantum system.

It would be possible to determine c0 and c1 with increasing accuracy by copying the unknown state ψ , and measuring the copies individually. However, it is impossible to copy | i an unknown quantum state, as dictated by the no-cloning theorem [Die82, WZ82]. It is, of course, possible to copy known quantum states, but this means we already know c0 and c1.

Measurement and the interpretation of quantum superpositions

It appears from the previous discussion that the measurement process has a sudden impact on the quantum system, i.e. the wavefunction “collapses”. It is precisely this collapse that has disturbed many physicists, especially regarding the interpretation of quantum superpositions. If the quantum state collapses to only one of these states in the superposition, then maybe the qubit was in that state all along? Maybe there are some local hidden variables that predetermine the measurement outcome? In 1964, John Bell proposed a novel experimental set-up which would prove or dis- prove such local hidden variables [Bel64]. Alain Aspect performed this experiment in 1980 [AGR81] and a signiﬁcant number of reﬁned experiments have been performed since then, all with the same conclusion: There cannot exist any local hidden variables, so that we must accept the picture of superpositions and especially entanglement, as strange as it seems.

Measurement and quantum parallelism

What does this all mean in the context of quantum parallelism? Suppose the state of the quantum system before measurement was ψ = c0 f(0) + c1 f(1) . Then, using the | i | i | i 2 arguments from the previous paragraph we measure f(0) with probability c0 and f(1) | | 2 with probability c1 . What we measure is the answer to one of the inputs only, not all of | | them simultaneously. If we had n qubits in some superposition, we would only measure one 22 CHAPTER 2. THEORY OF QUANTUM COMPUTING

of the 2n terms, and worse, we do not even know which input our measurement corresponded to. It thus appears that the massive quantum parallelism is not accessible to us because we cannot measure all of the 2n results of the function evaluation! This is in itself true, but is not detrimental for quantum computation. Fortunately we can construct special quantum algorithms which allow us to determine some global properties of some function f(x) faster than is possible on classical machines. But before going into details about quantum algorithms, we next discuss brieﬂy some notions of complexity theory to better quantify what we mean by “speeding up computations”.

2.1.4 Complexity theory

To understand how fast quantum computers are compared to classical machines, we must compare their speed in a way that is technology independent. Complexity theory provides this comparison by analyzing the minimum amount of resources required (time, memory, and space) to solve a problem of size n. For example, adding two numbers can be done using (n) elementary operations such as the nand gate. The problem scales linearly with O n. Factoring on the other hand requires (en1/3 ) elementary operations using currently O known algorithms [Knu98], and therefore scales exponentially with n. Computer scientists generally group problems into two categories; one containing problems that scale polynomially and another with problems that scale exponentially. Problems that require a polynomial amount of resources with increasing problem size are called efficient whereas problems that require an exponential amount of resources are called inefficient or intractable. Efficient and inefficient algorithms both require an infinite amount of resources for in- finitely large problem sizes. So why is one considered efficient and the other intractable? To get a better picture of intractability, consider the following example. Suppose some problem scales as 2n, and that for n = 1000, it takes one second to solve it. Then, for n = 1001, it would take two seconds and so forth. For n = 1016, it would already take just over an hour, for n = 1024, it would take over a year! In fact, we would need 35 billion years to solve the problem when n = 1060!! In other words, even though our problem size only increased by a few percent, the problem became impossible to solve in real time. Conversely, if the problem scaled as n4 for example, the same increase in problem size (going from n = 1000 to n = 1060) only increases the computation time by 25% instead of 1020%. Quantum algorithms allow quantum computers to perform a remarkable feat: Certain 2.1. QUANTUM COMPUTING VS. CLASSICAL COMPUTING 23

problems, that are intractable on any classical machine become tractable on a quantum computer. Factoring integer numbers which is believed but not proven to be intractable classically, is an example of such a problem. This discovery could have profound implications from a complexity theory point of view. Because an efficient quantum version exists, maybe there is also an efficient algorithm for factoring numbers classically? If not, then why is there a discrepancy between the classical and quantum world? We do not provide answers to these ongoing research questions in this dissertation. Instead, we simply want to point out some larger picture implications of quantum computers. Besides factoring, there exist other quantum algorithms which offer algorithmic speed- ups, and we discuss several of them in detail in section 2.3. One question that arises at this point is, however, whether these computational gains are still possible on real physical quantum computers, which are subject to errors. This is the focus of the next subsection.

2.1.5 Quantum error correction

Decoherence errors

The fragile quantum superpositions which we create for quantum computations slowly decay or decohere into only one of the terms of the superposition because of unavoidable interaction with the environment9. This process is equivalent to saying that the information stored in this superposition slowly leaks out into the environment. In fact, the environment can be regarded as a measurement device that weakly measures and alters the quantum state. This decay process is called decoherence and the time during which superposition states are preserved is called the coherence time. Decoherence eﬀects can be detrimental for quantum computations because they destroy our carefully created superpositions [Unr95]. We can circumvent decoherence eﬀects by either performing our entire computation within the decoherence time (passive control), or apply special error correction schemes (active control). We focus our discussion on the active control part next, even though all of our experiments took the passive control approach.

Quantum error correction

Quantum errors are much more diﬃcult to correct than classical errors. Classically, we can redundantly encode information and perform error correction by looking at the bits. But as

9We will discuss decoherence eﬀects in more detail later in this section (section 2.4) 24 CHAPTER 2. THEORY OF QUANTUM COMPUTING

discussed above, we cannot copy an unknown quantum state, and furthermore we cannot measure the quantum state because it collapses, and information is lost. Finally, quantum errors can be more than simple bit flip errors - we can have phase flip errors, or even more drastically, quantum bits could be completely lost! For these reasons, it was believed for a long time that the gains resulting from quantum parallelism are lost due to unavoidable decoherence effects, and practical quantum computers appeared hopeless. Two crucial discoveries, by Peter Shor [Sho95] and independently Andrew Steane [Ste96], provided the crucial breakthrough which gave hope that practical quantum computers could indeed become feasible: quantum error correction. Quantum error correction is very similar to the classical case consisting of encoding, the application of a noisy process, decoding, and finally error correction. The encoding process on a classical computer consists of redundant encoding, leading to reduced error probabilities. Assume we were to send one unencoded bit through a noisy channel, which flips the bit with probability p. Then obviously, the probability of receiving the wrong message is p, if we just send a single bit. Now assume we send an encoded bit

0L through the same channel with the following encoding scheme:

0L 000 (2.25) 7→

1L 111 (2.26) 7→ where 0L and 1L is a representation for one logical qubit. At the end of the channel, the probability that exactly one of the three bits is flipped is 3p(1 p)2. Suppose we do a − majority voting, so that 001 0L, 010 0L, and 100 0L (and similarly 011 1L, 7→ 7→ 7→ 7→ 101 1L, and 110 1L). Even though one bit has flipped, we still obtain the correct 7→ 7→ message. The probability that two bits have flipped during the transmission is 3p2(1 p). − Using our majority voting scheme above, we would recover the wrong logical bit. Similarly, if all three bits are flipped we also obtain the wrong message with probability p3. Overall then, the probability of obtaining the wrong answer is 3p2(1 p) + p3. If p < 1/2, then the − resulting error probability is reduced using this encoding scheme. The quantum bit flip code is actually very similar. We first encode one logical qubit into multiple physical qubits. If an error occurs, then the quantum state is in a different subspace of Hilbert space. We can measure in which subspace of Hilbert space the state is 2.2. QUANTUM GATES 25

located without measuring the state directly. This tells us about which error occurred and correct it10. In addition to the bit flip code, there also exist a quantum phase flip code as well as general codes that protect against arbitrary errors (bit flip, phase flip, or even the removal of a qubit). We do not detail these codes or how they work here, but instead just mention that they do exist, and that they will be extremely important for implementing large, scalable quantum computers.

Fault tolerant computing

Progress towards quantum error correction schemes culminated in a surprising and remarkable result [ABO97, ABO99, Kit97a, Kit97b]:

Provided the error rate per elementary quantum operation is below a certain threshold, and given a source of qubits, each in a known pure state, it is possible to perform arbitrarily long quantum computations.

Over the past several years, several estimates of the threshold level place it somewhere 4 6 between 10− and 10− , depending on the assumptions made about the errors. Recently 3 however, Andrew Steane calculated the threshold to be about 10− [Ste02]. Experimentally, this number appears frighteningly small. However, the average error rate on a classical 18 computer nowadays is incredibly small, around 10− - it is so small that error correction schemes are not even used! Conversely, during the days of vacuum tubes, people were employed around the clock solely to replace burned out vacuum tubes because the lifetime of vacuum tubes is short. Modern electronics has reduced this rate by many orders of magnitude since then. There is thus hope that there might be ways to engineer an appropriate quantum system to operate below the threshold level.

2.2 Quantum gates

In this section we describe a set of unitary gates with which we can construct any arbitrary quantum circuit. After quickly reviewing the concept of universality, we describe a set of universal gates which are straightforward to implement on an NMR quantum computer. This is followed by showing several techniques to decompose a complex unitary gate into a

10For further reading on quantum error correction we refer to [Got97, NC00]. 26 CHAPTER 2. THEORY OF QUANTUM COMPUTING

sequence of gates from the universal set. We gradually introduce the notation for quantum circuits throughout this section.

2.2.1 Universal quantum gates

Universality

A very useful result from classical information theory is that any boolean logic circuit can be implemented from a ﬁnite set of boolean functions. This set is known as a universal set of gates. The nand gate is in itself universal, and so is the nor gate. By extension, the nand and nor gate together must then also form a universal set. Since the toffoli gate is just a reversible nand gate, it is also in itself universal. We therefore see that there does not exist a unique set of universal gates but instead a multitude of them. Similar to the classical case, there also exists a set of universal gates for quantum computation. Any arbitrary unitary operation can be approximated to arbitrary extent from a ﬁnite universal set of quantum gates. As in the classical case, there is no single set that is universal, but here we focus on the following set because all of its gates are straightforward to implement on our quantum computer: Arbitrary single qubit operations and the controlled-NOT (cnot) gate form a universal set of quantum gates11. We begin constructing this universal set by describing single qubit gates.

2.2.2 Single qubit gates

As described in section 2.1.2, unitary evolutions are generated by Hamiltonians. Single qubit gates are generated by single term Hamiltonians that only involve one qubit. We indicated earlier that an arbitrary state ψ can be written as a column vector. Let us | i formalize this description to facilitate the description of quantum gates. Let us rewrite the arbitrary state ψ = c0 0 + c1 1 as: | i | i | i

c0 ψ = (2.27) | i c1

Now, ψ is a column vector of two entries, c0 and c1 which are the amplitudes of the 0 | i | i and 1 states respectively. The matrix representation of ψ is then just a row vector with | i h | 11Actually, the set of arbitrary single-qubit operations is not ﬁnite but we use this formulation because of its direct application to NMR quantum computation. Note that we could make this set ﬁnite by limiting it to Hadamard gates and π/8 phase gates. Together with cnot operations, it would still form a universal set. 2.2. QUANTUM GATES 27

the same entries, but taking their complex conjugates.

A single qubit unitary matrix acting on this state is simply a 2x2 matrix U:

ψ = U ψ (2.28) | ifinal | iinitial

For example, consider the not gate that takes 0 1 and 1 0 : | i 7→ | i | i 7→ | i 0 1 UNOT = (2.29) 1 0

By applying UNOT as shown in Eq.2.28, we get:

0 1 c0 c1 ψ final = = (2.30) | i 1 0 c1 c0

We can see that the amplitudes of the 0 and 1 states are switched, as expected. | i | i

The not gate is not the only single qubit gate we can apply. In general, any single qubit rotation takes the form iα U = e Rnˆ(β) (2.31) where Rnˆ(β) corresponds to a rotation of the state vector ψ on the Bloch sphere in Fig. | i 2.1 around the axisn ˆ = (nx, ny, nz) over an angle β. Mathematically, we can deﬁne Rnˆ as

βnˆ ~σ Rnˆ(β) = exp i · = cos(β/2) σI i sin(β/2)[nxσx + nyσy + nzσz] (2.32) − 2 −

where ~σ = (σx, σy, σz), with σx, σy, σz denoting the Pauli matrices and σI the identity matrix:

0 1 0 i 1 0 1 0 σx = , σy = − , σz = , σI = (2.33) 1 0 i 0 0 1 0 1 − The Pauli matrices satisfy the following useful relationships:

2 2 2 σxσy = iσz , σyσz = iσx , σzσx = iσy , σx = σy = σz = σI (2.34)

There are three important single qubit rotations - thex ˆ,y ˆ andz ˆ-rotations. These are given 28 CHAPTER 2. THEORY OF QUANTUM COMPUTING

by: β β cos( 2 ) i sin( 2 ) Rxˆ(β) = cos(β/2) σI i sin(β/2) σx = − , (2.35) − i sin( β ) cos( β ) − 2 2 β β cos( 2 ) sin( 2 ) Ryˆ(β) = cos(β/2) σI i sin(β/2) σy = β − β , (2.36) − sin( 2 ) cos( 2 ) i β e− 2 0 Rzˆ(β) = cos(β/2) σI i sin(β/2) σz = β . (2.37) − 0 e 2 Note that incidentally, the not gate up to an overall phase can be constructed by applying

Rxˆ(180◦). From thesex ˆ,y ˆ andz ˆ-rotations, we can implement any rotation about any arbitrary rotation axis, because we can write any U as:

iα U = e− Rz(β)Ry(γ)Rz(δ) (2.38)

We actually do not require the ability to perform explicitz ˆ-rotations because we can generate them from concatenatingx ˆ andy ˆ rotations:

Rzˆ(β) = Rxˆ(90◦) Ryˆ(β) Rxˆ( 90◦) (2.39) − where time goes from right to left (i.e. the rotation over 90 is applied ﬁrst). Thus, − ◦ arbitraryx ˆ andy ˆ-rotations are suﬃcient to implement any arbitrary single qubit rotation U.

One additional and important single qubit gate is the hadamard gate, deﬁned as:

H = Rxˆ(180◦) Ryˆ(90◦) = Ryˆ(90◦) Rzˆ(180◦) (2.41)

As we have shown, single qubit rotations can be generated by Pauli matrices. But what happens when have multiple qubits? When we are given n multiple qubits, but wish to i apply only a single qubit operation aroundn ˆ on qubit i, we denote this rotation by Rnˆ(β). 2.2. QUANTUM GATES 29

i This rotation can be generated by a concatenation of Pauli matrices σj where j is either x, i y, or z and σj is the short hand notation for an n-fold Kronecker product of σI matrices except for the i-th location, where we have σj. For example, when we have three qubits and wish to apply a single qubitx ˆ rotation on qubit three, then:

β(σI σI σx) β(IIX) R3(β) = exp i ⊗ ⊗ = exp i (2.42) xˆ − 2 − 2 where IIX is just the short-hand notation for I I σx and I is the 2 x 2 identity matrix. ⊗ ⊗ We can describe these single qubit rotations via the quantum circuit language, ﬁrst introduced by Deutsch [Deu89] and shown in Fig. 2.3. The horizontal wires denote the individual qubits, and time goes from left to right.

(a) (b)

Figure 2.3: Quantum circuit representation of (a) an arbitrary single qubit rotation U, and (b) the not gate.

2.2.3 Two-qubit gates

Now that we have shown a mathematical description for single qubit operations, we only need to describe the two-qubit cnot (controlled-not) gate, and we are ready to implement any arbitrary n qubit operation. The truth table for the two-qubit cnotij gates are shown in Fig. 2.4 where i is the control bit and j the target bit. In other words, bit j is ﬂipped only if bit i is in the logical state 1 (or the quantum state 1 ). | i Input Output Input Output 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 0 1 1 1 0 1 0 1 1 1 0 1 1 0 1

(a) (b)

Figure 2.4: Truth tables for (a) the cnot12 gate, and (b) the cnot21 gate. 30 CHAPTER 2. THEORY OF QUANTUM COMPUTING

We derive the matrix representation of the cnot12 gate by determining how it acts on an arbitrary two-qubit state ψ . Similar to Eq. 2.27, ψ is now a column vector with four | i | i entries: c00   c01 ψ initial =   (2.43) | i  c10     c11 

Using the truth table from Fig. 2.4 and applying cnot12 to ψ , the output is: | i

c00 c00     c01 c01 ψ = = U (2.44) final   cnot12   | i  c11   c10       c10   c11 

The unitary matrices for the cnot gates are then:

1 0 0 0 1 0 0 0     0 1 0 0 0 0 0 1 U = and U = (2.45) cnot12   cnot21    0 0 0 1   0 0 1 0       0 0 1 0   0 1 0 0 

There are several extensions of the cnot gates, and we describe them here just brieﬂy. One important two-qubit gate is the swap gate, which swaps two qubits, i.e. 00 00, 01 10, 7→ 7→ 10 01, and 11 11. This gate can be entirely constructed out of three cnot gates: 7→ 7→

swap12 = cnot12 cnot21 cnot12 (2.46)

Besides the cnot, we can in fact implement any controlled U operation. The quantum circuit representation for a few important two-qubit gates are shown in Fig. 2.5.

Compared with single-qubit gates, which can be written in a tensor product form using Kronecker products (see for example Eq. 2.42), two-qubit gates cannot in general be written in such a notation. The reason for that is directly related to the problem of entanglement. Single qubit operations are local operations and can never increase the entanglement in a system. Two-qubit operations are non-local and can increase entanglement. For exam- 1 ple, applying a cnot12 gate on the unentangled state ψ = ( 00 + 10 ) produces the | i √2 | i | i entangled state ψ = 1 ( 00 + 11 ). | i √2 | i | i 2.2. QUANTUM GATES 31

(a) (b) (c) (d)

Figure 2.5: Quantum circuit representation of (a) a controlled-not gate, (b) a zero- controlled-not, (c) a swap gate, and (d) a controlled-U gate. The symbol denotes the control qubit - the controlled operation is only executed if the qubit is• in the 1 state. The symbol denotes the zero-controlled qubit, i.e. the operation is only executed| ifi the control qubit◦ is in the state 0 . | i

2.2.4 Remarks on unitary gates

We have discussed and shown a set of universal gates that allow us to implement any arbitrary n qubit unitary gate. A quantum algorithm corresponds to a particular unitary evolution, which we can construct by concatenating the individual gates from the universal set. In this section we discuss a few rules that facilitate the process of designing and simplifying a sequence of unitary evolutions.

Simplifying a sequence of unitary operations

Assume for now you are given some sequence of individual unitary evolutions applied one after another. The overall resulting unitary evolution can be calculated as:

U = UiUi 1Ui 2 ...U2U1 (2.47) − −

In light of Eq. 2.28, the ﬁrst unitary gate that we apply, U1, must be on the right hand side, and last, Ui, on the left. What procedures exist to simplify this sequence? For example, can we rearrange this sequence such that two consecutive evolutions, Uk and Uk 1 (for − 1 < k i) cancel out? ≤ To answer this question, we need to investigate some of the properties of unitary matrices. In general, UkUk 1 = Uk 1Uk, so the prospects for rearranging the given sequence − 6 − of unitary operators are poor. However, there are instances when the order of two unitary matrices does not matter. This is the case when the Hamiltonians that generate the 32 CHAPTER 2. THEORY OF QUANTUM COMPUTING

unitary evolutions commute: [Hk,Hk 1] = HkHk 1 Hk 1Hk = 0. The following is a − − − − non-exhaustive list of operations that commute, along with some examples:

1. Unitary evolutions that act on diﬀerent qubits commute.

Single qubit operations acting on diﬀerent qubits. • Two- or multiple-qubit gates that do not involve the same qubits. • Two- or multiple-qubit gates that have the same control but diﬀerent target • qubits.

2. All diagonal operators commute.

Allz ˆ-rotations commute. • All singly or multiply controlledz ˆ-rotations commute. • 3. Some sequences of unitary evolutions can be replaced by a simpliﬁed version.

The sequence HXH can be replaced by Z where H is the hadamard • gate, X and Z are the Pauli σx and σz matrices. If any of the control qubits of a controlled operation are in the 0 state, the gate • | i can be omitted. If all of the control qubits of a controlled operations are in the 1 state, the gate • | i is always executed so the controls can be omitted.

When two or more unitary operators commute, it is not just possible to simplify a sequence of unitary operations, but it can also lead to a reduction of the experimental time - something that could be important especially with regards to decoherence errors: When two unitary operators commute, they can also be applied at the same time. In several of the experimental sections, we explicitly show several examples that led to a signiﬁcant simpliﬁcation of a sequence of unitary operations. In general however, no quantum compiler scheme exists to date that optimizes some given sequence of unitary operations.

2.2.5 Multiple (n > 2) qubit gates

We have just discussed how to simplify a given sequence of unitary operations, but how do we obtain this sequence of single and multiple qubit operations in the ﬁrst place? A general 2.2. QUANTUM GATES 33

procedure for a wide variety of multiple qubit gates is given in [BBC+95], and here we focus our discussion on describing a few examples of multiple qubit gates, which are of particular interest in our experiments. We begin by simply showing the decomposition of the three-qubit toffoli gate from Fig. 2.2c into just single and two-qubit gates. The resulting quantum circuit is shown in Fig 2.6 where i i q 2 q −2 V = " # (2.48) i i q −2 q 2 2 which has the property that V = Unot. Controlled-not gates with n 1 control qubits has − been shown to require (n) elementary (single and two-qubit) gates, if we allow additional O scratch qubits [BBC+95].

V V V

Figure 2.6: Quantum circuit representation of a toffoli gate, and its decomposition into two-qubit gates.

Another very useful gate is the fredkin gate, which in itself is universal for classical circuits. It is a controlled swap gate, and its quantum circuit representation is shown in Fig. 2.7. Because it is a controlled-swap gate, it can be decomposed by applying the three

Figure 2.7: Quantum circuit representation of a fredkin gate, and its decomposition into toffoli gates. This decomposition can be further simpliﬁed, as shown here, and explained in the text. cnot gates that implement the swap gate (Eq. 2.46), and controlling each one of them 34 CHAPTER 2. THEORY OF QUANTUM COMPUTING

with an additional qubit. Upon further inspection, the resulting circuit consisting of three toffoli gates can be simplified by removing the first control of the first and last toffoli gates. Then, if the first qubit is in the 0 state, two cnot operations are executed, which | i cancel each other out. If the the first qubit is in the 1 state, three cnot operations are | i applied, which result in a swap. This therefore, corresponds to a controlled-swap operation. On occasion, exact multiple qubits gates need not be implemented. For example, we might not care about the phases of the individual entries in the unitary matrix. Such scenarios are important, especially when we are working with quantum states that are not in a superposition. We have made use of this and simplified some of the quantum gates that we implemented.

Eﬃciency of approximating unitary evolutions

We have stated in this section that a finite set of universal gates can implement any arbitrary unitary transform. This is only true in the sense that we can approximate any arbitrary unitary transform to within any error. What we have not discussed is the efficiency for this approximation procedure. It turns out that the approximation of most unitary transforms requires exponentially many gates (see f.ex. [NC00]). Fortunately, the gates required for the known quantum algorithms can be efficiently approximated or even exactly implemented from the universal set of quantum gates that we describe here.

2.3 Quantum algorithms

In the introduction we said that certain problems can be solved using less resources on a quantum computer than on a classical machine. But before describing such problems in more detail, we briefly touch on the historical background of quantum algorithms. The first quantum algorithm was developed by David Deutsch and Richard Jozsa, in 1992 [DJ92] and later refined by Cleve, Mosca and Tapp [CEMM98, Tap98]. The initial algorithm can solve a mathematical problem, known as Deutsch’s problem, in two steps on a quantum computer - an exponential speed-up over the best classical algorithm. The modified version can actually solve the problem in just a single step! This was the first quantum algorithm to demonstrate the usefulness of quantum parallelism. In 1994, Peter Shor showed that integer numbers can be factored in only polynomial time [Sho94b, Sho97]. This was the breakthrough discovery that jump started the field of 2.3. QUANTUM ALGORITHMS 35

quantum computation. Shor’s algorithm is actually based on the algorithm discovered by Simon [Sim94, Sim97] which also exhibits an exponential speed-up over classical algorithms. In 1996, Lov Grover showed that a search on an unsorted database only requires the square-root of the time on a quantum computer compared with a classical machine [Gro97]. After these algorithms had been developed, a significant effort was initiated to discover more algorithms - with few results. Very recently however, in 2000, Ed Farhi and Tad Hogg presented a completely new paradigm to solve real and relevant problems, using quantum adiabatic evolution [FG00, Hog00]. While the scaling behavior of these algorithms is not yet known, they are still a remarkable discovery because they offer new insight into how quantum resources could be used to solve complex mathematical problems.

2.3.1 The Deutsch-Jozsa algorithm

Outline

David Deutsch and Richard Jozsa provided the first quantum algorithm which offers an exponential speed-up over classical ones [DJ92]. The algorithm solves Deutsch’s problem with certainty. This problem can be understood as follows: You are given a black box or oracle f with n input bits and exactly one output bit. The output is either the same value (0 or 1) independent of the input, or 0 for half of the inputs, and 1 for the other half. This problem is also known as a promise problem, because we are promised that the function f is either constant or balanced. The task is then to determine whether the function is constant or balanced, solely by providing some input bits and observing the output. How many function calls are necessary to determine with certainty whether f is constant or balanced classically? We simply supply several inputs at random. If the output bit differs for any of the inputs we know for sure that f is balanced. But if we are unlucky, after having tested the function for half the inputs, 2n/2, the function could still be balanced. Only after testing 2n/2 + 1 different inputs do we know with certainty if the function is constant. Therefore, this problem scales exponentially classically. Deutsch’s and Jozsa’s intuition was to put all of the inputs into a superposition, and to let the wrong answers destructively interfere while the correct answers should constructively interfere. They found that this allows us to solve this problem with just a single function call. Note that we don’t care about the specifics about the function f, for example which inputs corresponds to which output. All we want to determine is if the function is constant 36 CHAPTER 2. THEORY OF QUANTUM COMPUTING

n n n 0 H x x H Uf 1 H y y f(x)

ψ 0 ψ 1 ψ 2 ψ 3

Figure 2.8: Quantum circuit outline for the Deutsch-Jozsa algorithm.

or balanced. This is called a global property of the function, and it is properties like these for which quantum algorithms may oﬀer an advantage.

Quantum circuit description

The initial quantum circuit as proposed by Deutsch and Jozsa has been modiﬁed (by Cleve [CEMM98] and Tapp [Tap98]) to the scheme we show here. It is outlined in Fig. 2.8. The initial state for this algorithm is: n ψ0 = 0 ⊗ 1 (2.49) | i | i | i n where 0 ⊗ denotes that all n qubits of the ﬁrst register are in the state 0 . We often leave | i | i this n-fold Kronecker product implicit. The qubit in the second register is initialized to 1 . | i We then apply the hadamard transformation on all n + 1 qubits to get:

2n 1 − x 0 1 ψ1 = | i | i − | i (2.50) | i X √ n √ x=0 2 2

The ﬁrst register is now in an equal superposition of all possible 2n 1 inputs. The − second register is also in a superposition. Next we apply Uf : x, y x, y f(x) where | i 7→ | ⊕ i stands for addition modulo 2 to get: ⊕ 2n 1 − x 0 f(x) 1 f(x) ψ2 = | i | ⊕ i − | ⊕ i (2.51) X n | i x √2 √2

Whenever the f(x) = 0 the second register is not changed, however when f(x) = 1, it is 2.3. QUANTUM ALGORITHMS 37

changed to ( 0 1 ). In other words, whenever f(x) = 1 we apply a phase ﬂip to the − | i − | i second register. This can be conveniently rewritten as:

2n 1 − ( 1)f(x) x 0 1 ψ2 = − | i | i − | i (2.52) | i X √ n √ x=0 2 2

Note that the second register in this notation is in the same state before and after the application of Uf , so we could even leave it out, by simply implementing f as x | i 7→ ( 1)f(x) x . Finally, we apply the hadamard transformation on the ﬁrst register again. − | i Let’s look at what happens in the two cases, when f(x) is constant or balanced.

When f(x) is constant, then the phase factor ( 1)f(x) is also constant, introducing only − an overall phase which is not observable. Therefore, the ﬁrst register is eﬀectively in the

same state before and after we apply Uf . The hadamard gates then simply transform the n state back to 0 ⊗ . | i When f(x) is balanced, we need to look at what happens to the transformation H x | i which can be done by ﬁrst calculating H xi : | i 0 + ( 1)xi 1 H xi = | i − | i (2.53) | i √2

This can be rewritten as ( 1)xiz z H xi = − | i (2.54) | i X √ z=0,1 2 where xiz is understood to be the logical and operation of xi and z. Therefore,

( 1)x1z1+...+xnzn z , . . . , z x z n z1,...,zn 1 n z( 1) · z H⊗ x1, . . . , xn = P − | i = P − | i (2.55) | i √2n √2n

Hence we can write ψ3 as: | i x z+f(x) ( 1) · z 0 1 ψ3 = − | i | i − | i (2.56) X X n | i x z √2 √2

n When we measure the ﬁrst register, the amplitude of the 0 ⊗ state (i.e. z = 0) is | i ( 1)f(x) because x 0 = 0. We can now see that if f(x) is balanced then ( 1)f(x) = 0 Px − · Px − because there are as many numbers of x for which f(x) = 0 as there are for which f(x) = 1. n Hence the amplitude for the 0 ⊗ state is equal to zero. Using Eq. 2.56 we can also verify | i 38 CHAPTER 2. THEORY OF QUANTUM COMPUTING

the case when f(x) is constant and see that the amplitude can be either +1 or 1 depend- − ing on the speciﬁc constant value that f(x) takes on. In summary, if after executing the quantum circuit of Fig. 2.8, which calls the function f(x) only once, the ﬁrst register gives n the state 0 ⊗ , then f is constant, and otherwise f is balanced. | i

Signiﬁcance

The Deutsch-Jozsa algorithm is the first algorithm that solves a problem exponentially faster on a quantum computer than on a classical machine. This is a remarkable discovery on its own, but there are some caveats: First, there are no known applications for this problem. Second, this problem can be solved classically with high probability. For example, for a balanced function with randomly chosen inputs, the probability of obtaining the same output each time decreases exponentially. Thus, we can already make an exponentially good estimate of whether the function is balanced or constant. The significance of this algorithm is not really its application or usefulness but rather its clear demonstration of how a quantum computer could solve certain problems exponentially faster than classical machines. It is relatively easily understood and built the foundation for all quantum algorithms that followed. This problem was incidentally used in the first ever experimental demonstration of a quantum algorithm [CVZ+98, JM98].

2.3.2 Grover’s algorithm

Outline

In 1996, Lov Grover invented an algorithm to search unsorted databases in less time on a quantum computer [Gro96, Gro97] than on a classical machine. Searching an unsorted database is also known as an unstructured search. For example, ﬁnding one number out of an array of numbers sorted by increasing values is a structured search. Finding the value when the array is unsorted is an unstructured search. The problem that Grover considered can be understood as follows. Suppose you are given a black box f with n inputs and one output. The oracle returns f(x) = 0 for all inputs x except for one special x0 for which it returns f(x0) = 1. Given that all we are allowed to do is to query the oracle using several inputs, the task is to determine the special element x0. Classically, we can solve this problem by simply supplying diﬀerent inputs until the output is 1. This procedure requires on the average [N(N + 1)/2 1]/N N/2 function − ≈ 2.3. QUANTUM ALGORITHMS 39

calls where N = 2n is the size of the problem. Thus, the problems scales as (N). This is O also the most eﬃcient procedure for classical unstructured searches. Similar to the Deutsch-Jozsa algorithm, Grover suggested to put all of the inputs to the oracle into an equal superposition. He showed that you can use this superposition in combination with additional unitary gates to solve this promise problem with high probability in time (√N), a polynomial improvement over the classical algorithm. O The oracle function call of this problem is very similar to the Deutsch-Jozsa algorithm: 0 1 x, y x, y f(x) where y is initialized to | i−| i. Since here f(x) can only take on | i 7→ | ⊕ i | i √2 values 0 or 1 as before, the state of y again does not change by the oracle function, similar | i to Eq. 2.52. In the description of the Deutsch-Jozsa algorithm, we did not omit this second register. Here, we explicitly omit it by performing the transformation x ( 1)f(x) x | i 7→ − | i and show a quantum circuit consisting of just the ﬁrst register.

Quantum circuit description

We begin by initializing n qubits to the state:

ψ0 = 0 (2.57) | i | i

where we have now left implicit that the ﬁrst register consists of n qubits. We next apply hadamard transforms on all n qubits so the state transforms to:

2n 1 − x ψ1 = | i (2.58) | i X √ n x=0 2

We now apply the following subroutine π√N/4 times: d e First, we apply the oracle function call f(x) such that x ( 1)f(x) x . This step | i 7→ − | i flips the phase of the term x0 . Second, we apply hadamard transformations on all qubits | i again, followed by flipping the phase of only the 0 element, i.e. x x except for x = 0 | i | i 7→ | i in which case 0 0 , and finally another hadamard transformation on all n qubits. | i 7→ −| i The application of f(x) is often simply called a phase flip. The second step, consisting of two hadamard transformations on all qubits and the phase inversion of the 0 element, | i is also referred to as inversion about average, because it flips the amplitude of each element about the average value of all amplitudes. After applying the subroutine (phase flip and 40 CHAPTER 2. THEORY OF QUANTUM COMPUTING

000 001 010 011 100 101 110 111 equal superposition of all 8 terms

average oracle function call (phase flip of the 110 element)

inversion about average

oracle function call (phase flip of the 110 element)

inversion about average

Figure 2.9: Illustration of Grover’s algorithm for the case N = 8 (n = 3) and x0 = 110. The diagrams show the amplitude of the eight terms 000 through 111 . The algorithm begins with an equal superposition of all 8 terms. Next,| thei amplitude| of thei 110 is ﬂipped (oracle function call), followed by an inversion about the average of all amplitudes.| i One Grover iteration consists of an oracle function call and the inversion about average. For N = 8, the amplitude of the x0 element reaches almost unity after just 2 iterations. | i 2.3. QUANTUM ALGORITHMS 41

inversion about average) π√N/4 times, the qubits are in the state x0 with high proba- d e | i bility. To facilitate our understanding of the algorithm, consider the illustration drawn in Fig. 2.9. The two steps (the phase ﬂip or oracle call and inversion about average) are called one Grover iteration. Initially, all of the amplitudes are equal, but with each added Grover

iteration, the amplitude of the x0 element is ampliﬁed, while the amplitude of all remain- | i ing terms is reduced. This process continues up to a maximum value of this amplitude. Continued application of Grover iterations would decrease the amplitude again. In fact, the

amplitude of the x0 element varies sinusoidally with the number of Grover iterations. We | i show this behavior experimentally in section 6.3 where we demonstrate a 3-qubit Grover algorithm.

Scaling behavior and signiﬁcance

How does this algorithm scale with increasing array size N? It has been shown [Gro97] that we require (√N) Grover iterations, i.e. oracle function calls and inversion about average O steps. The remaining question is the scalability of the oracle function calls. Because in real life no oracles are available, we need to implement f(x) ourselves. For many problems, this would require (N) operations, which is in the same complexity class as the classical O algorithm. However, there are some useful problems for which the construction of f(x) scales. Speciﬁcally, these are problems for which we can easily compute f(x) but not the 1 inverse, f − . For example, consider this instance of a satisﬁability problem: Suppose we

want to know which values the three bits x1, x2, and x3 should take in order to satisfy

x¯2(¯x1x3 + x2). This rather simple example can be simpliﬁed tox ¯1x¯2x3 which is satisﬁed for

x1 = 0, x2 = 0, and x3 = 1. In general, a solution to this problem can’t be easily obtained by inverting f(x). Instead we have to test each possibility until the right one is found. On a quantum computer, we can construct such a boolean function using logic gates and build the oracle function call f(x). The exact scaling of this construction depends on the specific instance of f(x), but in general, it can be constructed such that the quantum computer offers a polynomial speed-up compared to classical machines. Grover’s algorithm is not just limited to the case where only one answer is obtained. This is especially useful because many satisfiability problems have more than one solution. If there are M solutions than we require ( N/M) Grover iterations [BBHT98]. However, O p this requires prior knowledge of M. Fortunately though, M can be efficiently calculated 42 CHAPTER 2. THEORY OF QUANTUM COMPUTING

( (√N)) via quantum counting [BBHT98, BBC98]. Furthermore, Grover’s algorithm is O the best possible oracle based search algorithm [BBBV97]. In summary, Grover’s algorithm, in combination with quantum counting, oﬀers a polynomial speed-up for searching unstructured databases, compared with classical machines. In sections 6.3 and 6.7 we present an implementation of a 3 and 2 qubit Grover Search algorithm respectively, using two diﬀerent quantum systems.

2.3.3 Order-ﬁnding and Shor’s algorithm

In 1994 Peter Shor discovered a quantum algorithm for prime factorization and for computing discrete logarithms [Sho94b, Sho97]. This discovery represents a remarkable breakthrough because it offers an exponential speed-up over all known classical deterministic and probabilistic algorithms, for an important mathematical problem. Since factoring is considered intractable classically, it is extensively used for encrypting data, like RSA encryption for example. A quantum computer running Shor’s algorithm could hence break most of the modern day encryption schemes12. Shor’s algorithm was later generalized to a problem called order-finding, which in turn was realized to be part of the even more general Abelian hidden-subgroup problem [Kit95]. All of these problems offer an exponential speed-up on a quantum computer, which is made possible by one key feature, which we discuss next: The quantum Fourier transform.

The quantum Fourier transform

The quantum Fourier transform (QFT), closely resembles the classical Fast Fourier transform (FFT), but can be computed exponentially faster. The FFT takes a string of N complex number as inputs, xj, and produces a string of N numbers as output, yk:

N 1 1 − 2πijk/N y = xje (2.59) k √ X N j=0

When the input string is periodic with period r, then yk is periodic with period N/r. In other words, the FFT inverts the period r, as illustrated in Table 2.1 for N = 8 (the output has been rescaled for clarity): In case r divides N with a remainder, the inversion is only

12Quantum cryptography, however, provides quantum encryption schemes that are secure even against attacks using quantum computers. Hence, a useful quantum computer would not spell the end of secure communication. 2.3. QUANTUM ALGORITHMS 43

period r Input Output inverted period N/r 8 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 4 1 0 0 0 1 0 0 0 7→ 1 0 1 0 1 0 1 0 2 2 1 0 1 0 1 0 1 0 7→ 1 0 0 0 1 0 0 0 4 1 1 1 1 1 1 1 1 1 7→ 1 0 0 0 0 0 0 0 8 7→

Table 2.1: Example of the FFT for N = 8. approximate. Furthermore, the FFT turns shifts in the input string into phase factors in the output string as seen in Table 2.2. The QFT performs exactly the same transformation

Input Output 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 7→ 1 0 i 0 1 0 i 0 0 0 1 0 0 0 1 0 7→ 1 0 −1 0− 1 0 1 0 0 0 0 1 0 0 0 1 7→ 1 0 −i 0 1 0 −i 0 7→ − −

Table 2.2: Example of the phase shifting property of the FFT for N = 8, and r = 4 as the FFT, but the complex numbers are stored in the amplitude and phase of the terms in a superposition. For example, c00 00 + c01 01 + c10 10 + c11 11 stores the complex | i | i | i | i numbers c00, c01, c10, and c11. In general, we require n = log2N qubits to store the string of N complex numbers. But how do we implement the QFT? Let us look at this question by ﬁrst writing down the QFT mathematically. As before, let us label the states in decimal rather than binary notation. The QFT transformation acting on a computational basis state j is: | i N 1 1 − j e2πijk/N k (2.60) | i 7→ √ X | i N k=0

We can rewrite this after some algebra:

2πi0.jn 2πi0.jn 1jn 2πi0.j1j2...jn 0 + e 1 0 + e − 1 ... 0 + e 1 j1, . . . , jn | i | i | i | i | i | i (2.61) | i 7→ 2n/2 where 0.jnjn 1 . . . j1 is the binary representation of decimal numbers smaller than 1: −

n jn 2jn 1 2 j1 0.jnjn 1 . . . j1 2− + 2− − + ... + 2− (2.62) − 7→ 44 CHAPTER 2. THEORY OF QUANTUM COMPUTING

Let us now simply reverse the outputs bits so that qubit jn becomes qubit j1 and so forth. Then, Eq. 2.61 changes to:

2πi0.j1 2πi0.j2j1 2πi0.jnjn 1...j1 0 + e 1 0 + e 1 ... 0 + e − 1 j1, . . . , jn | i | i | i | i | i | i (2.63) | i 7→ 2n/2

From this equation we can immediately see how to implement the QFT. The ﬁrst term is a

180◦ rotation on qubit 1 controlled by qubit 1, which can be done by a hadamard gate on qubit 1. The second term is a 180◦ rotation on qubit 2 controlled by qubit 2 (hadamard gate applied to qubit 2), and a 90◦ rotation on qubit 2 controlled by qubit 1 (this is done via a controlledz ˆ-rotation). The third term is similarly a hadamard gate on qubit 3, a

90◦ rotation on qubit 3, controlled by qubit 2, and a 45◦ rotation on qubit 3, controlled by qubit 1. For n qubits, n hadamard gates are required and n(n 1)/2 controlled rotations. − Thus, the QFT can be implemented using (n2) elementary one and two-qubit gates. In O contrast, the best classical algorithm for computing the discrete Fourier transform on 2n elements requires (n2n) elementary operations. As an example, the quantum circuit for a O n = 3 qubit inverse QFT is drawn in Fig. 2.10, following Coppersmith’s proposal [Cop94]. For the iQFT, the operations are performed in reverse, and the output is the complex conjugate of what it would be for the QFT. For the known quantum algorithms, it is of no consequence whether we use the iQFT or QFT. In fact, we have used the iQFT for all of our implementations. Hence, for simplicity we drop the distinction between the iQFT and QFT for the remainder of the text, and just refer to the operation as the QFT.

j 3 H j 1

j 2 90 H j 2

j 1 90 45 H j 3 Figure 2.10: Quantum circuit outline for the 3-qubit QFT. The most signiﬁcant input bit is qubit 3 whereas the most signiﬁcant output bit is qubit 1. We can explicitly reverse the ordering of the qubits by applying a series of swap gates, or implicitly by simply keeping this reverse order in mind when applying additional operations following the QFT. 2.3. QUANTUM ALGORITHMS 45

Order-Finding

We now show the application of the QFT in the order-ﬁnding algorithm. The order of a permutation π on M = 2n elements can be understood as follows. Imagine M rooms and one-way corridors connecting the rooms with exactly one entrance and one exit in each room. A corridor is permitted to loop back to the same room it came from. The setup ensures that going from room to room, eventually we end up in the same room we started from. The order r of the permutation π is then deﬁned as the minimum number of transitions needed to return to the starting room. The task is to determine r solely by trials of the type “make x transitions using π starting from room y and check which room you are in”. Such a problem is again described by a black box oracle which simply outputs πx(y). An illustration of this problem has been shown in Fig. 2.11 for M = 4. In this example

0 1 2 3

Figure 2.11: Illustration of an M = 4 order-ﬁnding problem. In this example, the order r of the permutation π is r = 1 for y = 2, and r = 3 for y = 0 , 1 , or 3.

π(0)1 = 3, π(0)2 = 1, and so forth. How many oracle queries are necessary to determine r given some probability of success? Richard Cleve showed that the number of oracle queries for this problem increases expo-

nentially with m = log2M [Cle00]. In contrast, the same problem and a given probability of success only requires a ﬁxed number of oracle function calls. There is thus an exponential gap between the quantum and classical algorithm. We now explain the steps of this algorithm using an M = 4 example: We require a total of 5 qubits, 3 qubits in the ﬁrst register and 2 in the second13. The second register

is initialized to the state y1y0 or y for short, where y1y0 is the binary representation | i | i 13Technically, we require twice as many qubits in the ﬁrst register as in the second, but this is not necessary in this example 46 CHAPTER 2. THEORY OF QUANTUM COMPUTING

of y. This is the room number we start from. The first register is initialized to an equal superposition by applying hadamard gates on all the qubits in the first register. The first register represents the number of transitions x we wish to perform. Thus, in decimal notation and suppressing normalization, the state of the quantum computer is now:

ψ1 = ( 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 ) y1y0 (2.64) | i | i | i | i | i | i | i | i | i | i

We next query the oracle by evaluating πx(y), and store the result in the second register. In other words, we wish to perform:

x y x πx(y) (2.65) | i| i 7→ | i| i where x is now in a superposition of all possible values. This step can be implemented via a x 4x 2x x sequence of three controlled operations because x = 4x2 +2x1 +x0 and π (y) = π 2 π 1 π 0 . 1 2 14 Let us say that y = 2, π (2) = 1, and π (2) = 2, for example . The quantum state ψ1 is | i now transformed to:

ψ2 = 0 2 + 1 1 + 2 2 + 3 1 + 4 2 + 5 1 + 6 2 + 7 1 (2.66) | i | i| i | i| i | i| i | i| i | i| i | i| i | i| i | i| i

= ( 0 + 2 + 4 + 6 ) 2 + ( 1 + 3 + 5 + 7 ) 1 (2.67) | i | i | i | i | i | i | i | i | i | i Next, we apply the QFT on the ﬁrst register to obtain:

ψ3 = ( 0 + 4 ) 2 + ( 0 4 ) 1 (2.68) | i | i | i | i | i − | i | i

Now, if we were to measure the ﬁrst register we would obtain either 0 or 4 . Since the | i | i only possible measurements are multiples of N/r (0 or 4 in this example), we can determine the period r from this measurement (r = 2 in this case). In general, the outcomes are not as simple compared with what we have shown in this example. In general, r is not a power of two or may not even divide N, and hence it will be necessary to append the QFT by a continued fraction expansion to estimate the period r from the outcome cN/r with c being some random integer [Kob94, HW60]. This requires at least twice as many qubits in the ﬁrst register as in the second to ensure a high enough probability of success. This was in

14From this we already know that r = 2 because we are back in the same room we started from. But for the sake of the argument, let’s proceed with the example. In general, this scenario becomes very unlikely as M becomes arbitrarily large. 2.3. QUANTUM ALGORITHMS 47

fact not met in this small example, but was also not necessary. Our small example clearly illustrates the usefulness of the QFT. Consider what would happen if we were to ignore the QFT and simply measure the first register. Then, we would obtain any of the outcomes 0 through 7 with equal probability - no useful information is thus obtained by the measurement. We could also argue for the following procedure. Perform the experiment once, and measure the second register and call that measurement b. Then perform the experiment again and measure the second register. If the measurement is b, then also measure the first register and note that value, otherwise do nothing. Repeat this last step several times. This procedures allows us to extract enough useful information from the first register to obtain the order r, but it is inefficient (it scales exponentially with increasing M).

0 H

0 H QFT

0 H

y1 π (y) π2(y) π4(y) y0

Figure 2.12: Illustration of an M = 4 order-ﬁnding circuit.

Let us summarize the algorithm for order-ﬁnding. We require a total of 3m qubits where

m = log2M and M is the number of elements (rooms). We shall use n = 2m qubits in the ﬁrst register and m in the second. The algorithmic steps are outlined in Fig. 2.12 and described below:

1. Initialization

0 y | i| i 48 CHAPTER 2. THEORY OF QUANTUM COMPUTING

n 2. Create equal superposition using H⊗

2n 1 1 − x y 7→ √ n X | i| i 2 x=0

3. Oracle function call, πx(y)

2n 1 1 − x πx(y) 7→ √ n X | i| i 2 x=0

4. Apply the QFT

2n 1 2n 1 1 − − e2πixk/N k πx(y) 7→ √2n X X | i| i x=0 k=0

5. Measure the ﬁrst register

6. (Perform continued fraction expansion, either on a classical machine or on a quantum computer) 2 2n 1 2πixk/N The probability of measuring outcome k depends on − e . Only when k is Px a multiple of N/r do we get substantial probabilities (when k is not a multiple N/r the probability of measuring that outcome is zero or very small). In section 6.5, we present the first implementation of the order-finding experiment, using M = 4. We next show a specific instance of the order-finding problem: Shor’s algorithm.

Shor’s algorithm

Shor’s algorithm was historically developed before order-finding, and is used to decompose integer numbers into their prime factors. We here write it in the context of order-finding. Shor’s algorithm works by finding the period of the function f(x) = ax mod N where N is the number we wish to factor and a is a randomly chosen small number less than N, and coprime with N (i.e. N and a cannot share any factors other than 1)15. The permutation πx(y) that is implemented is: πx(y) = axy mod N (2.69)

15The algorithm fails when N is even or a prime power. 2.3. QUANTUM ALGORITHMS 49

where we simply set y = 1. Also, ax mod N is called modular exponentiation.

The implementation of Shor’s algorithm follows the order-finding procedure exactly with the permutation being ax mod N. We thus also require two registers, the first containing n and the second m qubits. Since ax mod N gets stored in the second register and by definition has to be a number between 0 and N 1, m = log N . The first register must − d 2 e be at least twice as large (n = 2m). This ensures that the probability of correctly obtaining the factors of N after classical post-processing (continued fraction expansion) is sufficiently large to avoid inefficient repetition of the algorithm.

In the order-finding experiment, we are given some oracle function πx(y), but here we need to implement it ourselves. We show that ax mod N can be implemented efficiently, and the procedure for doing so is very similar to Fig. 2.12. Following standard classical circuit techniques, the modular exponentiation is constructed by utilizing the identity ax = n 1 2 − xn 1 2x1 x0 a − . . . a a , where xk are the binary bits of x. The multiplication of the second n 1 x x 2 − xn 1 2x1 x0 register by a mod N can be written as 1 a mod N = a − . . . a a 1 mod N. × n 1 The modular exponentiation can be transformed to 1 ax mod N = [a2 − xn 1 ... [a2x1 [1 × − × ax0 mod N] mod N] ... mod N]. In words, we first multiply 1 by a modulo N, if and 2 only if x0 = 1; then we multiply the result by a modulo N if and only if x1 = 1 and so 2n 1 forth until we finally multiply by a − modulo N if and only if xn 1 = 1. Thus, modular − exponentiation is reduced to n = 2m (or (m)) multiplications modulo N, each controlled O 1 2n 1 by just a single qubit xk where 0 k n 1. The numbers a , . . . , a − mod N by which ≤ ≤ − we need to multiply can be found efficiently on a classical computer by repeated squaring. Multiplication of m-bit numbers as shown here take (m2) operations, and since we need to O perform (m) of them, the modular exponentiation requires (m3) operations. In general, O O the controlled multiplications may require at most a polynomial number of ancilla qubits. Thus more than 3m = 3 log N qubits may be needed in general. d 2 e

Together with the QFT, we can eﬃciently implement each part of the quantum algorithm, and factoring integer numbers in only polynomial time! The quantum circuit from Fig. 2.12 also outlines Shor’s algorithm where we now replace y 1 and πx(y) ax mod N. 7→ 7→ We experimentally demonstrate in section 6.6 the smallest meaningful instance of Shor’s algorithm to ﬁnd the prime factors of N = 15. 50 CHAPTER 2. THEORY OF QUANTUM COMPUTING

2.3.4 Adiabatic quantum algorithms

Since the discovery of Shor’s [Sho94b] and Grover’s [Gro97] algorithms, the quest of ﬁnding new quantum algorithms proved a formidable challenge. Recently however, a novel algorithm was proposed, using adiabatic evolution [FG00, Hog00]. Despite the uncertainty in its scaling behavior, this algorithm remains a remarkable discovery because it oﬀers new insights into the potential usefulness of quantum resources for computational tasks.

Outline

The evolution of a quantum state during an adiabatic quantum algorithm is determined by a time-dependent Hamiltonian that slowly changes from an initial (beginning) Hamiltonian

Hb to some ﬁnal (problem) Hamiltonian Hp. Suppose we are given some time-dependent

Hamiltonian H(t) where 0 t T , and at t = 0 we start in the ground state of Hb. ≤ ≤ By varying H(t) slowly, the quantum system remains in the ground state of H(t) for all 0 t T provided the lowest two energy eigenvalues of H(t) are never degenerate [Mer76]. ≤ ≤ Now suppose we can encode an optimization problem into Hp. Then the state of the quantum system at time t = T represents the solution to the optimization problem.

Specifically, adiabatic algorithms optimize for the value x which minimizes some function f(x). Therefore, we could hope to use adiabatic evolution for satisfiability problems, or unstructured (and even structured) searches. The encoding of the problem into the Hamiltonian is achieved as follows. Let an n-qubit Hamiltonian be defined as

2n 1 − Hp = f(x) x x (2.70) X | ih | x=0 so that the eigenenergies of the eigenstates of Hp correspond to the function f(x). Now, if we let H(t) = (t/T )Hp +(1 t/T )Hb and start in the ground state of Hb, then by varying the − Hamiltonians slowly enough, we end up in the ground state of Hp at time t = T . The ground state of Hp is a computational basis state because Hp is diagonal and it also directly gives us the value of x which minimizes f(x). When implementing some optimization problem, we must make sure that Hp can be encoded eﬃciently which may not necessarily be the case. Fig. 2.13 sketches the evolution of Hb and Hp. 2.3. QUANTUM ALGORITHMS 51

Performance

2 The total run-time T of the adiabatic algorithm scales as gmin− where gmin is the minimum separation between the lowest two energy eigenvalues of H(t) [FG00]. It is the scaling behavior of gmin that ultimately determines the success of adiabatic quantum algorithms. Classical simulations of this scaling behavior are hard due to the exponentially growing size of Hilbert space. In contrast, suﬃciently large quantum computers could simulate this behavior eﬃciently. Ed Farhi has shown that for at least up to 15 qubits, one instance of an adiabatic algorithm (randomly generated hard instance of the exact cover problem) scales polynomially [FG00]. However, since the Taylor series approximation of an exponential is also polynomial, it is not clear whether this instance scales polynomially in general.

Remarks on implementation

Smoothly varying some time-dependent Hamiltonian appears straightforward but it con- trasts with the traditional picture of discrete unitary operations including fault tolerant quantum circuit constructions. It may also not be possible to experimentally vary the systems Hamiltonian smoothly. Fortunately, we can approximate a smoothly varying Hamilto- nian using methods of quantum simulations [Tro59] and recast adiabatic evolution in terms of unitary operations. Discretizing a continuous Hamiltonian is a straightforward process and changes the run time T of the adiabatic algorithm only polynomially. For simplicity, let the discrete time Hamiltonian H[m] also be a linear interpolation from some beginning Hamiltonian

H[0] = Hb to some ﬁnal problem Hamiltonian H[M] = Hp such that H[m] = (m/M)Hp +

(1 m/M)Hb. The unitary evolution of the discrete algorithm can be written as: −

U = U = e i((1 m/M)Hb+(m/M)Hp)∆t (2.71) Y m Y − − m m where ∆t = T/(M + 1), and M + 1 is the total number of discretization steps. The adiabatic limit is achieved when both T,M and ∆t 0. Fig. 2.13 also shows how → ∞ → the continuous adiabatic evolution can be approximated by discrete steps. We describe an implementation of a 3-qubit adiabatic optimization problem in section 6.8. This is the ﬁrst demonstration of an adiabatic algorithm to date, and also represents the ﬁrst demonstration of a full search algorithm (no oracle function calls or ancilla qubits 52 CHAPTER 2. THEORY OF QUANTUM COMPUTING

strength

H p

H b

Figure 2.13: Illustration of the evolution of H(t) for the adiabatic algorithm with H(t) = (t/T )Hp + (1 t/T )Hb (solid line). The vertical axis represents the strength of the terms in H(t), and the− horizontal axis represents time. This continuous evolution can be approximated by discretizing the Hamiltonian and applying a zeroth order interpolation (dashed line).

were required).

2.3.5 Quantum simulations

Simulating physical systems on quantum computers has been conjectured by Feynman [Fey82] long before the invention of the first quantum algorithm. Quantum simulations do not gain significant attention even though they represent an important practical application. For example, almost every new product that is introduced on the market today, like cars, has been simulated in some form on a computer. Simulating quantum systems is thus likely provide a very productive avenue in the future. The process to simulate one quantum system on another typically involves mapping the quantum states from one system to the other, letting them evolve under some specifically designed Hamiltonian which simulates the dynamics of the system under investigation, and then mapping the states back to the original system. Specific procedures have been developed for several problems: 2.3. QUANTUM ALGORITHMS 53

Approximation of eigenenergies and eigenvectors of some Hamiltonian (for example, • calculating the energy levels of an atom, or ﬁnding the energy splitting of several Cooper pairs in a semiconductor) [AL99, WBL02]

Simulating the dynamics of many-body systems [AL97]. •

Studying quantum chaos [GS01]. •

2.3.6 Perspectives

There exist three known categories of quantum algorithms which offer a computational speed-up over classical algorithm: (1) order-finding (algorithms that involve the QFT), (2) search algorithms, and (3) quantum simulations. These algorithms were invented in the mid 1990’s and after intense effort, a new type algorithm was invented recently: (4) adiabatic optimization algorithms. If adiabatic algorithms turn out not to scale, it would be somewhat disappointing if no other applications of quantum computers are found. Nonetheless, there are several important practical applications of quantum computers, and they might even ultimately play a key role in simulating whether adiabatic algorithms do indeed scale. If adiabatic algorithms scale polynomially, that would mean a significant breakthrough and would lead to broad applications of quantum computers. From a more fundamental point of view, should a quantum computer be more powerful than classical machines? The strong Church-Turing thesis states that all universal computers are polynomially equivalent. Thus, if we are successful in building a quantum computer that can factor in polynomial time, and no efficient classical algorithm can be found, then the Church-Turing thesis would have to be rewritten. But why then does this gap exist? Where does it come from? Similarly, if we fail in building quantum computers, and the Church-Turing thesis holds, we stand to learn a fundamentally new insight into physics. At this point, it appears that there is indeed an exponential gap between quantum and classical computes as illustrated by the Deutsch-Jozsa algorithm, and the order-finding algorithm. But whether we can experimentally demonstrate this is unclear. On a final note on quantum algorithms, quantum computers do not offer speed-ups for many common tasks, such as word processing. But this is perfectly acceptable because these tasks can already be done efficiently on a classical computer. 54 CHAPTER 2. THEORY OF QUANTUM COMPUTING

2.4 Decoherence

In section 3.1.5 we discuss how decoherence is an important factor to consider for building real quantum computers. In this section we describe a mathematical model for describing and simulating decoherence eﬀects16. We use the operator-sum representation given by:

ρ EkρE† (2.72) 7→ X k k where Ek is an operator acting on the Hilbert space of the system. Since the output is still a density operator, the trace has to be conserved (see Eq. 2.7). Therefore, the Ek must satisfy the completeness relation: E E = I (2.73) X k† k k But what is the physical meaning behind these equations? Physically, the decoherence process transforms ρ to:

EkρEk† ρk = (2.74) T r(EkρEk†) with probability

p(k) = T r(EkρEk†) (2.75)

This operator-sum representation is actually general in the sense that it embodies all processes that can act on a quantum system, both unitary and non-unitary (like decoherence). When the process is unitary, then the summation Eq. 2.72 consist of simply one term

E0 = U. But when the process is non-unitary than we are given several Ek. We now review several of non-unitary processes.

2.4.1 Energy dissipation

The quantum operation known as amplitude damping arises from energy dissipation - energy exchange between the quantum system and the environment, or a bath. Examples of energy dissipation are spontaneous emission of an atom, spins thermally equilibrating with an environment, and scattering of a photon in an interferometer or cavity. We describe ﬁrst amplitude damping when the environment is at zero temperature and then when it is at

16This discussion is heavily based on Nielsen’s and Chuang’s description [NC00]. 2.4. DECOHERENCE 55

ﬁnite temperature.

Amplitude damping

Amplitude damping for a single qubit can be described by two operators:

ρ E0ρE† + E1ρE† (2.76) 7→ 0 1 where 1 0 0 √γ E0 = and E1 = (2.77) 0 √1 γ 0 0 − Physically, this equation means the following. If the qubit is in the ground state 0 , then | i it stays in its ground state. However, if it is in the excited state 1 , it decays to the ground | i state with probability γ, loosing a quantum of energy to the environment. A single qubit in an arbitrary state is transformed to:

a b 1 (1 γ)(1 a) b√1 γ − − − − (2.78) b c 7→ b √1 γ c(1 γ) ∗ ∗ − − Generalized amplitude damping

When the environment is at ﬁnite temperature, we must generalize Eq. 2.76 to:

3 ρ EkρE† (2.79) 7→ X k k=0 where

1 0 E0 = √p (2.80) 0 √1 γ − 0 √γ E1 = √p (2.81) 0 0 √1 γ 0 E2 = 1 p − (2.82) p − 0 1 0 0 E3 = 1 p (2.83) p − √γ 0

The physical interpretation here is similar. If the qubit is in the ground state 0 , then | i it is lifted to the excited state with probability γ(1 p). If the qubit is in the excited − 56 CHAPTER 2. THEORY OF QUANTUM COMPUTING

state, it decays to the ground state with probability γp. The parameter p is related to the temperature of the environment and the energy diﬀerence between the ground and excited state (if p = 1, we retain amplitude damping at zero temperature). When the qubit completely equilibrates with a bath at ﬁnite temperature, it reaches the mixed state:

p 0 ρ = (2.84) ∞ 0 1 p − This thermal equilibrium state is important, and we revisit its importance in section 3.1.3 when we describe initialization requirements for quantum computers.

The eﬀects of general amplitude damping can be visualized by analyzing the transformation of an arbitrary state vector:

(rx, ry, rz) rx 1 γ, ry 1 γ, rz(1 γ) + γ(2p 1) (2.85) 7→ p − p − − −

t/T In many physical systems, the parameter γ takes on the form γ = 1 e 1 where T1 − − is a characteristic time constant which corresponds to the lifetime of an excited state (the idea of this time constant was ﬁrst introduced in NMR [Blo46]).

2.4.2 Phase randomization

Phase randomization manifests itself in the loss of coherence, or the phase relationship, between the two energy eigenstates, 0 and 1 . Physical examples of phase randomization | i | i processes are photons scattering randomly as they travel through a waveguide, or ﬂuctua- tions of the magnetic ﬁeld that lead to phase randomizations of nuclear or electron spins.

Suppose a qubit is in the state ψ = c0 0 + c1 1 and a small Rz(θ) rotation is applied | i | i | i iθ/2 iθ/2 with random θ. The quantum state is transformed to ψ = c0e 0 + c1e 1 . The | 0i − | i | i relative phase between the two basis states 0 and 1 has now changed by θ radians. Let | i | i us call this random operation a phase kick. Let us also suppose the phase kick angle θ is well-represented by a Gaussian distribution with 0 mean and variance 2λ. Then the resulting density matrix is given by:

1 ∞ θ2/4λ ρθ0 = Z Rz(θ)ρRz†(θ)e− dθ (2.86) √4πλ −∞ 2.4. DECOHERENCE 57

Letting ρ = ψ ψ we can show that ρ ρ as: | ih | 7→ θ0 λ a b a be− (2.87) λ b∗ c 7→ b∗e− c

The oﬀ-diagonal elements of the density matrix hence decay exponentially over time, and hence this eﬀect is also called phase damping. Since the diagonal entries of the density matrix do not change under phase damping, the loss of coherence occurs without net loss of energy.

In the operator-sum representation, the operators of the phase damping process are given by:

1 0 E0 = √γ (2.88) 0 1 1 0 E1 = 1 γ (2.89) p − 0 1 − λ where γ = (1 + e− )/2. Phase damping is hence equivalent to a phase ﬂip (a Rz(180◦) rotation) with probability 1 γ. −

In many physical systems, λ increases linearly over time, λ = t/T2. Like T1, the charac-

teristic time T2 was also ﬁrst introduced in NMR [Blo46]. In addition to phase randomiza- sys tion T2, the system may also undergo systematic dephasing T2 . The information about the erroneous evolution for systematic dephasing is known and not lost to the environment like

T2. Hence, we can often correct systematic dephasing. For example, spin-echo techniques can reverse dephasing of spins due to an inhomogeneous magnetic ﬁeld. The combined sys eﬀect of both T2 and T2 is often described as:

1 1 1 = + sys (2.90) T2∗ T2 T2

For multiple qubits, phase damping can have a much greater impact the more qubits are entangled with each other and the stronger the degree of entanglement. For example, a phase kick on a single qubit results in a state 1 (e iθ/2 0 + eiθ/2 1 ). However, if we have √2 − | i | i two qubits undergoing phase kicks, then the entangled state 1 ( 00 + 11 ) decays much √2 | i | i faster to 1 (e iθ 00 + eiθ 11 ). In general, the decoherence rate for maximally entangled √2 − | i | i n qubits, is n times higher. 58 CHAPTER 2. THEORY OF QUANTUM COMPUTING

2.4.3 Remarks on amplitude and phase damping

We can learn about the coherence times of any system by precisely measuring the characteristic lifetimes T2 and T1. Since T2 has no effect on the diagonal entries of the density matrix, we can measure T1 by observing the decay rates of the diagonal entries of the density matrix. However both T2 and T1 impact the off-diagonal values of the density matrix and hence it is not easy to obtain a clean measurement of T2, unless T1 T2. Often, we find in NMR that T1 T2 but this is not necessarily the case for all systems! This operator-sum model of decoherence can be extended to multiple qubits. We do this explicitly in section 6.6 when we describe the results of our experimental implementation of Shor’s factoring algorithm.

2.4.4 Other sources of decoherence

Qubit erasure

The qubit itself may sometimes be a very fragile object (not just the quantum state of a qubit), and could get erased or lost through some process. For example, an atom might escape from an optical trap. This error can be corrected provided we can replace the missing state by some arbitrary state, which can subsequently be corrected through error correction. As we describe in section 3.1.4, it is however acceptable if a qubit gets destroyed during the measurement process.

Information leakage outside the qubit manifold

Some physical implementations of quantum computers use a two-level subsystem of some larger Hilbert space. For example, a trapped ion typically has more than just two energy levels, but only two of them are used as the physical qubit. In this case, it is possible that transitions from the subsystem to the larger manifold can occur. Obviously, this leads to errors during our computation. In our description of the experiment in section 6.7 we only use a subsystem of a higher dimensional Hilbert space for a computation, and hence our experiments are susceptible to exactly this type of error. Furthermore, in section 7.4 we use tools from NMR quantum computing to help reduce the leakage out of the qubit manifold in a Josephson phase qubit. Having additional levels outside the qubit manifold may not always be a negative aspect. In some systems, we purposely take the state of the qubit out of its subsystem to some other 2.5. SUMMARY 59

subsystem, in order to facilitate some other operation.

2.5 Summary

Quantum mechanics and information theory have long been considered separate entities, but recently, the two fields merged and created the field of quantum computing and quantum information. We have described basic elements of this field in this section. Let us summarize some key results:

1. Information can be represented by any system, including a quantum system, that has at least two distinguishable states.

2. Computation can be performed reversibly without energy consumption.

3. Quantum systems containing n qubits can in general only be represented by 2n complex variables.

4. Arbitrary unitary transforms can be generated from a ﬁnite universal set of quantum gates.

These results have been the theoretical highlights of the quantum computing ﬁeld, and nicely illustrate the enormous potentials a quantum computer could have. The ideas resulted in the discovery of quantum algorithms which allow us to solve certain problems faster on a quantum computer than on a classical machine. The most well known quantum algorithms are:

1. Deutsch-Jozsa algorithm ( (2n) classically versus (1) quantum mechanically) O O 2. Grover’s search algorithm ( (2n) versus (2n/2)) O O 3. Order-ﬁnding ( (2n) versus (1)) O O 4. Shor’s factoring algorithm ( (2n1/3 ) versus (n3)) O O 5. Adiabatic algorithms (scaling behavior unknown)

Despite this remarkable theoretical promise, can a quantum computer still outperform a classical computer in the presence of noise and errors? Rather surprisingly, the answer is yes: 60 CHAPTER 2. THEORY OF QUANTUM COMPUTING

Eﬃcient and arbitrarily long quantum computations can be performed, provided the error rate per quantum gate is below some threshold value.

Decoherence effects can be mathematically represented by using the operator-sum approach. Specifically, the decoherence processes of energy dissipation (amplitude damping) and phase randomization (phase damping) was described using this approach, and is used to simulate decoherence effects in our experiments. We use this description also to simulate decoherence effects in a Josephson Junction qubit (see section 7.4). We have now seen the enormous potential of a quantum computer, even when subject to noise and errors. But how do we build a useful quantum computer? What are the requirements? It turns out that there exist five practical requirements to built practical quantum computers. The next chapter discuss these requirements in detail. Chapter 3

Implementation of quantum computers

This chapter focuses on the physical requirements to build a real quantum computer. These are the five criteria by DiVincenzo [DiV00]. We describe each requirement in detail in each subsection: (1) a system of qubits (subsection 3.1.1); (2) a universal set of quantum gates (subsection 3.1.2; (3) an efficient initialization procedure (subsection 3.1.3); (4) the ability to measure the qubits (subsection 3.1.4); (5) long coherence times (subsection 3.1.5). I then present a summary of results and prospects for different quantum computing proposals.

3.1 Requirements

The effort to built real physical quantum computers culminated into a concise list of minimal requirements for quantum computation. We describe these criteria in detail in this section. But first we briefly estimate how many qubits are needed for a quantum computation to outperform a classical one.

Estimated size of a useful quantum computer

The entire discussion in section 2 dealt with general discussions about how a quantum computer could outperform classical machines. But how many qubits are actually needed to achieve this feat? Let us look at this question by determining the number of qubits

61 62 CHAPTER 3. IMPLEMENTATION OF QUANTUM COMPUTERS

needed to factor a 400 digit number, using Shor’s algorithm1. From the discussion of Shor’s algorithm, we can see that a total of 3log 10400 4000 qubits are required. Several hundreds 2 ≈ of scratch qubits are probably required to implement the modular exponentiation. In the context of error correction, Steane’s 7-qubit code encodes one logical qubit using 7 physical qubits. Realistically, one may need several layers of encoding. Thus, several tens or even hundreds of thousands of qubits may be required to factor a 400 digit number. Compared with the largest experimental implementation of a quantum computer to date (7 qubits), this number looks hopelessly large. Fortunately, there are a variety of smaller interesting problems which require far fewer qubits. Most of these are all simulations of quantum systems. For example, 50 to 100 qubits may be suﬃcient (ignoring error correction) to simulate a complex quantum system which would be intractable classically [AL99]. The same range could also provide much better insight into the scaling behavior of adiabatic algorithms [FG00]. If we can design a system with a few hundred qubits which we can control suﬃciently well, then we might already have a useful quantum computer! So how do we design a quantum computer with several hundreds of qubits, and possibly tens or hundreds of thousands of qubits? What are the requirements to build a quantum computer?

The ﬁve criteria

Continued eﬀort to build a real quantum computer culminated into a concise list of requirements. These are the ﬁve criteria of David DiVincenzo [DiV00]:

1. a scalable system of well-characterized qubits.

2. a universal set of quantum gates.

3. an eﬃcient procedure to initialize the quantum system to a known state.

4. the ability to perform qubit-speciﬁc measurements.

5. long coherence times compared with the average time for a logic gate.

We shall describe each of these criteria in some detail in the following subsections.

1Classically, this problem would require more time than the current age of the universe even on the fastest supercomputer. 3.1. REQUIREMENTS 63

3.1.1 System of qubits

As mentioned in chapter 2, any physical system with at least two different and distinct states can serve as a bit. A system of qubits could thus be built out of several two-level quantum systems. Some natural systems are, for example, spin-1/2 particles, or polarized photons. However, we could also choose higher-level quantum systems and only work with two of these levels. For example, we could just use two of the energy levels of an atom. Therefore, the choice for the qubits is rather flexible, and proposed qubits can be either Fermions or Bosons and range from trapped atoms and ions to nuclear and electron spins and to magnetic flux and photons. Based on the previous paragraph a quantum computer could take on a variety of different physical forms. But whatever the system might look like, it needs to be scalable. The cost for adding an additional qubit must not be exponential while simultaneously still meeting all of the remaining four criteria. For example, for each added qubit the system should not incur an exponential overhead when implementing single qubit operations. The question of scalability is important and needs to be addressed for each of the five criteria.

Alternatives to qubits

In principle, we could use quantum systems with more than two states and use all of the states, instead of just using two as described above. For example, we could use a four level system as a 2-qubit quantum computer, provided the remaining criteria are met. This approach yields no computational advantage from a complexity theory standpoint, but they might offer some remarkable alternatives to two-levels systems, as we shall describe in section 4.7. To be absolutely clear, a single 2n-level system cannot be efficiently used as an n- qubit quantum computer. This is because the number of energy levels has to increase exponentially. If we assume that only finite energy is available then the difference between the energy levels has to decrease exponentially. Thus, the qubits become exponentially less distinguishable, and so this system is not scalable. Yet, several higher order system coupled (much like several spin-1/2 systems that are coupled) can scale efficiently and can lead to several intriguing advantages (see section 4.7). Another approach is to use continuous variables, such as the momentum of a particle for quantum computation. This idea has been theoretically explored, especially with regards 64 CHAPTER 3. IMPLEMENTATION OF QUANTUM COMPUTERS

to entanglement, among other aspects. However, in the presence of errors, we can longer distinguish between continuous variables with certainty so that the precision of these systems is limited by noise. Hence, it appears that computers require the discretization of continuous variables to perform computations.

3.1.2 Quantum gates

In section 2.1.2 we described that the evolution of a closed quantum system is determined by its Hamiltonian. A quantum computation is performed by controlling the Hamiltonian so that the evolution corresponds to the computational steps of an algorithm. Remarkably, we also found that this transformation can be achieved from a ﬁnite set of unitary gates. We described a set that includes arbitrary single-qubit rotations and the cnot gate. We focus on the requirements for this set of universal gates and then touch upon other possible scenarios.

Single-qubit gates

Implementing single-qubit gates requires selective control over at least some of the qubits in our system. The most straightforward scenario would be one where we have time-dependent control over each of the qubit-selective Hamiltonians. Suppose we can time-dependently control each of the Hamiltonians that generate qubit-selective rotations (Eqs. 2.35 and 2.36). Then we can generate any arbitrary single-qubit rotation at any point in time. In fact, this is the setup we are using for NMR quantum computing, but it is not unique.

Suppose we can only control the Hamiltonians that generate qubit-selective rotation for some of the qubits. If we can still perform swap operations between the qubits such that an unselected qubit is linked to one that can be selected, we can still implement arbitrary single qubit operations: We swap the state of a qubit to one that we can address, apply the single qubit gate, and then swap the quantum state back to the initial qubit. In such a network, we need to be careful that an appropriate coupling network exists and that the overhead associated with the swap operations is at most polynomial. But the details of this depend on the speciﬁc network and nature of the coupling between the qubits. 3.1. REQUIREMENTS 65

Single qubit gates: Cellular automata

There does exist another scalable and general approach to the scenario where only some of the qubits can be individually addressed which was proposed by Lloyd [Llo93], using cellular automata. He showed that one logical qubit can be encoded by three physical qubits using the following architecture:

D ABC ABC ... ABC (3.1) − − − − where, from the standpoint of control and measurement, all qubits A are completely indistinguishable from each other, and so are qubits B and C. Each unit cell ABC represents one logical qubit, but the physical information is only stored in B. Let us brieﬂy describe how this setup can serve as a universal quantum computer.

All A, B, and C are initialized to 0 except for D which is in 1. Assuming appropriate nearest neighbor couplings only, we can perform swap operations and move the 1 to qubit A of any arbitrary unit cell.

So then all A, B, and C start oﬀ in the state 0 except for the unit cell i in which A is set to 1. Then if we perform a controlled-U operation of A on B, then we only perform the operation in unit cell i so that we implement the operation U on qubit i. Given that only nearest neighbor interactions exist, we can move the special index of A to index i + 1 by applying the sequence swapABswapBC swapABswapAC . Using similar sequences we can move the special index to anywhere within the array, and thus we can implement any arbitrary single qubit operation U.

A controlled-U operation of qubit i on qubit i 1 can be performed using the following − sequence: fredkinA,BC swapAB cc UAB,C swapAB fredkinA,BC where cc UAB,C is a − − controlled-controlled-U operation on C only if A and B are in the 1 state. Together with | i the single qubit operations we can now implement any arbitrary quantum logic circuit.

We can use qubit D as the read-out qubit as well by simply continuing to shift information to qubit D and measuring. While this set-up allows arbitrary quantum computation, its overhead for computation is tremendous because a signiﬁcant amount of swap or controlled-swap operations have to be performed. 66 CHAPTER 3. IMPLEMENTATION OF QUANTUM COMPUTERS

Two-qubit gates

Two-qubit gates require at least a two-body interaction between qubits. All of the natural couplings are two-body interactions (no three-body interaction has yet been seen). The exact form of this interaction does not matter, provided it can generate evolutions that together with single-qubit operations lead to a universal set of quantum gates. Similar to the discussion of Lloyd’s cellular automata idea above, not all of the qubits need to be coupled. As long as there exists a path that connects any two qubits, arbitrary two-qubit gates between any pairs of qubits can be implemented. Most of the experiments we show in chapter 6, have a coupling network in which all of the qubits are coupled to each other. This is extremely rare in nature, and is not a likely scenario for larger implementations of quantum computers. Another possible coupling alternative is for the qubits to be coupled via an external bus. The external degree of freedom introduced by this bus facilitates two-qubit gates, similar to intermediary qubits in a linear coupling network. The three diﬀerent coupling networks are sketched in Fig. 3.1. A real quantum computer might use one of these architectures or some combination of them.

1 5 (b) 1 2 3 4 5

2 4 BUS (c) 3 1 2 3 4 5 (a)

Figure 3.1: Three coupling network architectures for ﬁve qubits. (a) All qubits are coupled with each other. (b) The qubits are only coupled via nearest neighbor interaction. (c) Coupling of the qubits via a bus.

Universal quantum computing solely with two-qubit gates

At this point, we would like to point out that very strictly speaking, single qubit gates may not be necessary for universality. In fact, “almost any” two-qubit gate is universal [DBE95, Llo95]. However, almost none of the naturally occurring two-particle Hamiltonians can generate such universal two-qubit quantum gates because they exhibit too much symmetry. 3.1. REQUIREMENTS 67

It is possible however to use just the Heisenberg interaction as a generator for universal gates at the cost of a factor of three in the number of qubits (for encoding) [DBLW00].

Universal quantum computing solely with single-qubit gates and non-linear elements

Universal quantum computing appears to require some form of non-linear interaction (i.e. two-qubit terms in the Hamiltonian) to implement two-qubit gates. However, these gates are not necessary. One can still perform universal quantum computations using linear interactions (single-qubit gates) along with measurements and classical feedback [KLM01]. Entanglement is created via a probabilistic scenario, and used via a combination of single- qubit gates and teleportation [GC99] to create two-qubit gates. We point out that this proposal was developed in the context of optical quantum computers, where non-linear interactions between qubits (which are photons here) are diﬃcult to achieve without too much optical absorption. Nonetheless, this scheme could potentially ﬁnd useful applications in other schemes as well.

Accuracy of quantum gates

In light of the fault-tolerant threshold theorem we want the quantum gates to be executed with high fidelity, i.e. the resulting unitary transform should match the desired transform closely. Sometimes this may not be possible but in this case it often feasible to “unwind” the undesired evolution following the application of the gate. In section 5.4 we specifically show how to improve some unitary gates which inherently also undergo an erroneous unitary evolution. Not correcting the erroneous unitary evolution can have the same detrimental effects as decoherence errors. Of course, we can perform error correction provided the error is small enough, but the overhead associated with this is large. If the error is too large, then error correction is not even useful.

3.1.3 Initialization

So far, the description of quantum algorithms assumes that we can initialize our quantum system to a well known pure state. It doesn’t matter which state this is provided we can convert it to the logical 0 state in polynomial time. For classical bits, this is easy to do | i 68 CHAPTER 3. IMPLEMENTATION OF QUANTUM COMPUTERS

because we can easily reset the bits. But this is not so in the quantum setting. In section 2.1.1 we introduced the idea of a mixed state which occurs when the qubits can be in one of several states ψi . This concept is important when we let the qubits | i equilibrate at some temperature T . Suppose that the states 0 and 1 of the qubits are | i | i separated by some energy ∆E. If ∆E kBT where kB is Boltzman’s constant, then the qubits equilibrate to the 0 state with high probability. However, when ∆E kBT or in | i ≈ the extreme case ∆E kBT , we need to use the description of mixed states:

ρeq = exp ( /kBT ) / (3.2) −H Z where is a normalization factor. At room temperature kBT 26meV which is much Z ≈ larger than ∆E for almost all spin based quantum computer proposals and superconducting qubits. The initialization procedure thus appears to require temperatures in the mK range - something that is diﬃcult or even impossible for some proposals. Very remarkably however, quantum computations can still be performed on mixed states via eﬀective-pure states or pseudo-pure states.

Eﬀective-pure states

The idea of eﬀective-pure states was developed in the context of NMR quantum computing but is general and applicable to other systems as well [GC97, CFH97]. The density operator for n qubits in an eﬀective pure state ρeff is given by:

1 0 ρeff = − I + 0 ψ ψ (3.3) 2n | ih |

n n where 0 is the polarization of the qubits (see section 4.4.3), I is the 2 x 2 identity matrix and ψ is a known quantum state such as 000 ... 0 . We can verify that ρeff is still a mixed | i | i 2 state because T r(ρeff ) < 1 when 0 < 1. For traceless observables, the identity portion I does not produce any signal and does not even evolve under unitary transforms U because

UIU † = I. Therefore, any system in an eﬀective-pure state undergoes the same dynamics as one in a pure state and produces the same output signal (except for a proportionality factor 0). If it is not possible to reduce the temperature of the system enough to create a pure state, 3.1. REQUIREMENTS 69

then one has to convert the mixed thermal equilibrium state (Eq. 3.2) into an eﬀective- pure state. It is important that this procedure neither involves an exponential overhead nor that 0 decreases exponentially with increasing n. Unfortunately, the methods that have been developed thus far to create eﬀective-pure states all scale exponentially. But as we show in chapter 4, it is nonetheless possible to use these schemes to demonstrate the proof-of-principle of quantum computing.

Schulman-Vazirani cooling scheme

Do other schemes exist that permit us to obtain pure qubits from a set of partly mixed qubits, instead of eﬀective-pure qubits? The answer to this question was believed to be ’no’, so it came as a surprise when it was shown that one can eﬃciently obtain k pure qubits from n partly mixed qubits [SV99]. This cooling scheme is truly a surprising answer. It is a scalable algorithm for obtaining k pure qubits from n partly mixed qubits [SV99]. The number of partly mixed qubits required is linear in k (k n), and the number of operations is quasi-linear, (klogk). ∝ O The idea behind this algorithm is based on thermodynamic considerations: Redistribute the entropy of the qubits such that the entropy of a subset of the qubits approaches zero while the entropy of the remaining qubits increases. Overall, of course, the total entropy has to be conserved. The binary entropy function, H, is given by:

H(p) = plog p (1 p)log (1 p) (3.4) − 2 − − 2 − where p is the probability of being in one of two states ( 0 or 1 in our case). Then, | i | i according to the cooling scheme, we begin with n partly mixed qubits with total entropy nH(pth) where pth is the probability of ﬁnding the qubits in the 0 state. We wish to | i obtain k pure qubits, with total entropy kH(1). The remaining n k qubits should then − have maximum entropy, which occurs when they are totally mixed, i.e. (n k)H(1/2). − Then, by conservation of entropy:

nH(pthermal) = kH(1) + (n k)H(1/2) (3.5) −

From Eq. 3.4 and Eq. 3.5, we ﬁnd the maximum number of pure qubits kmax to be:

kmax = (1 H(pth)) n (3.6) − 70 CHAPTER 3. IMPLEMENTATION OF QUANTUM COMPUTERS

In the limit of high temperature, compared with the energy splitting ∆E between the 0 and | i 1+0 1 state, we know that pthermal is almost but not quite equal to 1/2, i.e. let pthermal = . | i 2 Then, we can approximate kmax as:

2 kmax n (3.7) ≈ 0

From this it becomes clear that that k scales linearly with n, and that it scales polynomially with 0. Schulman and Vazirani then also showed that their scheme is optimal and that it achieves the entropic bound in the limit of large n. Of course, if 0 is only on the order of 5 10 10− (as it is in NMR at room temperature), then we require 10 qubits to obtain a single pure qubit. This scheme is hence not very practical in the high temperature limit, unless other means for increasing the polarization are found. First steps towards that direction were taken by Verhulst et al. [VLV+01], to increase the average polarization of two qubits in liquid state NMR at room temperature by a factor of roughly 10. We describe in detail other methods to create eﬀective pure states in section 4.4 when we discuss state initialization for NMR quantum computers. In section 6.4 we present the NMR implementation of an elementary building block of this cooling scheme.

3.1.4 Measurement

Any computation is meaningless unless we can read the final answer. In section 2.1.3 we showed that no single measurement can give us complete information about an unknown quantum state. However, we also showed in section 2.3 that after running quantum algorithms, the final state of the qubits we are interested in is in some computational basis state. This state can be easily measured using the projective measurement bases 0 , 1 . | i | i Other bases are fine too because we can always apply an appropriate unitary transform before measurement that changes the bases.

Strong and weak measurements

Before going into other details, let us describe two diﬀerent measurement processes - strong and weak measurements. The measurement process inherently requires coupling of the quantum system to the measurement apparatus which is generally a macroscopic device. 3.1. REQUIREMENTS 71

The measurement process than collapses the quantum state and projects it onto the measurement basis. This can happen either instantaneously or slowly. In other words, the measurement is either strong or weak.

An example of a strong measurement is driving the 0 1 state of an atom which can | i ↔ | i be in two energy levels, 0 and 1 . Then, if it is in the 1 state, it emits a photon which | i | i | i can subsequently be detected. If the atom was in the 0 , no signal is detected. If the atom | i was in a superposition of c0 0 + c1 1 , then the state instantaneously collapses to either 0 | i | i | i 2 2 or 1 and with probability c0 we do not detect a signal, and with probability c1 we do. | i | | | | Clearly, when the measurement is strong, we need to decouple the measurement apparatus from the quantum system during the computation, requiring the ability to switch it on and oﬀ.

However, if the information leaks out of the quantum system slowly, it is possible to leave the measurement apparatus on all the time. Thus, the qubits only gradually decohere during the computation. At the same time however, we cannot learn much about the state of the qubit. Consequently, weak measurements usually require signal averaging either by measuring a large ensemble of identical quantum computers (ensemble averaging) or by repeated execution of the same algorithm on the same quantum computer and averaging the results (time averaging). Speciﬁcally, the NMR techniques we describe in chapter 6 use averaged measurements over an ensemble of identical quantum computers.

In light of the order-finding algorithm (section 2.3.3), or specifically Shor’s algorithm, averaged measurements pose a specific problem. Recall that in the order-finding algorithm, the measurement is some integer multiple of N/r, i.e. we measure cN/r for some integer c. The value of c is random, and hence time averaged or ensemble averaged measurements give us no meaningful information about N/r. Can we still run the order-finding algorithm when our computer only allows weak measurements? Fortunately, the answer to this question is yes. Recall that in general, the quantum computation is appended by a continued fraction expansion which allows us to extract r from any cN/r with high probability. If we were to simply perform this computation on the quantum computer, the answer is r each time with high probability. Therefore, time or ensemble averaged measurements of the qubits after the continued fraction expansion give us the answer r. Such a procedure is related to derandomization, and is a useful procedure for all quantum algorithms that give probabilistic outputs. 72 CHAPTER 3. IMPLEMENTATION OF QUANTUM COMPUTERS

Remarks on measurements

In this section we have thus far described how strong and weak measurements are both acceptable for quantum computations (provided none of them incur exponential overhead). We now detail other considerations about the measurement process. It is perfectly acceptable that the measurement process destroys the measured qubits. Because the measurement collapses the state of the qubit, it is no longer entangled to the rest of the qubits. Hence we can perform any local operation on this qubit, including throwing it away. It is not necessary to have the capability of measuring all of the qubits, as long as we can measure at least one of them while being able to eﬃciently swap the states between two qubits. In the cellular automata description of Seth Lloyd, we can only measure a single qubit. Since we can transfer the state of any qubit to that read-out qubit, we can hence measure each of the qubits. This comes at the cost of a large polynomial overhead however. Finally, we also do not need to measure the qubits during the computation. If we were asked to measure during the computation but no subsequent operation depended on the measurement outcome, then it can be left out altogether. If a subsequent operation did depend on the measurement outcome, we can use controlled operations and omit the measurement.

3.1.5 Coherence

In section 2.4 we provided a model of decoherence acting on a single qubit. We also brieﬂy described the characteristic T1 and T2 lifetimes. In order for decoherence to have only a small impact, we want the quantum system to have a coherence time τc which is much greater than the average length of a quantum logic gate τop. With respect to the threshold theorem for fault-tolerant quantum computing we want

3 τop/τc > 10− (3.8)

This equation should be taken with a grain of salt since τc includes contributions from a variety of decoherence process (described in section 2.4), and τop is just the average time of a wide variety of quantum logic gates.

Note that we emphasize on the fact that all we care about is the ratio of τop and τc and not just τop itself. This means that the actual physical time of a logic gate on the 3.2. STATE OF THE ART 73

System τc τop nop 2 8 3 6 5 14 Nuclear spin 10− 10 10− 10− 10 10 −3 − 7 −4 Electron spin 10− 10− 10 + 1 14 13 Iontrap (In ) 10− 10− 10 8 14 6 Electron Au 10− 10− 10 − 10 13 3 Electron GaAs 10− 10− 10 − 6 9 3 Quantum dot 10− 10− 10 5 14 9 Optical cavity 10− 10− 10 0 4 4 Microwave cavity 10 10− 10

Table 3.1: Crude estimates for decoherence lifetimes (τc), operation times (τop), and maximum number of operations nop. quantum computer is irrelevant as long as it is much shorter than the decoherence times.

Of course, τop directly tells us about the “processing speed” of the quantum computer, which might be significantly slower than classical computers. However, since certain problems on a quantum computer belong to a different complexity class, even with a “lousy” time τop of 50 Hz, a quantum computer could outperform the best classical computer for a certain size problem. For example, to factor a 400-digit number, it could require about 200, 000 quantum operations (a reasonably good estimate). Suppose that error correction would increase this by a factor of 100. The resulting twenty million operations would take 5 days on a quantum computer, assuming a speed of only 50 Hz. This is still significantly better than the time it would take a classical computer using presently known algorithm, namely the age of the universe (assuming a 1 GHz clock speed).

3.2 State of the art

We have described in the previous section the requirements to build a physical quantum computer. A wide variety of proposals for quantum computers now exist, but their description is beyond the scope of this thesis2. Instead, in Table 3.1 we just provide a table listing the average operation time τop, the decoherence time τc, and an estimate of the maximum 3 number of operations nop for several systems . Note that despite containing eight systems, only three fundamental qubit representations are given (spin, charge, photon).

2We however brieﬂy explain superconducting qubits as well as the ideas behind ion trap quantum computing in chapter 7. 3This table is taken from Nielsen and Chuang’s book [NC00]. 74 CHAPTER 3. IMPLEMENTATION OF QUANTUM COMPUTERS

Despite the wide variety of possibilities, it is not clear which of these systems could eventually lead to a practical and scalable system (if there was, there wouldn’t be that many proposals). However, the wide variety of proposals is certainly encouraging.

3.3 Summary

Modern day technology can meet any one or even several of the ﬁve requirements exceedingly well, however, but not all ﬁve requirements simultaneously:

1. Thousands of quantum dots serving as qubits have been produced on a single chip using semiconductor technology.

2. Very precise two-qubit gates have been implemented using natural coupling between neighboring qubits ([CVZ+98]).

3. Atoms have been cooled to their internal ground state ([DBIW89, RLM+00]).

4. Lifetimes of several days have been achieved for spins in solids ([FG59]).

While the advances for each of these areas is remarkable on its own, realizing a system that meets all five criteria simultaneously signifies a tremendous experimental challenge. It is difficult to control the qubits (manipulate, measure and initialize) while simultaneously isolating them (long lifetimes). The ideal “candidate” optimally balances these opposing requirements. To date, most of these systems have only demonstrated Rabi oscillations of a single qubit. Besides an implementation of a two-qubit Deutsch-Jozsa algorithm using trapped ions [GRL+03], only one other quantum system has been capable of implementing even small quantum algorithms: nuclear spins in liquid solution manipulated by magnetic resonance techniques. We explain this system in detail in the next two chapters. Chapter 4

Liquid-state NMR quantum computing

In this chapter we investigate how nuclear spins in molecules in liquid solution satisfy each of the ﬁve requirements for building a quantum computer, and we discuss the scalability of this method. We concentrate our discussion on spin-1/2 nuclei, but we introduce the idea of higher-order spins in the last section. We begin by explaining the system of qubits and how to diﬀerentiate between qubits (section 4.1). This leads to a discussion of single-qubit gates (section 4.2). Following the introduction of the nature of the couplings between the qubits, we proceed to explain two- qubit gates (section 4.3). We then discuss several initialization procedures of the qubits (section 4.4), and present the measurement procedure (section 4.5). This is followed by a review of some decoherence mechanisms for our system and how to minimize them (section 4.6). Finally, we present how higher-order spins could be used for quantum computation as well (section 4.7).

4.1 System of qubits

The qubits in NMR quantum computing are implemented by the spin of appropriate nuclei and subjecting them to a strong and static magnetic ﬁeld B0. We focus our attention heavily on spin-1/2 nuclei because they have two discrete energy levels. Good spin-1/2 nuclei are 1H, 13C, 15N, 19F, and 31P. Spin-0 nuclei are not magnetic and hence they are not detectable with NMR. Higher-order spins (e.g. spin-3/2 or spin-7/2) can have more than two

75 76 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

non-degenerate energy levels and are suitable to serve as one or even more qubits. In section 4.7 we introduce the idea of higher-order spins in more detail. An in-depth understanding of spin-1/2 nuclei, however, helps facilitate the discussion of high-order spins.

4.1.1 Single-spin Hamiltonian

The Hamiltonian of a single spin-1/2 particle subject to a static magnetic ﬁeld B0 along thez ˆ-axis is given by [Fre97, EBW87]

~ω0/2 0 0 = ~γB0Iz = ~ω0Iz = − (4.1) H − − 0 ~ω0/2

1 where γ is the gyromagnetic ratio of the nucleus, ω0/2π is the Larmor frequency of the spin , and Iz is the angular momentum operator in thez ˆ direction. The traditional notation in NMR employs the use of angular momentum operators. They are related to the Pauli matrices by 2Iz = σz, 2Iy = σy, and 2Ix = σx. In order to better relate to chapter 2, we exclusively use the Pauli matrix notation from here on out. The interpretation of Eq. 4.1 is that the spin has two-discrete energy eigenstates - 0 | i (the spin is aligned with the magnetic ﬁeld, i.e. “spin-up) and 1 (the spin is anti-aligned | i with the magnetic ﬁeld, i.e. “spin-down”). Furthermore, the 0 state, whose energy is | i given by 0 0 0 , has ~ω0 less energy than the 1 state. This scenario is called Zeeman h |H | i | i splitting and is illustrated in Fig. 4.1.

1 hω 0

Figure 4.1: Zeeman splitting for a spin-1/2 particle subject to a static magnetic ﬁeld.

The evolution of a spin-1/2 subject to a magnetic ﬁeld can be calculated using Eq. 2.11, i t/~ e 0 . On the Bloch sphere this evolution causes an arbitrary spin state c0 0 + c1 1 − H | i | i to precess about thez ˆ-axis with frequency ω0, as indicated in Fig. 4.2. For visualization purposes, imagine a spinning top precessing about the gravitation axis. In typical modern

1 Later, we often call ω0 the Larmor frequency. 4.1. SYSTEM OF QUBITS 77

spectrometers, the magnetic ﬁeld B0 is on the order of 10 Tesla, so that the Larmor frequency is on the order of GHz or hundreds of MHz which is in the radio-frequency regime.

B 0

Figure 4.2: Precession of a spin-1/2 particle about the axis of a static magnetic ﬁeld with frequency ω. The precession can be visualized on the Bloch sphere by assuming that the magnetic ﬁeld is along thez ˆ direction.

4.1.2 Distinguishing the qubits

In order to build an n-qubit quantum computer, we require n distinguishable qubits unless a cellular automata approach is used. We could achieve this if each nuclear spin had a diﬀerent Larmor frequency.

Heteronuclear spins

Different species of nuclei (or heteronuclear spins) can be spectrally distinguished because they generally have a different gyromagnetic ratio γ, and hence also a different Larmor frequency2. Table 4.1 summarizes several Larmor frequencies at a magnetic field strength of 11.74 Tesla. nucleus 1H 13C 15N 19F 31P ω0 500 126 51 470 202 −

Table 4.1: Larmor frequencies [MHz] of several species of nuclei, at 11.74 Tesla

Homonuclear spins

In case we are given the same species of nuclear spins (homonuclear spins) in a molecule, they can still be spectrally distinguished, if their bonding is diﬀerent, which results in a chemical

2This gyromagnetic ratio can be negative as is the case for 15N. 78 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

shift σi. The electron clouds can shield different nuclei of the same species by different amounts from the external magnetic field. Since the Larmor frequency is proportional to the magnetic field seen by the nucleus, nuclei in different chemical environments have slightly different resonance frequencies. The chemical shift of a nucleus is typically expressed in units of parts per million (ppm) of its Larmor frequency. At first, this appears to be an obscure measure, but it is actually very convenient. The chemical shift is independent of the applied magnetic field. This allows us to characterize a molecule without specifying the magnetic field strength. The chemical shift can be translated into frequency units via 6 ∆ω = δω0/10 where δ is the chemical shift in ppm and ω0 is the Larmor frequency of the nucleus. For protons (1H), the chemical shift can be up to 10ppm, for fluorines (19F) 200 ppm, and for carbons (13C) 200 ppm. For typical magnetic field strengths, the chemical shift is in the range of several to tens of kHz. Mathematically, the single spin Hamiltonian for a molecule with n homonuclear spins is given by:

n n i i i 0 = ~(1 σ )γB0σz/2 = ~ω σ /2 (4.2) H − X − − X 0 z i=0 i=0

Fig. 4.3 shows two molecules and describes which homonuclear spins have a chemical shift. The degree of asymmetry of the molecule and the electronegativity of the atoms within the molecule strongly inﬂuence chemical shifts.

F F F F C C C C F F F Br (a) (b)

Figure 4.3: Structure of (a) tetrafluoroethylene. This molecule has four equivalent fluorine nuclei and hence they all have the same chemical shift. (b) Bromotrifluoroethane has three inequivalent fluorine nuclei. The two fluorines on the left are not free to rotate about the double bond between the two carbon nuclei and hence they each have a different chemical environment with respect to the bromine. The third fluorine has an even different chemical environment compared to the previous two. If the bond between the two carbons was only a single bond, the two fluorines on the left would be free to rotate about the axis of the bond and hence their chemical shift would be the same due to molecular tumbling. 4.2. SINGLE-QUBIT QUANTUM GATES 79

In general, we ideally desire a molecule consisting of n heteronuclear or homonuclear spins (with non-zero and non-overlapping chemical shifts) so the Hamiltonian can be described by: n i i 0 = ~ω σ /2 (4.3) H − X 0 z i=0 i where each of the n spins can be spectrally distinguished from all others (all ω0 are diﬀerent).

4.2 Single-qubit quantum gates

4.2.1 Rotations in the xˆ-yˆ plane

Single-qubit rotations are implemented by applying an electromagnetic ﬁeld at frequency

ωrf along a fixed axis perpendicular to B0, with the magnetic field being of strength B1. The Hamiltonian due to the RF-field3 is given by [Fre97, EBW87]:

rf (t) = ~ω1 (cos(ωrf t + φ)σx/2 + sin(ωrf t + φ)σy/2) (4.4) H − where ω1 = γB1, and φ is the phase of the oscillating ﬁeld, which allows us to choose betweenz ˆ ory ˆ-rotations (see below).

In practice, we apply an RF-field that oscillates with frequency ωrf along a fixed axis. This is not the same as the rotation that is described in Eq. 4.4. However, the oscillating field can be decomposed into two counter-rotating fields one of which rotates at frequency

ωrf in the same direction as the spin. We call this component the B1 ﬁeld. The other ﬁeld rotates in the opposite direction and results in a tiny shift of the Larmor frequency, also known as the Bloch-Siegert shift [BS40]. In section 5.2 we discuss artifacts of spin selective pulses which are similar to the Bloch-Siegert shift.

Evolution under an RF ﬁeld

We can now investigate the evolution of a spin due to the RF-ﬁeld at or near the Larmor frequency ω0. The dynamics however are very complicated when described in the laboratory reference frame. A description at a reference frequency that rotates about thez ˆ-axis at or

near the Larmor frequency ω0 greatly simpliﬁes this picture. This is the rotating reference frame.

3We use RF-ﬁeld here because the electromagnetic ﬁelds are typically in the radio-frequency (RF) range. 80 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

Suppose we apply an RF-field with ωrf = ω0 and let the coordinate frame also rotate about thez ˆ-axis at the same frequency. Then the B1-field appears to be along a fixed axis and the Hamiltonian is given by

rot = ~ω1 (cos(φ)σx/2 + sin(φ)σy/2) (4.5) Hrf −

Within the rotating frame, a vector on the Bloch sphere simply rotates about this ﬁxed axis with frequency ω1 - this precession is called nutation in the NMR language. We can control the direction of this axis by adjusting the phase φ of the RF ﬁeld. Within the laboratory frame, the spin spirals towards the xy-plane. Since B0 is much stronger than

B1, the spin undergoes signiﬁcantly more precession cycles about thez ˆ-axis before it reaches the xy-plane. The dynamics as seen from the rotating and laboratory frame are sketched in Fig. 4.4a and b.

y y ω 1 ω 1 x x

(a) (b)

Figure 4.4: Dynamic evolution of a spin subject to an on-resonance RF ﬁeld (a) in the rotating frame and (b) in the laboratory frame.

Single-qubit rotations

By controlling φ we can implementx ˆ ory ˆ-rotations or anything in-between. The angle of rotation is controlled by adjusting the duration of the RF ﬁeld:

θ = ω1tpw = γB1tpw (4.6) 4.2. SINGLE-QUBIT QUANTUM GATES 81

The parameter tpw is the pulse width or simply the duration of the RF ﬁeld. A properly calibrated pulse with the right phase can thus implement rotations about thex ˆ ory ˆ-axis or about an arbitrary axis in thex ˆyˆ-plane. We can calibrate Rx(90◦) pulses, which we denote simply by X. Similarly, the rotation Ry(90◦) is denoted by Y . If we perform a rotation 2 twice as long (Rx(180 )), it is denoted as X . A rotation about xˆ is written as Rx( 90 ) ◦ − − ◦ or simply as X¯. Note that in the context of writing down a sequence of pulses, we use this notation, which is not to be confused with the notation of Eq. 2.42 where we used X to denote the

σx matrix. We wish to point out that only the relative phase between pulses applied on the same spin matters. Hence the ﬁrst pulse of any pulse sequence is irrelevant, and the phase of all future pulses must be properly set with respect to the ﬁrst one.

RF ﬁeld applied oﬀ-resonance

We have just described the evolution of spins in the rotating frame when the RF ﬁeld is applied on resonance. Let us brieﬂy discuss what happens when we apply radiation

oﬀ-resonance (ωrf = ω0). The Hamiltonian becomes 6

rot = ~∆ωσz/2 ~ω1 (cos(φ)σx/2 + sin(φ)σy/2) (4.7) Hrf −

where ∆ω = ω0 ωrf . The spin now precesses with frequency −

2 2 ω10 = q∆ω + ω1 (4.8)

about an axis that is tilted away from thez ˆ-axis by an angle

θ = arctan(ω1/∆ω) (4.9)

The picture of the rotation axis being tilted towards thez ˆ-axis when applying radiation off-resonance is an important visual aid when we consider artifacts of single qubit rotations. Furthermore, off-resonance pulses are not necessarily unwanted effects - they can be used to generate rotations that lie outside thex ˆyˆ-plane, but we have only used such pulses to remove undesired artifacts during simultaneous soft pulses (see section 5.3). 82 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

4.2.2 Rotations about the zˆ axis

The third general type of single-qubit rotation, thez ˆ-rotation can be implemented in NMR in several diﬀerent ways. We next brieﬂy outline these procedures.

1. Concatenatingx ˆ andy ˆ-rotations

We have shown in section 2.2.2 (Eq. 2.39) thatz ˆ-rotations can be implemented by a sequence of onlyx ˆ andy ˆ-rotations. For example,

Z = XY X¯ = Y X¯Y¯ (4.10)

with time going from right to left. We have used this concatenation of several rotations to achieve another in the ﬁrst experiment (section 6.3). However, there are two experimentally more convenient approaches.

2. Compression ofz ˆ-rotations

Once we obtain a sequence of operations consisting of single-qubit operations and time delays which implement our quantum circuit (we discuss details on how to design such sequences in section 4.3), we aim at moving allz ˆ-rotations to the end of the sequence. For example: XZ = XY X¯Y¯ = ZY¯ (4.11)

We thus moved thez ˆ-rotation past thex ˆ-rotation by replacing it with ay ˆ-rotation. Sincez ˆ-rotations commute with the natural Hamiltonian of nuclear spins in solution, we can movez ˆ-rotations past time evolutions as well4. We can gather allz ˆ-rotations at the end of the sequence and then implement a singlez ˆ-rotation.

3. Implicitly absorbingz ˆ-rotations

This procedure eﬀectively implements the same technique as we just described but is performed diﬀerently. Here, we create an additional reference frame besides the rotating frame. The operation Z is then implemented by shifting the phase of the

additional reference frame by 90◦. This is equivalent to changing the phase of the

pulse. Thus, all future pulses are now applied 90◦ out of phase with respect to the initial rotating frame. For example, XZ applies a single RF pulse, but oﬀset by a

4We have not discussed coupling between nuclear spins yet, but it suﬃces to say here thatz ˆ-rotations and coupled evolution between spins commute. 4.2. SINGLE-QUBIT QUANTUM GATES 83

phase of 90◦ degrees. This pulse corresponds eﬀectively to ay ˆ-rotation in the rotating frame (see Eq. 4.11). This procedure does not require us to rewrite the sequence of operations as was the case in the previously described technique, and hence it is even more practical. We use this technique in the later experiments.

In fact, this procedure is so simple to implement thatz ˆ-rotations become almost trivial. We hence try to convert as manyx ˆ andy ˆ-rotations toz ˆ-rotations to save experimental time. For example, XY can be simpliﬁed via XY = XY XX¯ to ZX. In this example, the experimental time is reduced by half.

4. Jumping the carrier frequency

This procedure effectively implements az ˆ-rotation. If the rotating frame is exactly on-resonance with the Larmor frequency of the spin, a spin appears to be fixed on the Bloch sphere in the rotating frame. If we were to move slightly off-resonance for some time, the spin begins to accumulate some phase in the rotating frame, just like az ˆ-rotation. If we calibrate the amount by which we go off-resonance as well as the duration, we can implement arbitraryz ˆ-rotations. We use this method in section 6.8.

Of course there exit other possibilities such as producing an extra RF field along thez ˆ- axis. With current probe head designs in NMR however, this is rather difficult to achieve, especially if we desire a high degree of field homogeneity. Nonetheless, there are a variety of different approaches which may each have advantages over another depending on the specifics of the experiment.

4.2.3 Selective excitation using shaped RF pulses

We can spectrally distinguish the individual spins of a molecule given they are heteronuclear spins or homonuclear spins with chemical shift. We left implicit thus far that the selective spin rotations are achieved by long pulses tuned to the resonance (Larmor frequency) of the spins. While this technique can provide sufficient selectivity even when chemical shifts are small, we can improve the selectivity by shaping the RF pulses, called shaped pulses or soft pulses [Fre97, Fre98]. Soft pulses are designed to excite spins only within a limited bandwidth while minimizing excitation outside this window. These pulses begin with a very small B1 field amplitude, gradually increase the field to some maximum value, and finally decrease it to a small value again. Experimentally, pulse-shaping is done by dividing the 84 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

pulse into a few tens to hundreds of discrete slices each with constant amplitude and phase, and by changing the amplitude and/or phase from slice to slice. We can use the Fourier transform as a good guide to estimate the excitation proﬁle of a shaped pulse, but only for small spin rotations. The Fourier transform applies to systems with a linear response (an increase in input power leads to a proportional increase in output power), but the spin response is sinusoidal. However, for small spin rotations, we can approximate sinx x which is linear and hence we can apply use Fourier analysis. For ≈ large spin rotations, the Fourier transform still gives us a rough picture, but to obtain an exact solution we have to use another method. We next show how to calculate the exact response for spins when applying a shaped pulse5. According to the description in the previous paragraph, we shape pulses by dividing the pulses into several slices (N), each with constant amplitude and phase, and by changing the amplitude and/or phase from slice to slice. Thus, the Hamiltonian is constant for each slice, and hence we can use Eq. 2.11 to calculate the unitary matrix for each slice. The constant RF Hamiltonian in the rotating frame for each slice i for some amount of oﬀ-resonance ∆ω, is similar to Eq. 4.7:

rot = ~∆ωσz/2 ~ω1,i (cos(φ)σx/2 + sin(φ)σy/2) (4.12) Hrf,i −

The overall unitary evolution is given by

N 1 − U = UN 1UN 2 ...U1U0 = Ui (4.13) − − Y i=0

i rot t − Hrf,i i where Ui = e− ~ with ti denoting the duration of each slice i. We can then apply the resulting unitary evolution to some input state and predict thex ˆyˆ andz ˆ-magnetizations of the spin. We then repeat the procedure for a different amount of off-resonance, ∆ω. The resulting excitation profiles6 for several pulse shapes are shown in Fig. 4.5.

Since we chose to use a large rotation angle of 180◦ we can verify that the excitation proﬁle of a rectangular shaped pulse is not simply equal to the Fourier transform of the time proﬁle. Furthermore, it is also not very selective. The gaussian shape [BFF+84] on the other hand is very selective because it has negligible excitation outside this excitation

5This model forms the basis to predict several interesting artifacts due to shaped pulses (section 5.2) 6The code which calculates the excitation proﬁle for diﬀerent pulse shapes is shown in Appendix A 4.2. SINGLE-QUBIT QUANTUM GATES 85

1 4000 Z Rectangular 0.8 XY 3000 0.6 2000 0.4

1000 0.2

0 0 [kHz] 1 ω −0.2 −1000

normalized amplitude −0.4 −2000 −0.6

−3000 −0.8

−4000 −1 0 200 400 600 800 1000 −6000 −4000 −2000 0 2000 4000 6000 time [µ s] ∆ ω [Hz]

1 4000 Z Gaussian 0.8 XY 3000 0.6 2000 0.4

1000 0.2

0 0 [kHz] 1 ω −0.2 −1000

normalized amplitude −0.4 −2000 −0.6

−3000 −0.8

−4000 −1 0 200 400 600 800 1000 −6000 −4000 −2000 0 2000 4000 6000 time [µ s] ∆ ω [Hz]

1 4000 Z Hermite 180 0.8 XY 3000 0.6 2000 0.4

1000 0.2

0 0 [kHz] 1 ω −0.2 −1000

normalized amplitude −0.4 −2000 −0.6

−3000 −0.8

−4000 −1 0 200 400 600 800 1000 −6000 −4000 −2000 0 2000 4000 6000 time [µ s] ∆ ω [Hz]

1 4000 Z IBURP 0.8 XY 3000 0.6 2000 0.4

1000 0.2

0 0 [kHz] 1 ω −0.2 −1000

normalized amplitude −0.4 −2000 −0.6

−3000 −0.8

−4000 −1 0 200 400 600 800 1000 −6000 −4000 −2000 0 2000 4000 6000 time [µ s] ∆ ω [Hz]

Figure 4.5: Time profile (left) and excitation profile (right) for several pulse shapes all of the same length and calibrated to result in a 180◦ rotation when applying the RF field on resonance. The excitation profile shows thez ˆ, andx ˆyˆ-component of the spin magnetization, starting from thez ˆ-axis. 86 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

window, at the cost of higher peak power. The hermite 180 shape [War84] is similar to the gaussian shape but it has a flatter excitation window and requires an even higher peak power. The iburp shape [GF91] has been specifically designed to invert spins within a certain bandwidth. In general, each pulse shape has its own strengths and weaknesses, and we next briefly overview some criteria related to the design of pulse shapes.

Pulse shape design

There are several criteria relevant to choosing a particular pulse shape (in no particular order):

selectivity: Selectivity is deﬁned as the product of the excitation bandwidth times the • pulse width. A lower number is more selective.

peak power: The peak power required for a pulse shape. A pulse with a smaller peak • power is less demanding for the electronics.

transition region sharpness: The width of the region between the unselected and • selected frequency region. A smaller region has a sharper excitation window.

refocusing property: Pulse shapes can be designed to refocus the eﬀect of couplings • between the spins during the pulse (more on couplings and refocusing in section 4.3). A pulse with better refocusing properties has a ﬂatter excitation window.

robustness: Some shapes are more robust to inhomogeneities in the B1 magnetic ﬁeld • and miscalibrated pulse widths.

universality: Some pulses are designed assuming the spins are in a particular state • (for example all along thez ˆ-axis). Other pulses perform the same rotation about the same angle independent on the input state of the spins.

We revisit the topic of shaped pulses and artifacts related to using shaped pulses in chapter 5. In section 7.4 we show how shaped pulses are useful for other implementations of quantum computers, speciﬁcally Josephson phase qubits. 4.3. TWO-QUBIT QUANTUM GATES 87

4.3 Two-qubit quantum gates

4.3.1 Interaction Hamiltonian

Our description of nuclear spins has not included the interaction between diﬀerent nuclei of a molecule which is required when we wish to build two-qubit gates. Nature provides two diﬀerent interactions between nuclear spins:

magnetic dipole-dipole interaction: This interaction is similar to the interaction be- • tween two bar magnets close to each other. It takes place through space and is inversely proportional to the cube of the spatial separation between the two spins, and the relative positioning of the magnetic moment vector of the two spins. Both intramolecular dipolar couplings between spins of the same molecule and intermolecular dipole couplings between spins of diﬀerent molecules exist. However, when the molecules are dissolved in an isotropic liquid, these dipolar couplings are averaged away due to rapid tumbling of the molecules.

J-coupling: This interaction is also known as scalar coupling and is mediated through • shared electrons in the chemical bond between the spins. The magnetic ﬁeld seen by one spin is perturbed by the state of the electron cloud which interacts with another spin. The coupling strength depends on the element and isotope of the nuclei and generally decreases with the number of chemical bonds separating the nuclei. It is this interaction that we are interested in.

J-coupling

J-coupling can be described in slightly more detail as follows. The nuclear spin interacts with its neighboring electrons. These in turn interact with other neighboring electrons, which can then inﬂuence other nuclear spins. In order to obtain strong J-couplings, the nuclear-electron and electron-electron interaction need to be strong. The nuclear-electron interaction depends on the overlap of the electron wave function with that of the nucleus. The wave function of the nucleus is much more localized than the electron wave function and hence the overlap is largest when the electron wave function is also localized at the nucleus. An electron s-orbital is spherically symmetric and much more localized at the center than the dumbell-shaped p-orbital, which has a node at the nucleus. We thus expect systems with p-orbitals to result in a smaller J-coupling than systems with 88 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

s-orbitals. Similarly, the electron-electron interaction depends on the overlap of the electron wave functions. We thus expect increased J-couplings for double-bonds.

The strength of the J-coupling is furthermore dependent on additional parameters such as the gyromagnetic ratio of the nuclei, and the geometry of the molecule. However, it is independent of the applied magnetic ﬁeld B0.

The Hamiltonian for scalar J-coupling of a molecule containing n spin-1/2 nuclei is given by: n n i j i j i j i j J = ~ πJijσ σ /2 = ~ πJij(σ σ + σ σ + σ σ )/2 (4.14) H X · X x x y y z z i

When the condition ωi ωj 2πJ applies, the spectra are also said to be first order. | − | Let us briefly explain why this approximation is valid. Considering just the two-qubit case for the time being, we can write the system Hamiltonian as a sum of Eqs. 4.14 and 4.3 to obtain: ω1 + ω2 + πJ 0 0 0   ~ 0 ∆ω πJ πJ 0 H =  −  (4.16) 2  0 πJ ∆ω πJ 0   − −   0 0 0 ω1 ω2πJ  − − where ∆ω = ω1 ω2. Let us consider the subsystem which has non-zero off-diagonal terms. − We observe that this subsystem is given by:

~ ∆ω πJ πJ H = − (4.17) 2 πJ ∆ω πJ − − ~ This can be rewritten as H = (∆ωσz + πJσx πJI) where σz and σx are the Pauli 2 − matrices, and I is the 2 x 2 identity matrix. This is very similar to the Hamiltonian that results when a spin-1/2 is excited off-resonance (see Eq. 4.7). Hence, when ∆ω J, we | | can effectively ignore the off-diagonal elements. As a result, we are left overall with the 4.3. TWO-QUBIT QUANTUM GATES 89

diagonal matrix:

ω1 + ω2 + πJ 0 0 0   ~ 0 ∆ω πJ 0 0 H =  −  (4.18) 2  0 0 ∆ω πJ 0   − −   0 0 0 ω1 ω2πJ  − − For n spins, we repeat the same argument for each qubit pair i and j, with the net result i j i j that the σxσx + σyσy contributions can be ignored.

This assumption was valid for all of our experiments using spin-1/2 nuclei. The full Hamiltonian of n nuclear spins in an isotropic solution and with ﬁrst order spectra is then

n n i i i j = ~ω σ /2 + ~ πJijσ σ /2 (4.19) H − X 0 z X z z i=0 i

The interpretation of this equation is that in addition to the static magnetic ﬁeld, each spin “sees” another magnetic ﬁeld along zˆ produced by neighboring spins. The resonance ± frequency ωi of each spin i is shifted by Jij/2 when spin j is in the 0 state, and by +Jij/2 − | i when spin j is in the 1 state. A molecule with two spins that are coupled with strength J | i would then have two resonance frequencies for each spin. We can associate each with the state of the other spins, either 0 or 1 . We discuss the interpretation of NMR spectra in | i | i much more detail in section 4.5.

The coupling strength J can be a few hundred Hertz for couplings through a single bond, and a few Hertz or tenths of Hertz for couplings through three or four bonds. The coupling need not be positive and the sign can be measured via two-dimensional experiments [BMG+87] or one-dimensional experiments using decoupling techniques. The sign cannot be determined from a single one-dimensional experiment.

When thinking about the spin dynamics of this total Hamiltonian in the rotating frame on-resonance with each spin the ﬁrst term vanishes, and we are left with just left with the second term in Eq. 4.19.

The spin interaction Hamiltonian forms the basis of all two-qubits gates for NMR quantum computation. For a coupled two-spin system, the time evolution operator is given 90 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

by:

iπJt/2 e− 0 0 0   0 e+iπJt/2 0 0 U (t) = exp iπJσ1σ2t/2 = (4.20) J z z  +iπJt/2  −  0 0 e 0    iπJt/2  0 0 0 e− 

We use this operator extensively in the following discussion.

4.3.2 The controlled NOT gate

Because single-qubit operations and the cnot gate form a universal set of unitary gates, we discuss the cnot gate here, and explain a two qubit example.

One possible implementation avoids using UJ (t) altogether by very narrowly exciting 2 only the transition at ω + J12/2. This excites and inverts spin 2, only if spin 1 is in 1 0 | i (after [CPH98a]). However, for multiple qubits this technique may not be very practical because we would need to invert half of the spectral lines for the spin. Furthermore, when some of the spectral lines overlap, this technique even become impossible.

The second technique involves direct use of the evolution UJ (t). This is illustrated in Fig. 4.6. First, a selective rotation on spin 2 about thex ˆ-axis is applied. This pulse covers 2 1 the frequency range ω J12/2 but not ω . The spin is rotated to the yˆ-axis. Second, we 0 ± 0 − 2 simply wait 1/2J12 seconds. Since the rotating frame is with respect to ω0, spin 2 moves clockwise if spin 1 is 1 , and anti-clockwise if it is 0 . After 1/2J12 seconds, spin 2 is along | i | i xˆ if spin 1 is 1 , and x if it is 0 . We now apply a rotation on spin 2 about the yˆ-axis − | i | i | i − so that in the end spin 2 is ﬂipped only if spin 1 is in 1 . | i

y x X 1/2J Y

Figure 4.6: Visualization of the cnot12 operation of two spins using the Bloch-sphere 2 representation. Spin 2 is shown on the Bloch-sphere which rotates aboutz ˆ at ω0/2π. Spin 2 is left untouched if spin 1 is 0 (solid line), and ﬂipped if spin 1 is 1 (dashed line). | i | i 4.3. TWO-QUBIT QUANTUM GATES 91

This visual representation is a nice illustration of what happens when the spin is in a classical state ( 0 or 1 but not in a superposition). When the spin states are superposition | i | i states or even entangled, this picture breaks down. In fact, the given sequence of operations is not even correct for arbitrary input states. The unitary matrix implemented by the sequence above is:

1 0 0 0   0 i 0 0 U¯ = Y¯ U (1/2J )X = (4.21) cnot12 2 J 12 2    0 0 0 1     0 0 i 0  − with time, going from right to left. This unitary transformation is similar to Ucnot12 from

Eq. 2.45, but not quite the same. We can obtain Ucnot12 by adding phase shift rotations (ˆz-rotations): 1 0 0 0   0 1 0 0 U = Z Y¯ U (1/2J )X Z¯ = (4.22) cnot12 1 2 J 12 2 2    0 0 0 1     0 0 1 0 

4.3.3 Refocusing J-couplings

In this thesis, I invested a large amount of eﬀort into designing appropriate refocusing schemes for our systems. Following the outline of this section, the design of such a scheme is not diﬃcult. But when adding the complication that J-coupling is active during the application of long spin selective pulses, such schemes may no longer refocus the couplings between the spins. This section here only outlines the basics of refocusing schemes, but we revisit the more complex scenario in section 5.4.

Basic idea

The procedure for performing a cnotij operation using UJ (t) with multiple qubits requires us to selectively turn on only the Jij coupling, even though all of the natural J-couplings are always turned on. This can be done in NMR using refocusing schemes, which refocus the

undesired evolutions by applying 180◦ spin selective rotations interlaces with time evolution intervals. The basic idea is the following. Suppose we are given two coupled spins. When we apply 92 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

2 2 a time evolution operator sandwiched between X1 or X2 rotations, the resulting evolution behaves like a negative time evolution:

2 2 2 2 X UJ (t)X = UJ ( t) = X UJ (t)X (4.23) 1 1 − 2 2

Applying a time evolution, followed by a negative time evolution leads to no net evolution:

2 2 2 2 UJ (t)X1 UJ (t)X1 = I = UJ (t)X2 UJ (t)X2 (4.24)

Hence, the coupling between the two spins has been refocused.

00 01 01 00 01 00 y 10 11 x 11 10 11 10

00 00 01 y 01 10 10 x 11 11

Figure 4.7: Bloch-sphere representation of a refocusing scheme with n = 3 spins, with time going left to right, and the spins (1,2, and 3) are ordered from top to bottom. All spins start out in thex ˆyˆ-plane. The refocusing scheme begins with a time evolution UJ (t), followed 2 by X3 (a 180◦ rotation on spin 3 aboutx ˆ), and finally another time evolution UJ (t). The first row indicates what would happen to spin 1 when it points along yˆ and the remaining spins are along zˆ. The four possible states for the remaining two spins− 2 and 3 are also drawn for clarification.± Similarly, row 2 indicates what would happen to spin 2 if it was pointed along yˆ and the remaining spins along zˆ, and so forth. The couplings J13 and − ± J12 are refocused while the coupling J12 is turned on. Hence spin 3 ends up alongy ˆ. To flip 2 spin 3 back to yˆ and complete the refocusing scheme, we need to apply a final X3 pulse. Spin 1 ends up− along xˆ, depending only on the state of spin 2, but not 3. Similarly, spin 2 also ends up along ±xˆ depending only on the state of spin 1, but not 3. ± 4.3. TWO-QUBIT QUANTUM GATES 93

A 3-spin refocusing scheme

We can show how refocusing works using the illustration for three spins drawn in Fig. 4.7. 2 The sequence that is applied is UJ (t)X3 UJ (t). From the ﬁgure, we can see that the couplings

J13 and J23 are refocused, and hence spin 3 does not undergo any phase evolution (it is just 2 ﬂipped, which can be undone by completing the refocusing scheme with a ﬁnal X3 pulse).

Spins 1 and 2 both undergo a net evolution under the coupling J12.

General n-spin refocusing schemes

Refocusing schemes have been generalized to multiple qubits. The most straight forward 2 scheme [LBCF99] is to divide the evolution UJ (t) into two equal segments and apply X3 2 in the middle. Each segment is then again divided two equal parts with a X4 pulse in the middle, and so forth. This scheme however, requires a factor of two in the number of refocusing pulses for each added spin (assuming all spins are coupled with each other), and hence scales exponentially. It is thus not very practical for a large number of qubits. There are other constructions which scale polynomially and make use of simultaneous spin selective rotations [LCYY00, JK99].

+ + + + + + + + 1 + + + + + + + + 2 + − − + + − − + 3 + − − + − + + − 4 + + − − + + − − 5 + + − − − − + + 6 + + + + − − − − 7 Figure 4.8: Refocusing scheme for a 7 spin system in which all spins are coupled to each other. A + or indicates whether the spin points in its original direction or whether it is − ﬂipped. The black rectangles denote 180◦ instantaneous spin rotations, ﬂipping the spins (+ to or vice versa). All couplings except for J12 are refocused. A coupling acts “forward” in time− when both spins point in the same direction (++ or ), and “backwards” in time when both spins point in the opposite direction (+ or +).−− − − 94 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

Consider for example the refocusing scheme for 7 spins drawn in Fig. 4.8 using simultaneous pulses and a total of eight time evolution segments. The basic idea is that the total evolution consisting of positive and negative time evolution should sum to zero. When both spins have the same sign in the diagram, the coupling is “forward” (positive) and when it diﬀers, the coupling is “backward” (negative). Spins 1 and 2 are never pulsed and hence they undergo T2∗ dephasing while the remaining spins are only subject to T2. The inhomogeneous dephasing can be refocused by applying the scheme from Fig. 4.8 twice, and applying 180◦ refocusing pulses on spins 1 and 2 in-between the two segments and at the end. We use such a full refocusing scheme later in section 6.6.

Simplifying refocusing schemes

Refocusing schemes can oftentimes be significantly simplified. This is the case when we know, for example, that certain spins are along zˆ or when we no longer care about a ± certain subset of spins. This is a common situation in early or possibly late stages of a quantum computation. Fig 4.9 gives a simplified refocusing scheme which applies when we know that spins 3 through 7 are along zˆ. A phase evolution of a spin along zˆ is ± ± of no consequence. This fact was used during the initialization scheme in the experiment described in section 6.6.

+ + + + − − − − 1 + + + + − − − − 2 + + + + + + + + 3 + + + + + + + + 4 + + + + + + + + 5 + + + + + + + + 6 + + + + + + + + 7 Figure 4.9: Simpliﬁed refocusing for a 7 spin system, useful when spins 3 through 7 are along zˆ. The coupling J12 is active, and the couplings J1i and J2i are refocused. All remaining± couplings have no measurable eﬀect on the quantum state of the system. 4.4. INITIALIZATION 95

Pulse sequences

A general cnot gate acting on two out of n spins thus requires a refocusing scheme. Re- gardless of whether we can simplify such a refocusing scheme or not, it boils down to the repeated application of RF pulses and delay times. Because cnot gates can be reduced to single qubit rotations and controlledz ˆ-rotations, and because any quantum circuit can be reduced to single-qubit and cnot gates, every quantum circuit can be reduced to a repeated application of RF pulses and delay times. This is what is referred to as the pulse sequence. These elementary instructions are like the machine language of the NMR quantum computer!

Coupling network considerations

In general, cnot operations for systems consisting of multiple qubits can incur a significant overhead due to refocusing or simply due to the unavailability of direct couplings. This is illustrated in Fig. 3.1 (a) and (b). If two spins are not directly coupled, at most 2(n 2) swap operations must be imple- − mented. Each swap includes refocusing of the nearest neighbor couplings which requires a number of pulses that is linear in n. If all spins area coupled with each other (very unlikely or even impossible in larger systems), we have to apply (n) time evolution segments and (n) pulses per segment, O O resulting in a scheme that scales as (n2). From a computer science point of view, any O polynomial overhead is considered efficient. Yet from an experimental point of view, these overheads can be significant given the limited state-of-the-art in experimental quantum computing. It is hence important to minimize the number of refocusing pulses or swap operations.

4.4 Initialization

A substantial portion of this section is a review of existing techniques. However, my own contributions enter in the design for the temporal labeling initialization schemes of section 4.4.6, speciﬁcally the parts on using linearly independent permutations and product operators. 96 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

4.4.1 Thermal equilibrium

In section 2.3 we described several quantum algorithms and assumed the initial state to be in some known pure state. Let us now focus the discussion on what the most easily accessible state in NMR is: the thermal equilibrium state. Here, the spins are thermally equilibrating with an environment at some finite temperature. The density matrix is then given by: β 0 e− H ρth = (4.25) Z β where β = 1/kBT and is the normalization constant = T r(e 0 ). The typical field Z Z − H 5 strength in NMR is on the order of 10 Tesla, so that ~ω0/kBT 10 1 for a proton ≈ − with a field of 11.7 Tesla. The thermal equilibrium state can thus be approximated by using a first order Taylor series expansion. An n-spin system in thermal equilibrium can then be written as: n ρth 2− [1 β 0] (4.26) ≈ − H

We have not included the J-coupling between the spins in this approximation. Since ω0 is on the order of MHz while J is on the order of tens to hundreds of Hz this remains a valid approximation. A single spin in thermal equilibrium with the environment is then simply approximated by: 1 1 + 0 0 ρth (4.27) ≈ 2 0 1 0 − where 0 is the spin polarization

0 = ~ω0/2kBT (4.28)

The 2n states of n spins occur with almost equal probability. We do not know which state the n-spins are really in. This situation is very diﬀerent from what we desire - a perfectly well known state (for example, the 00 ... 0 state). We next describe several | i schemes that allow us to change the NMR thermal equilibrium state into an eﬀective pure state that we can use for quantum computation.

4.4.2 Schulman-Vazirani cooling

We have described this eﬃcient cooling scheme in section 3.1.3. The scheme transfers the polarization between the spins to increase the polarization of a subset of spins while 4.4. INITIALIZATION 97

decreasing it for the rest [SV99]. Several iterations of this procedure are likely required to obtain a subset of spins that are perfectly pure. We described that the number of pure qubits, k, is linear with the total number of partly mixed qubits n. Furthermore, the algorithm to obtain k pure qubits is quasi-linear, (nlogn). However, the overhead O required for this procedure is inversely proportional to the square of the thermal polarization 5 10 (Eq. 3.7). For current thermal polarization 0 10 , this scheme would thus require 10 ≈ − partly mixed spins which is unreasonably high for nuclear spins at room temperature with current techniques. Despite this drawback, this scheme allows one to make the following statement:

The highly random initial state of room temperature spins represents no fundamental obstacle to scalable quantum computation.

Hyperpolarization

To turn this technique into a practical one, we desire to increase the thermal polarization of nuclear spins. If we can increase the polarization to unity, we might not even need this cooling scheme at all. The most obvious way to increase polarization appears to be a reduction of temperature. Temperatures in the milli-Kelvin regime would give us very high polarizations. However, at those temperatures the sample would be frozen. Hence the molecules are no longer able to tumble around, re-introducing dipolar couplings that are averaged out in liquids. Intramolecular dipolar couplings complicate the spin dynamics while intermolecular dipolar couplings broaden the spectral lines. Several proposals by Yamamoto’s and Cory’s groups exist to address these complications [YY99, LGD+00, CLK+00] using solid-state NMR techniques, but they are often experimentally challenging. Another approach to increasing thermal polarization has been achieved using nuclear spins in liquids via SPINOE cross-relaxation techniques [VLV+01]. While this experiment is a remarkable step forward, the thermal polarization is still several orders of magnitude away from unity. In another experiment, two protons have been polarized to an estimated value of 0.1 using para hydrogen [HBG00]. These are hydrogen molecules where the two protons (1H) are in the singlet state. By reacting these para hydrogen molecules with some other precursor, we can build quantum computer molecules with very high polarizations (possibly up to 0.5). However, ﬁnding suitable molecules for “conventional” NMR quantum computing is 98 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

challenging enough, and the added requirement that the molecule must be easily formed from a precursor and H2 signifies a substantial hurdle. Other hyperpolarization techniques exist, including dynamic nuclear polarization (DNP) [Jef63] and chemically induced DNP (CIDNP). In DNP, we use free radicals and transfer electron spin polarization to the nuclear spins. Since the magnetic moment of the electron spin is 1800 times stronger than that of nuclear spins, the polarization of electrons is similarly higher. However, these techniques have not yet been demonstrated in connection with quantum computing. The state of the art of these hyperpolarization techniques are still far away from producing fully polarized, or significantly polarized spins useful for quantum computing. Even though continued work may significantly improve these techniques, we must look at other techniques if we wish to study quantum computing at room temperature today.

4.4.3 Eﬀective pure states

We showed in section 3.1.3 how effective pure states can be used for quantum computation. We mentioned that these are useful because a density matrix proportional to the identity matrix does not produce a measurable signal. The observable signal in NMR is proportional to the difference between two populations, instead of the populations themselves. Hence any part of a density matrix that is proportional to the identity matrix can be ignored during the measurement. Furthermore, the identity matrix does not evolve under unitary operations, i.e. U †IU = I. Consequently, for the purpose of quantum computing we only need to consider the deviation density operator ρdev - the component that deviates from the identity background: n ρ = I/2 + ρdev (4.29) where ρ is normalized of course (T r(ρ) = 1). Gershenfeld and Chuang [GC97] and independently Cory, Havel and Fahmy [CFH97, CPH98a] then realized that a density matrix of the form of Eq. 3.3 1 0 ρeff = − I + 0 ψ ψ (4.30) 2n | ih | can be used for quantum computation. The identity portion neither produces an output signal nor does it evolve under unitary transforms, and hence the output signal is proportional to the pure state ψ ψ . The density operator ρeff is thus also called an effective-pure | ih | 4.4. INITIALIZATION 99

state, or a pseudo-pure state. The goal is to create an effective-pure state for which all populations are equal (identity part) except for one which differs (deviation part), for example 00 ... 0 . | i The challenge is to turn the thermal equilibrium state ρth from Eq. 4.26 into an effective pure state ρeff . How is this done? Since the eigenvalues of a density operator ρ are conserved during unitary operations, and the eigenvalues of ρth and ρeff differ, we know that this procedure must contain non-unitary steps7. Three methods have been developed to create effective-pure states: Spatial labeling [CPH98a], logical labeling [GC97], and temporal labeling [KCL98]. These three methods are described in detail in the following subsections. Temporal and spatial averaging have been the most widely used techniques thus far. Several hybrid schemes exist that mix or combine several techniques into one. All three techniques as well as their hybrids suffer from one major drawback. They all incur an exponential cost either in the signal strength or the number of experiments involved. The fundamental reason for this is that they all select out the ground state population that is present at thermal equilibrium, and this population scales as n/2n. Obviously, such an overhead defeats the purpose of quantum computation, and are thus not useful for scalable systems. They are however useful to test quantum computations containing only a small number of qubits.

Note that from here on out we only work with the deviation density matrix unless stated otherwise.

4.4.4 Spatial labeling

Spatial averaging [CPH98a] averages the NMR signal spatially over a volume of the sample to equalize all populations except for the ground state population. This is done via magnetic field gradients which cause spins at different points in the sample to process at different frequencies, so that the phases are apparently randomized8. All non-diagonal entries of the density matrix are made zero in this fashion.

7We claimed that the Schulman-Vazirani cooling scheme shifts the entropy within a system - an inherently unitary process. However, the process of disregarding n k spins at the end (i.e. throwing them away) is non-unitary, and thus the Schulman-Vazirani scheme does− contain non-unitary steps. 8We say “apparently” here because we can unwind this dephasing by applying a magnetic field gradient in the opposite direction immediately afterwards, assuming the spins have not significantly diffused into different parts of the sample. 100 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

Spatial averaging can be best understood in the product operator formalism which uses the Pauli matrices. We stress that in the oeprator formalism, Z denotes the Pauli σz matrix, and not a 90◦ rotation. The thermal equilibrium state ρth can be written in the product operator form via: n ρ = ωi Zi (4.31) th X 0 i=0 where Zi is an n-fold Kronecker product of identity matrices except for the i-th location 2 where we use Z, a Pauli σz matrix. For example, Z of a three qubit system corresponds to I Z I. The desired eﬀective-pure state is the summation over all possible n-fold ⊗ ⊗ combinations of I and Z:

n n n ρ = Zi + ZiZj + ZiZjZk + ... + Z n (4.32) eff X X X ⊗ i=0 i

For example, the eﬀective pure state in which only the 00 ... 0 component diﬀers is given | i by ρeff = IIZ + IZI + ZII + ZZI + ZIZ + IZZ + ZZZ.

Spatial averaging transforms Eq. 4.31 into Eq. 4.32. For example, consider this procedure for two homonuclear spins [CPH98a]:

IZ + ZI R2 (60) 1 √3 x ZI + IZ IY (4.33) −→ 2 − 2 1 Gz ZI + IZ (4.34) −→ 2 R1 (45) √2 1 √2 x ZI + IZ YI (4.35) −→ 2 2 − 2 t(1/2J ) √2 1 √2 12 ZI + IZ + XZ (4.36) −→ 2 2 2 R1 ( 45) 1 1 1 1 1 y − ZI XI + IZ + XZ + ZZ (4.37) −→ 2 − 2 2 2 2 1 1 1 Gz ZI + IZ + ZZ (4.38) −→ 2 2 2

The term ZZ contains no net polarization, so that the total polarization is half as much as the thermal polarization. This scheme thus erases half of the thermal polarization. With 4.4. INITIALIZATION 101

each added spin, the polarization reduces by another factor of two, and hence the signal- to-noise ratio (SNR) scales as: S n (4.39) N ∝ 2n In general, the preparation sequence becomes unwieldy even though methods for the design of spatial averaging sequences exist [SOF00, SHC00]. Only one experiment needs to be performed but for a larger number of spins, signal averaging of the same experiment may be required. This technique has been successfully used by other groups but it was not used in my thesis work.

4.4.5 Logical labeling

Logical labeling permutes the populations of the thermal equilibrium state such that a subset of spins is in an eﬀective pure state [GC97, VYSC99]. The quantum computation is then performed on the subset of spins. We explain logical labeling using a three spin example. The diagonal of the thermal equilibrium density matrix for three homonuclear spins is given by:

ρth [3 1 1 1 1 1 1 3] (4.40) ∝ − − − − where this diagonal represents the populations of the state 000 , 001 , ..., 111 . Suppose | i | i | i we can swap the populations of the states 011 and 100 which can be done using a sequence | i | i of one and two-qubit operations. The density matrix changes into:

ρeff [3 1 1 1 1 1 1 3] (4.41) ∝ − − − −

Now, the subspace 000 , 001 , 010 , 011 is in an effective pure state. In other words, | i | i | i | i when spin 1 is 0 , the system is effective-pure. Performing logic operations within this | i subsystem allows two-qubit quantum computations (this may require decoupling spin 1 from the other spins). This procedure scales polynomially in terms of the number of effective pure spins k that can be obtained from n spins at thermal equilibrium, and in terms of the number of operations that are required. The number of equally populated states in thermal equilibrium 2 2 is equal to n!/[(n/2)!] so that we can obtain k = log2(1 + n!/[(n/2)!] ) effective pure spins. As n gets large, k/n tends to unity. The operations that are required to obtain k effective 102 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

pure qubits also scale polynomially with n. However, the signal strength decreases exponentially with the number of spins n. For large even n, the entries of ρth that are equal to each other are zero (for odd n, they are close n+1 to zero). The largest entry in ρth is n~ω0/2 2kbT which is also the maximum strength of

ρeq. Since only one experiment is performed, the noise is constant and independent of n. Hence the SNR scales as: S n (4.42) N ∝ 2n which is the same scaling behavior as for spatial labeling. Despite its elegance, this scheme has only been used in a few experiments [VYSC99] mostly because it means sacrificing some of the qubits. Since it is difficult to find suitable molecules with many spins, such scratch qubits are very “expensive”.

4.4.6 Temporal labeling

Temporal labeling consists of summing multiple experiments each of which underwent a diﬀerent permutation of the populations. Each experiment is designed such that the sum of the input states equals an eﬀective-pure state. Since quantum mechanics is linear, the sum of the resulting output states behaves as if the computation was done on a pure state. This becomes clear in the discussion that follows. Three variations of temporal labeling exist, two of which we have used in our experiments, and which I helped design: Linearly independent permutations, and the product operator approach.

Cyclic permutations

This ﬁrst method beautifully illustrates the idea of temporal labeling but is also complex to implement for multiple qubits. This scheme calls for a summation over 2n 1 experiments for − an n spin system. Each experiment performs a diﬀerent cyclic permutation. We illustrate this scheme using a two-qubit example. The thermal equilibrium state for two qubits is given by:

a   b ρ1 = ρth   (4.43) ∝  c     d  4.4. INITIALIZATION 103

We know what the populations are for a two-spin system, but we used arbitrary values a, b, c, and d to demonstrate that cyclic permutations also work for arbitrary population distributions. Let this state also be the ﬁrst of the temporal labeling experiments, ρ1. We now cyclically permute the last three populations without touching the ﬁrst. This is achieved by the sequence Uperm = Ucnot12 Ucnot21 on the thermal equilibrium state with time going from right to left. The resulting density matrix is given by:

a   d ρ2 = UpermρthUperm†   (4.44) ∝  b     c 

Similarly, if we perform the operation Uperm = Ucnot21 Ucnot12 on the thermal equilibrium state, we obtain: a   c ρ3 = UpermρthUperm†   (4.45) ∝  d     b 

Summing ρ1, ρ2, and ρ3 and letting e = b + c + d, we obtain

3a 3a e    −  e 0 ρeff = ρ1 + ρ2 + ρ3 =   = eI +   (4.46)  e   0       e   0  where I is the 4 x 4 identity matrix. This is an eﬀective pure state. What is remarkable about this scheme is that it does not matter what the actual populations are. Since the trace of the thermal deviation density matrix is zero, T r(ρth) = 0, we can never reach the situation where 3a e = 0. −

A quantum computation is now performed by creating ρ1, and performing the quantum computation on this state. We then create ρ2 and perform the same quantum computation, and then the same for ρ3. Finally, we sum all three experiments. This works because

Uρ1U † + Uρ2U † + Uρ3U † is the same as U(ρ1 + ρ2 + ρ3)U †, i.e the sum of the outputs is the same as performing the computation on the sum of the inputs. 104 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

But how does the SNR scale? The SNR of this cyclic permutation scheme scales exponentially. The ground state population from all 2n 1 experiments simply add up, and the − ground state population scales as n/2n for each experiment because we never touch it. The noise increase as the square root of the number of experiments √2n 1. hence the SNR − scales as S n 2n 1 n − = √l (4.47) N ∝ 2n √2n 1 2n − where l = 2n 1. This signal is the same as the one we would obtain if we had performed − l identical logical labeling experiments. Since the implementation of cyclic permutations becomes very complex for n > 2, we never used it in the experiments that we describe here. There are two variations of the cyclic permutation scheme that are more practical for larger n, and we discuss them next.

Linearly independent permutations

The method that we just described essentially averages out the difference between 2n 1 − populations, but this can be done in a different manner than cyclic permutations. For any n 2 1 linearly independent population distributions diag(ρi), we can find a set of weights − wi such that the weighted sum equals an effective pure state:

2n 1 − ρ = w ρ (4.48) eff X i i i=1

The number of experiments here is the same as for cyclic permutations, but each experiment has a signiﬁcantly easier sequence of operations. The main advantage is that we can sometimes approximate the eﬀective pure state in far fewer experiments than 2n 1. In − fact this was the case for our 3-qubit experiments from section 6.3 and 6.8 where I found a close approximation using only 3 experiments instead of 7. The largest disadvantage is that the SNR is suboptimal, compared with cyclic permutations. For l experiments, the SNR scales as:

l S n wi n Pi=1 √l (4.49) N 2n 2n ∝ l w2 ≤ qPi=1 i with equality only if all the weights are equal to 1. This is generally not the case, and in fact some of the weights could be negative, resulting in a poor SNR. Nonetheless, already for 4.4. INITIALIZATION 105

n = 3, the signiﬁcantly easier sequence of operations for the individual experiments as well as a fewer number of total operations outweighed the disadvantages. We use this scheme for two n = 3 experiments (section 6.3 and section 6.8) to create eﬀective pure states.

Product operator approach

Temporal labeling can be even further simplified by exploiting the structure of the thermal density matrix. This is done using the product operator formalism, already introduced in Eq. 4.31 and Eq. 4.32 for the thermal state and effective-pure state respectively. Assume we are given only homonuclear spins so that each term in the thermal density matrix has the same weight (see. Eq. 4.31). The idea then is to perform a series of cnot operations that transform the n terms in the thermal density matrix into a different set of n terms that are part of the effective-pure state. We then perform a different series of operations to create additional n new terms. Since 2n 1 terms are required, and we can − create n terms in each experiment, we need to perform (2n 1)/n to obtain all necessary d − e terms. We can cancel out additional terms by performing not operations. The following table illustrates how cnot and not operations affect the product operator terms:

II cnot12 II (4.50) −→ IZ cnot12 ZZ (4.51) −→ ZI cnot12 ZI (4.52) −→ ZZ cnot12 IZ (4.53) −→ I not I (4.54) −→ Z not Z (4.55) −→ −

As an example, let us explain a three qubit homonuclear example. The thermal density matrix is given by ρth = ZII + IZI + IIZ. The following three experiments create an eﬀective pure state ρeff = ZII + IZI + IIZ + ZZI + ZIZ + IZZ + ZZZ after summing over all terms:

cnot21 cnot32 ZZZ + IZZ + IIZ (4.56) 7→ cnot31 ZIZ + IZI + ZII (4.57) 7→ 106 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

cnot12 X3 ZII + ZZI IIZ (4.58) 7→ −

For heteronuclear spins each term in the thermal density matrix has a diﬀerent weight and hence we require 2n 1 experiments to obtain all terms in the eﬀective-pure state. Obviously, − this would be the same number of experiments as in the cyclic permutation scheme, but the experiments here would be much easier. If k of the n spins are homonuclear, we simply create k new terms in each experiment, requiring (2n 1)/k experiments. d − e This scheme can reduce the number of experiments by a factor of n, compared with cyclic permutations. The SNR is optimal because all the terms are added up in each experiment with equal weight: S n 2n 1/n n − = √l (4.59) N ∝ 2n 2n 1/n 2n p − In practice, it may sometimes be advantageous to use slightly more experiments than needed to simplify the preparation sequences, at a slight cost in SNR. We apply this scheme in section 6.5 for a 5-spin homonuclear system using 9 experiments (instead of the minimum number of 7), as well as in section 6.6 for a partly homonuclear 7-spin system. Deter- mining the optimal sequence of cnot operations even for 5 or 7 spins is non-trivial. I designed several versions of the 7-spin temporal labeling scheme, one of which we used in our experiments (see section 6.6).

4.5 Measurement

This section discusses the measurement procedure of nuclear spins in liquid solution. We begin by determining the physical properties of the measured signal: It is an induced oscillating voltage which we can Fourier transform. We then describe how the obtained spectra relate to the spin states 0 and 1 , followed by a discussion on the NMR signal-to- | i | i noise ratio and how it is aﬀected.

4.5.1 Interpretation of NMR spectra

The NMR signal

The signal of a single nuclear spin is too weak to be directly detected, and therefore NMR experiments are performed using a large ensemble of identical molecules, all dissolved in some liquid. The entire sample is subject to the same RF ﬁeld and hence each molecule 4.5. MEASUREMENT 107

undergoes the same operations9.

The same RF coil that produces the B1 ﬁeld (see section 6.1.3) is also used to detect the signal. The spins precess about thez ˆ-axis at their respective Larmor frequency. The oscillating transverse component of the magnetic moment induces a voltage in the RF coil. We can Fourier transform this signal to obtain a spectrum. The amplitude and phase of the spectral lines give us information about the respective spin states. Mathematically, we can write the induced voltage in the RF coil due to spin i as

i 0t/~ i 0t/~ i i Vi(t) = V0T r(e− H ρ(0)e H ( iσ σ )) (4.60) − x − y where V0 is the maximum signal strength (to be discussed shortly) and ρ(0) is the density matrix at the start of the measurement. We are free to choose the phase of the observable ( iσi σi ) as along as we are consistent. We assign a positive absorptive spectral line − x − y to correspond to the spin being along the yˆ-axis - this is the signal we obtain if apply a − Rx(90 ) rotation on a spin in the state 0 . Similarly, we choose a negative absorptive line to ◦ | i correspond to the spin being along +ˆy, and a positive and negative dispersive lines to a spin along +ˆx and xˆ respectively. Eq 4.60 is given in the lab frame, but we mix the signal with − i a reference oscillator at frequency ω to obtain expectation values of ( iσx σy). When 0 − − ρ(0) is mixed, as is the case here, the expectation is averaged over the statistical mixture of states. The signal observed is the excess of spins between the most populated states. Since a spin along the zˆ-axis of the Bloch sphere, corresponding to the computational ± basis states 0 , 1 , cannot induce a voltage in the RF coil, we have to change basis to obtain | i | i a measurement in the 0 , 1 basis. This is achieved using a Rx(90) read-out pulse. With | i | i the previously introduced conventions, a spin in the 0 state produces a positive absorptive | i signal following the read-out pulse, and a spin in the 1 state produces a negative absorptive | i signal.

Interpretation of the NMR measurement process

The measurement process has a few peculiarities which we address here. Speciﬁcally, we would like to address why we can measure both of the non-commutingx ˆ andy ˆ components of a quantum mechanical object (see Eq 4.60).

9 This requires extremely homogeneous magnetic B0 and B1 ﬁelds across the entire sample. In practice, all of the molecules undergo only approximately the same operation because of ﬁeld inhomogeneities. 108 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

The measurement process is a weak measurement (section 3.1.4). The RF coil is present at all times, but it is only weakly coupled to the spins, contributing only small amounts to decoherence. The other decoherence process (inhomogeneities in B0, interactions with other spins and the environment) are still active. The transverse component of the magnetic moment hence decays over time, usually exponentially. This decaying signal in the time it/T domain is called free induction decay or FID. The FID typically decays as V (t)e− 2∗ where

T2∗ includes the T2 decoherence processes as well as other systematic phase randomizations (section 2.4.2). The Fourier transform of this signal gives rise to a Lorentzian lineshape:

1 iω t/T2∗ V (t)e− (ω) 2 2 (4.61) 7→ F ∝ 1 + (ω ω0) − 1 + (ω ω0) − − where the two terms correspond to the absorptive and dispersive components which may or may not both be present at the same time. The linewidth at half height is given by

∆ω 1 ∆f = = (4.62) 2π 2πT2∗

Since the measurement is weak, only very little information about the individual spins leaks out of the system. However, because we are dealing with a large number of identical molecules, we can learn much more information about the average spin state than is possible with a measurement with a single spin. Due to the ensemble measurement we can simultaneously measure the expectation values of ( iσx σy) of each spin which is not − − possible using a single molecule. Hence, ensemble measurements can, in some sense, provide more information about the average spin state than a single projective measurement on a single spin. However, weak measurements also average certain outputs of quantum algorithms. It is possible though to modify these algorithms to circumvent this problem, as explained in section 3.1.4.

Interpretation of NMR spectra

Based on the previous discussion, we are now ready to describe NMR spectra and their interpretation with respect to quantum computation. As mentioned earlier, when we apply a read-out pulse on a single spin that is either in the 0 or 1 state, we obtain either | i | i a positive or negative absorptive line. This is the case for most quantum computations. However, other states such as superpositions give rise to diﬀerent signals. By designing a 4.5. MEASUREMENT 109

proper sequence of diﬀerent read-out pulses, we can reconstruct the full density operator of this single spin (see section 4.5.3). When we are given multiple coupled spins, the spectra display some ﬁne structure. As shown in section 4.3.1, the Larmor frequency of spin i shifts by Jij/2 when spin j is in − the 0 state, and by +Jij/2 when spin j is in the 1 state. Hence, the spectrum of two | i | i coupled spins, for example, displays two spectral lines for each spin, one corresponding to the neighboring spin being in the state 0 and another when it is in 1 . When we have | i | i three coupled spins and none of the couplings overlap, we obtain four spectral lines, each of which we can assign to the neighboring spins being in the states 00 , 01 , 10 , and 11 . | i | i | i | i

a 10 1010 1011 1110 1111 0010 0011 0110 0111 8 1000 1001 1100 1101 0000 0001 0100 0101 6 4 2 0 −50 0 50

Figure 4.10: Thermal equilibrium spectrum of a 5 spin molecule, used in one of our experiments (section 6.5). The frequencies are in units of Hz with respect to the Larmor frequency of that spin. The signal strength is in arbitrary units. Knowledge about the couplings (J12 < 0, J13, J14, J15 > 0 and J12 > J13 > J15 > J14) allows us to assign each line with a particular state of the remaining| | spins.| | | | |

In Fig. 4.10 we show the thermal equilibrium spectrum of a 5-spin molecule with non- 5 1 overlapping J-couplings such that we see 2 − = 16 resolved spectral lines for one of the spins. With knowledge about the J-couplings we can assign each line with a particular state of the remaining four spins. Since at thermal equilibrium the population difference between the 0 and 1 state is the same independent of the state of the remaining qubits, all 16 | i | i lines should be of equal strength. However, if we were in an effective-pure state of 00000 , | i only one line should be visible, namely the one we labeled as 0000 in the figure. We made heavy use of this fine structure of the spectra in many experiments described in chapter 6. If the spectra are not well-resolved, we can still obtain the answer of the quantum computation by integrating the full spectrum of each of the spins instead of each individual spectral line of a single spin. Since the answer of a quantum computation is generally in a 110 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

computational basis state (sometimes this may require a procedure similar to the continued fraction expansion; see section 3.1.4), we should see only one of the lines of a single spin if they are well-resolved. If they are not well-resolved, the location and amplitude of the line of a single spin does not permit us to obtain the answer with certainty. Instead, we simply look at the amplitude of the integrated spectrum (positive or negative corresponding to the state 0 or 1 of this spin). We then do the same for all remaining spins to obtain the answer. | i | i This is the most general procedure to obtain the answer of most quantum computations when we can measure each spin. In fact, we used this procedure in the experiment described in section 6.6.

4.5.2 Signal-to-noise ratio

We now brieﬂy discuss the signal-to-noise ratio (SNR) in NMR experiments. For a more detailed discussion, see [HR76, Hou96]. The SNR is simply the ratio of the signal over the noise.

signal •

The NMR voltage signal is proportional to the magnetization M0 in, which is proportional to the number of molecules (calculated by multiplying the volume V of the

sample and the concentration nc), times the number of equivalent nuclei Ne in each

molecule, times the polarization 0. The active sample area is calculated by taking the fraction of the coil volume occupied by the sample, η, and a factor K which depends on coil geometry, among others.

Because the signal is inductive, we also need to multiply by the Larmor frequency

(proportional to γB0). Since the signal decays exponentially with T2∗, the signal is hence also proportional to ∞ exp( t/T ) = T . A tall narrow line, gives a better R0 − 2∗ 2∗ signal than a short broad line. Hence the signal is proportional to:

2 2 S M0ηKγB0T ∗ = V ncNe0ηKγB0T ∗ = V ncNeγ B ηKT ∗/Ts (4.63) ∝ 2 2 0 2

where Ts is the temperature of the sample, and 0 was calculated by using Eq. 4.28.

noise • 4.5. MEASUREMENT 111

The noise is just the thermal Nyquist noise inside the coil, which is proportional to

the square root of (1) the temperature of the coil Tc, (2) the shunt resistance R of the coil , and (3) the band-width of the ﬁlter ∆f. Hence the noise is given by:

N TcR∆f (4.64) ∝ p

Typically, one thinks of the resistance being independent of the frequency. This is true for low frequencies, but for high frequencies, due to the skin depth effect, the current tends to flow only within a certain width of the conductor, which scales as 1/ω1/2. Hence, the resistance for large frequencies scales as ω1/2 as the current flows through smaller and smaller regions of the conductor. As it turns out, for typical NMR frequencies (hundreds of MHz), the skin effect must be included. Including the skin effect, the SNR can be summarized by

V n N γ7/4B7/4ηKT SNR c e 0 2∗ (4.65) ∝ Ts√Tc∆f

We would like to explicitly point out the 7/4 power dependence on the magnetic field B0. This is precisely the reason for the continued to effort to build spectrometers with larger magnetic fields, with spectrometers now commercially available at a proton frequency of 900 MHz.

4.5.3 Quantum state tomography

We brieﬂy described in section 4.5.1 that the full density operator can be reconstructed by performing several experiments each with appropriate unitary transforms just before the measurement. In this section we brieﬂy describe how this is done10. This procedure is called quantum state tomography [CGKL98]. The idea behind this is as follows. A straight measurement of a single spin without a read-out pulse gives us the observable ( iσx σy). This is a single quantum coherence (SQC) between the state 0 − − | i and 1 for this spin, i.e. we measure the element 0 1 of the density matrix. For multiple | i | ih | qubits the measurement of spin i gives us the SQC elements of spin i. For example, the

10We describe in section 6.7 how to reconstruct the density matrices for higher order spin systems. 112 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

SQC elements of a two-spin density matrix are

 · · × ·   · × · ·   · · · ×  and  × · · ·  (4.66)      × · · ·   · · · ×      · × · · · · × · for spin 1 and 2 respectively. Since the density matrix is Hermitian, we can reconstruct both the entries above and below the diagonal. Therefore, Eq. 4.66 only has two independent elements for each spin, corresponding to the area underneath the two spectral lines in each n 1 spectrum of the two spins. For n-spins systems, there are 2 − SQC elements, corresponding n 1 to the 2 − lines of each spin. To reconstruct the other coherences of the density matrix, we repeat the experiment several times, but with a different read-out procedure which rotate other coherences of the density matrix into observable positions. In an ideal experiment, each of the different experiments are compatible with each other, but in a real experiment this may not be so because of fluctuations in the homogeneity of the magnetic field over time or other changes in the experimental parameters. In this case we can still obtain a good estimate of the density matrix via a least squares fit. Quantum state tomography in general requires on the order of 4n experiments which is only practical for a small number of qubits. We have reconstructed the density matrix for the three qubit experiments described in section 6.3 and 6.8.

4.6 Decoherence

In this section discuss some physical mechanisms of decoherence in uncoupled spin systems. Decoherence in such systems is well described by a combination of longitudinal and transverse relaxation [Abr61, Sli96, Lev01] which are closely related to generalized amplitude damping and phase damping from section 2.4. In coupled spin systems, there are generally additional mechanisms for decoherence such as cross-relaxation and the Nuclear Overhauser Eﬀect (NOE), but we do not consider them here since they were not a large factor in our experiments. We begin by describing the main decoherence mechanisms, followed by a description on how minimize them, and ﬁnally we show some methods how to measure the characteristic time constants. 4.6. DECOHERENCE 113

4.6.1 Causes and eﬀects of decoherence

Relaxation of nuclear spins is caused by fluctuations in the magnetic field experienced by the spins. These can be microscopic or macroscopic fluctuations and contribute to energy exchange (T1) or phase randomization (T2). Typically, the time scale of these fluctuations determines which relaxation process is being activated. Low-frequency fluctuations typi-

cally give rise to phase randomizations while high frequency ﬂuctuations (at ω0, or higher harmonics, or even sums or diﬀerences between the Larmor frequencies of coupled spins i j ω ωo) give rise to energy exchange - this is just a rule of thumb however and should not 0 ± be regarded to be true all the time. We begin by listing the microscopic relaxation mechanisms [Lev01]:

intermolecular dipole-dipole interactions with nuclear spins. •

Intermolecular dipole-dipole couplings contribute to phase randomization (T2) and are modulated by molecular translation and rotation. In isotropic liquids short-range intermolecular interactions average to zero but long-range interactions are not averaged out due to diﬀusional motion.

intramolecular dipole-dipole interactions with nuclear spins. • Intramolecular dipole-dipole couplings ﬂuctuate due to molecular tumbling. It con-

tributes to T1 and depends on the distance between the two nuclei. In isotropic liquids, these are averaged to their isotropic values which may not be zero but are small.

inter- and intramolecular dipole-dipole interactions with electron spins. • If unpaired electrons are present (as in paramagnetic ions and free radicals), then this is a dominant interaction because the electron spin has a much higher magnetic moment than the nuclei.

chemical shift anisotropy (CSA). • The chemical shift can be anisotropic, and hence it fluctuates with molecular tumbling. This effect increases with the magnetic field strength because chemical shifts depend on the field strength.

spin-rotation • 114 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

A molecule consists of positive and negative charges. If the molecule rotates, the charges also rotate which corresponds to an electric current, generating a magnetic ﬁled. This induced magnetic ﬁeld interacts with the magnetic moments of the nuclei.

J-coupling anisotropy • Similar to CSA, the J-coupling can be anisotropic as well. It is often ignored and is very diﬃcult to distinguish from dipole-dipole coupling.

quadrupolar coupling • Nuclei with a spin greater than 1/2 do not have a spherically symmetric charge distribution. The nuclei interact with electric field gradients. Hence quadrupolar nuclei relax very fast because of molecular tumbling which lead to fluctuations in the electric field gradient.

This rapidly ﬂuctuating quadrupolar nuclei contributes to the relaxation of spin states of other nuclei.

chemical exchange • The molecule may rapidly undergo changes in its chemical structure either through internal or external reactions such that the chemical shifts of certain nuclei ﬂuctuate between two or more values.

In addition to these microscopic ﬂuctuations we could have macroscopic ﬂuctuations:

electromagnetic fluctuations • Noisy RF amplifiers or other equipment which emit electro-magnetic fields at ω0

shorten T1.

B0 and B1 fluctuations • The magnetic fields B0 and B1 can fluctuate and shorten the apparent T2. Such inhomogeneous broadening can in principle be reversed when diffusion rates are low.

Typically, one distinguishes between T2, the intrinsic transverse relaxation, and T2∗, which includes both the intrinsic relaxation and inhomogeneous broadening.

radiation damping • 4.6. DECOHERENCE 115

The combined magnetic moment of all spins can be large enough to induce a voltage in the RF coil which in turn generates a magnetic ﬁeld that rotates the spins. This is especially the case for very concentrated samples.

Reducing decoherence eﬀects

A significant portion of the relaxation mechanisms can be partially controlled by the experi- menter. It is crucial to control relaxation as much as possible even for small demonstrations of quantum computation. We can influence some microscopic relaxation processes by controlling molecular tumbling rates, by controlling the macroscopic relaxation mechanisms, and by carefully prepar- ing the NMR sample. Molecular tumbling can be influenced by controlling the following parameters:

solvent viscosity • Lowering the solvent viscosity leads to higher tumbling rates of the molecules. Su- percritical solvents are ideal because they combine the high solubility and density of liquids but with the low viscosity of gases. Sample preparation of supercritical solvents however, is not as straight forward. Furthermore, low viscosities also increase the diﬀusion rates.

temperature • Higher temperatures provide higher thermal energy for tumbling and generally also decrease viscosity.

molecule size • Small molecules typically have faster tumbling rates.

External relaxation mechanisms can be reduced by the following:

homogeneity in B0 • Invest a lot of time trying to make the static ﬁeld as homogeneous as possible through shimming (see section 6.1.2). This can also be improved by spinning the sample about

thez ˆ-axis, which averages away inhomogeneities of the B0 ﬁeld in thex ˆyˆ-plane. 116 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

homogeneity in B1 •

Design RF coils with very homogeneous B1 fields. Current RF coils are not the optimal design for producing homogeneous fields, because typical NMR experiments require a flexible procedure for changing samples. If we can loosen this requirement, better coils can be designed.

ampliﬁer blanking • Blanking of the ampliﬁers between RF pulses to reduce noise.

radiation damping • Radiation damping can be reduced by reducing the Q of the coil or the concentration of the sample.

We can also reduce relaxation eﬀects by following certain guidelines during the sample preparation and molecule selection:

Remove paramagnetic impurities such as oxygen from the sample • Avoid quadrupolar nuclei • Reduce the solute concentration to help lower intermolecular relaxation. • Also choose solvents with non-magnetic nuclei.

Avoid solvents which may lead to chemical exchange •

4.6.2 Measurement of decoherence times

We have used standard techniques to determine the T1 and T2 (and T2∗) relaxation times [Fre97].

1. T1 measurement The inversion recovery method is the standard procedure for a

clean measurement of the T1 spin-lattice relaxation time. When thez ˆ-magnetization is brought out of equilibrium, it relaxes via the diﬀerential equation:

dMz M0 Mz = − (4.67) dt T1 4.6. DECOHERENCE 117

where Mz is the time-dependentz ˆ-magnetization, and M0 is the equilibrium magnetization. This equation can be solved:

M0 Mz(t) = [M0 Mz(0)]exp( t/T1) (4.68) − − −

where Mz(0) is the initial value of thez ˆ-magnetization. We begin by inverting the

equilibriumz ˆ-magnetization (Mz(0) = M0), and then observe the decay to the − equilibrium value, given by

t/T1 Mz(t) = M0(1 2e− ) (4.69) −

This is achieved by applying the pulse sequence (note that time goes from left to right): X2 t X acquisition (4.70) − − − We first apply a 180 rotation to flip the spin from +ˆz to zˆ. Then the spin is ◦ − allowed to relax towards +ˆz for some time t. Finally, we apply a read-out pulse and measure the signal (the NMR lingo for measurement is acquisition). We apply this procedure for several values of t and fit the resulting signal (after phasing it to be

absorptive) to the form of Eq. 4.69 where we let T1 be the parameter. Sometimes, an additional ﬁtting parameter α is used instead of the value 2 in Eq. 4.69 because of RF inhomogeneities lead to spin magnetizations that are not completely inverted. This approach was not used in the experiments of this thesis.

t/T ∗ 2. T2∗ measurement Thex ˆyˆ-magnetization of the spin decays exponentially, e− 2 , as

mentioned in section 4.5.1. From Eq. 4.62, we can determine the the T2∗ by simply measuring the linewidth at half height. This is assuming that the line is indeed Lorentzian, which is not necessarily the case in inhomogeneous ﬁelds.

3. T2 measurement The measurement of T2 requires a method to refocus dephasing

due to magnetic ﬁeld inhomogeneities. This involves using a series of 180◦ refocusing pulses interlaced with delay periods, called a Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence. First, we apply a Y -pulse, which tips the spin to thex ˆ-axis. Then we apply the following subsequence k times:

(τ/4)X2(τ/2)X2(τ/4) (4.71) 118 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

We array k, and record the signal for each k. Typically, τ is on the order of 1-10 ms, short enough so that minimal diﬀusion takes place during the delay times. The refocusing pulses remove dephasing due to magnetic ﬁeld inhomogeneities. The signal decays exponentially with time t = kτ:

t/T2 M(t) = M0e− (4.72)

As we mentioned in section 2.4.3, T1 impacts the oﬀ-diagonal values of the density

matrix such that the signal decay that we observe includes both, T1 and T2. Often

T1 T2 so that the CPMG method gives good estimates of T2. Furthermore, B1 inhomogeneities lead to a measured T2 that includes these inhomogeneities. In principle, these eﬀects can be improved but are not done during the CPMG sequence.

Finally, the measurement of T2 is complicated for coupled spins, in particular when τ = 1/2J. In multiply coupled spin systems, it becomes difficult to choose an appropriate value of τ that is different enough from 1/2J. Different choices of τ can lead to

diﬀerent measured values for T2, reducing the meaning of the results. In addition, for homonuclear spin systems, the X2 pulses can be rather long. We have generally performed a spin echo experiment for our larger spin systems (with 5 and 7 qubits). This procedure consists of applying an X pulse on the spin, waiting a period τ/2,

applying a refocusing 180◦ pulse, waiting τ/2 and measuring the signal. We then take the absolute value of the peak integral and plot it as a function of time. The drawback of this method is that diﬀusion plays a larger role for larger times τ and

ﬁeld inhomogenities are no longer averaged out. We believe the measured T2 values

are nontheless close to the real T2 values because we can simulate and predict the observed spin dynamics rather well using our measured values (see section 6.6).

4.7 Higher-order spins

This entire chapter thus far has been dedicated to the description of using spin-1/2 nuclei of a molecule as qubits. We have already mentioned in chapter 3 that this is not a unique proposal and quickly touched on the idea of cellular automata. We also hinted at using higher-order spins. We now brieﬂy explain how these higher-order spins systems are useful 4.7. HIGHER-ORDER SPINS 119

for quantum computation.

4.7.1 System of qubits

The Hamiltonian for a quadrupolar system [Abr83, Sli96] can be written as:

2 e qQ 2 0 = ~ω0Iz (3I I(I + 1)) (4.73) H − 4I(2I 1) z − − where eq is the electric field gradient, eQ the quadrupole moment, Iz is thez ˆ angular moment operator, and I is the spin of the nucleus (f.ex. I = 3/2). The energy eigenvalues are given by: 2 e qQ 2 Em = ~ω0m + [3m I(I + 1)] (4.74) − 4I(2I 1) − − where m = 1 , 3 ,..., I when I is half-integer. When I is an integer m = 1, 2,..., I. ± 2 ± 2 ± ± ± ± Let us first consider the case when Q = 0. We have 2I + 1 energy levels, and the only allowed transitions correspond to ∆m = 1. Hence all of the transition energies are the ± same, resulting in only one spectral line at ω0. However, when Q = 0, the degeneracy is 6 lifted and the spectrum splits into 2I spectral lines, all split by the same amount, related to the quadrupolar coupling11. As an example, a spin-3/2 nucleus subject to a static magnetic field with nonzero quadrupolar coupling has four energy levels, all split by the same amount. Fig. 4.11a shows the schematic energy level diagram of a spin-3/2 nucleus and Fig. 4.11b plots the thermal equilibrium spectrum of a 133Cs nucleus in a liquid crystal (liquid crystals introduce a non-zero quadrupole coupling). We can relabel the four energy levels as 00 , 01 , 10 , and 11 and use them as a | i | i | i | i two-qubit system, similar to [KF00]. Analogous arguments can be made for other higher order systems.

4.7.2 Single and multiple-qubit gates

We here only focus on how to construct a set of universal gates for a spin-3/2 system. The tools and methods developed are general and can be transferred to other higher-order spin systems. In earlier sections we stated that arbitrary unitary gates can be constructed

11The levels are equally split only to first order. When including second order quadrupolar effects, the degeneracy in the splitting energy is lifted, but this effect is tiny. 120 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

8 x 10 11 ω 3 23

ω34 2 10 1 ω23 ω ω signal [arb] 12 34 01 0 ω12 00 −10 −5 0 5 10 frequency from Cs133 [kHz] (a) (b)

Figure 4.11: (a) Schematic energy level diagram for an I = 3/2 system with quadrupolar splitting. The energy levels correspond to the spin states I = 3/2, m = 3/2 , I = 3/2, m = 1/2 , I = 3/2, m = 1/2 , and I = 3/2, m = 3/2 , and| can be assignedi the| logical labels i00|, 01 , 10 , and− 11i . (b) Thermal| spectrum− ofi 133Cs displaying three of the allowed seven| i transitions,| i | i forming| i a spin-3/2 subsystem.

just from arbitrary single qubit rotations and the cnot gate. The cnot gate can be constructed using a controlled Z-rotation and hadamard gates. Hence, if we can show how to implement arbitrary single qubit gates and the controlled Z-gate, we can construct any unitary operation.

We begin this description by looking at the Hamiltonians that correspond to the allowed transitions in our spin-3/2 system. Similar to the spin-1/2 case, the Hamiltonians associated with the three transition frequencies ω12, ω23, and ω34 are:

H12 = 00 01 + 01 00 and H12 = i 00 01 i 01 00 , (4.75) x | ih | | ih | y | ih | − | ih | H23 = 01 10 + 10 01 and H23 = i 01 10 i 10 01 , (4.76) x | ih | | ih | y | ih | − | ih | H34 = 10 11 + 11 10 and H34 = i 10 11 i 11 10 (4.77) x | ih | | ih | y | ih | − | ih |

We can then calculate the unitary transforms generated by these Hamiltonians. By carefully concatenating the appropriate transforms, we can generate arbitrary single qubit rotations and the controlled Z-gate. The sequence of operations are summarized in Fig. 4.12. 4.7. HIGHER-ORDER SPINS 121

Two qubit spin−3/2 Two qubit spin−1/2 transition circuit model circuit model Two qubit spin−1/2 circuit model X α Zα X α Two qubit spin−3/2 Xα transition circuit model

Z X α/2 Xα α Zα Y2 Y2 Z3α/2 Xα

Figure 4.12: Illustration how to implement a universal set of quantum gates in an I = 3/2 system. Time goes from left to right. The ﬁrst and third column show qubit operations with the horizontal lines denoting the qubits, and the subscript denoting the rotation angle in units of π/2. Columns two and four show the corresponding transition selective operations for the I = 3/2 system, where the ﬁrst line denotes the transition at frequency ω12, and so forth. Single qubit Y -rotations are implemented similar to the shown X-rotations. From these, single qubit Z-rotations can be implemented. Together with the controlled Zα-gate, arbitrary 4x4 unitary gates can be implemented.

Similar to the description of section 4.2.2, we can implicitly absorb the transition selectivez ˆ-rotations into the phase of future pulses, and hence they require very little experimental eﬀort. Since the controlled Z-gate requires only transition selectivez ˆ-rotations, it too requires eﬀectively no experimental time. This is in stark contrast to spin-1/2 systems, where a cnot gate requires 1/2J seconds. This, comes at the cost of increasing the experimental time for single qubit rotations. The fact that certain gates require far less time could be advantageous for implementing certain algorithms faster compared with spin-1/2 systems [Fun01].

4.7.3 Initialization

The thermal equilibrium state is similar to the spin-1/2 case. There exists a large background identity population distribution, and some small deviation part. The deviation part 122 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

can be derived by using the fact that the population diﬀerence between neighboring energy levels is constant [KF00] (more on the thermal equilibrium state in section 6.7). This state can be turned into an eﬀective-pure state via the described pure state creation methods.

4.7.4 Measurement

The observable for higher-order spin systems is very similar to the spin-1/2 case. Instead of observing ( iσi σi ) as in Eq. 4.60, we now observe ( iHi Hi ) so that for I = 3/2 − x − y − x − y we have three observables:

12 12 O1 T r(ρ( iH H ) (4.78) ∝ − x − y 23 23 O2 T r(ρ( iH H ) (4.79) ∝ − x − y 34 34 O3 T r(ρ( iH H ) (4.80) ∝ − x − y where ρ is the density operator before measurement. In order to do quantum state tomography, we simply apply a series of unitary operations that allow us to observe diﬀerent terms of the density matrix. We illustrate this using a spin-3/2 system in section 6.7.

4.7.5 Coherence times

We have already described in section 4.6.1 that quadrupolar couplings reduce coherence times. Since we are using quadrupolar nuclei, we expect the coherence times to be much shorter than for carefully prepared molecules that only contain spin-1/2 nuclei. The typical coherence times are tens to hundreds of milliseconds, compared with several seconds for spin- 1/2 nuclei. Nonetheless, we can still use higher-order spins for small quantum computations, and we do so for a two-qubit Grover algorithm using a spin-3/2 subsystem of a higher-order spin in section 6.7. On a ﬁnal note about using higher-order spins as qubits: these systems are only useful when several of them are coupled together. A single spin-m/2 nucleus does not scale for arbitrary quantum computations, as already described in section 3.1.1. It is furthermore not clear whether coupled higher-order spins easily lend themselves to building bigger quantum computers than is possible with spin-1/2 nuclei. 4.8. MOLECULE DESIGN 123

4.8 Molecule design

The molecule choice is crucial for the successful implementation of most quantum computing experiments. The requirements for a good molecule follow from the five criteria we described in this chapter. We list a set of criteria directly related to using a spin-1/2 nuclei, but other alternatives can be considered based on higher-order spins or cellular automata, for example. First, we require a system of qubits. The most straightforward choice is to choose a number of spin-1/2 nuclei in a molecule that is equal to or larger than the number of qubits required. Good choices for qubits include 1H, 13C, 15N, 19F, and 31P. However, isotopic labeling is required to obtain sufficiently high concentrations of 13C, and 15N. We then require the frequency separation between each pair of qubits to be larger than their coupling strength, i.e. i j ω ω Jij (4.81) | 0 − 0| | | This choice ensures that the spectra are first order. This condition is automatically satisfied for heteronuclear molecules. In homonuclear systems we can achieve sufficiently large chemical shifts ranging from about 5 kHz to 100 kHz (at a field strength of 10 Tesla), depending on the nuclei. High γ nuclei such as 1H, 19F have a stronger signal (recall Eq. 4.65), but have shorter coherence times, compared with low γ nuclei (15N and 13C). Furthermore, we require a sufficiently strong coupling path between any pair of qubits such that the average time of a two-qubit gate is much shorter than the coherence times 19 T1 and T2 (we mentioned several aspects concerning coherence times in section 4.6.1). F and 13C nuclei tend to have stronger J-couplings. There are, of course, other proposals that relax some of the requirements. One example is the cellular automata idea described in section 3.1.2. The only unsolved problem here with respect to NMRQC is how to initialize the system to an effective-pure state at room temperature. Another example is to use higher-order spins. We explain and demonstrate experiments using higher-order spins later in sections 6.7 and 7.5.2. Finally, there are several other requirements related to “common sense”. We need the molecules to dissolve in some NMR solvent (chloroform, acetone, ether, D20, toluene, among others), and be stable for a reasonably long time. They must also be “easily” synthesized, affordable (isotopic labeling can be very costly) and non-toxic. There exist plenty of molecules that appear to have spectacular properties, but are unstable, toxic, or are very hard to synthesize. 124 CHAPTER 4. LIQUID-STATE NMR QUANTUM COMPUTING

4.9 The ”quantumness” of NMRQC

The idea of NMR quantum computing has raised several interesting and stimulating theoretical discussions about the “quantumness” of NMR quantum computing. In 1999, Schack and Caves showed that no NMR quantum computing experiment performed to date exhibited entanglement [SC99]. The reason for this is the low polarization at room temperature. Only for about 13 qubits there might be entanglement for typical room temperature polarizations [SC99]. However, it was believed that entanglement was a necessary requirement for the speed-up of quantum computers. What then does this mean for NMR quantum computing? If there is no entanglement present, then are NMR quantum computers still really quantum computers or just fancy classical computers? This idea led to an intense investigation of which resources actually speed-up quantum algorithms. The answer to this question is not fully known yet, but the current consensus is the following:

1. Entanglement appears to be a necessary resource.

This is a generally held belief, but it is not proven to be true at this point. For example, there exist quantum algorithms which have a speed-up over classical algorithms, but the quantum state during the computation is never entangled. It is believed that for algorithms like Shor’s algorithm however, entanglement is necessary. Whether or not entanglement is a necessary requirement for computational speed-ups or for which problems entanglement is necessary is still under intense debate. Interesting work to this extent has been done, for example, by Schack [SC99], Lloyd [Llo99], Meyer [Mey00], Laﬂamme [LCNV02], and Ekert [EJ98].

2. Entanglement is not suﬃcient.

Entanglement appears necessary but not suﬃcient. The Gottesman-Knill theorem [NC00] states that a quantum computation can be eﬃciently simulated on a classical computer, if it involves only the following elements: state preparations in the

computational basis, hadamard gates, phase gates (i.e. a 90◦ zˆ-rotation), cnot gates, Pauli gates, and measurements of observables in the Pauli group, together with the possibility of classical control conditioned on the outcome of such measurements. Since entanglement can be generated using these elements, and since this can also be eﬃciently simulated classically, it appears that entanglement is in itself not enough to obtain computational speed-ups, but rather the suﬃcient condition appears to be 4.10. SUMMARY 125

how entanglement is used.

To summarize, the power of a quantum computer appears to “take its origin in the properties of the evolution as well as the states it uses” [LCNV02].

4.10 Summary

The major point to take away from this chapter is that nuclear spins in liquid solution largely fulﬁll the ﬁve criteria to build a quantum computer. The discussion focused on spin- 1/2 nuclei but as we have seen in section 4.7, higher-order spins can also serve as useful qubits.

1. Coupled spin-1/2 nuclei in a molecule are well suited for qubits.

2. Single and multiple-qubit operations are implemented via RF pulses and delay times even when the coupling between the spins can not be explicitly controlled.

3. The system can be initialized to an eﬀective-pure state, currently only by incurring an exponential cost in either SNR or time (but this is not a fundamental problem).

4. Each qubit can be read-out spectroscopically by using a large ensemble of identical molecules.

5. Nuclear spins have long coherence times compared with the average gate duration.

Though large portions of this chapter have been a review, my own contributions entered mostly in the design for temporal labeling sequences, either through linearly independent permutations, or through the product operator formalism. I also spent considerable effort in designing appropriate refocusing schemes (f.ex. the scheme from Fig. 4.8), but more details are to follow in section 5.4 of the following chapter. This chapter only described the ideal methods with which an NMR quantum computer can be built. However, in real life we are often faced with artifacts and undesired evolution of the spins which can lead to significant errors if not corrected. The next chapter details solutions to several such undesired effects. Chapter 5

Tools of liquid-state NMR quantum computing

The previous chapter discussed the ideal behavior of nuclear spins in liquid solution. There are however several artifacts that arise when using shaped pulses that we have not yet described. These can lead to spin dynamics which significantly deviate from what was intended. In this chapter, we detail these artifacts and show how to correct them. The chapter begins with a discussion of how to excite spins off-resonance (section 5.1), followed by a description of undesired effects during single-spin rotations (section 5.2) and simultaneous-spin rotations (section 5.3) and how to remove them. We then describe a method to unwind J-coupled evolution active during long shaped pulses, and extend this method to several refocusing schemes (section 5.4). This is followed by a summary of our pulse sequence simplification techniques to minimize the number of applied pulses for arbitrary quantum circuits (section 5.5). This chapter contains several of my key contributions. Several of the described techniques and methods here are new to the field of NMR and have begun to find applications in traditional NMR experiments.

5.1 Excitation of spins oﬀ-resonance

In section 4.2.1 we described the dynamics of a spin that is excited on- and oﬀ-resonance.

Suppose the Larmor frequency ω0 of the spin diﬀers from the RF carrier frequency ωrf by ∆ω so that the RF pulse is oﬀ-resonance. How do we still excite the spin as if we were

126 5.1. EXCITATION OF SPINS OFF-RESONANCE 127

on-resonance? This is made possible by linearly incrementing the phase φ of the RF pulse during the application of the pulse at a rate dφ/dt = ∆ω. The frequency of the output signal is ∆ω

higher than ωrf . Mathematically, the RF Hamiltonian in Eq. 4.4 is replaced by:

dφ dφ rf (t) = ~ω1 cos ωrf t + φ0 + t σx/2 + sin ωrf t + φ0 + t σy/2 H − dt dt = ~ω1 cos [(ωrf + ∆ω) t + φ0] σx/2 + sin [(ωrf + ∆ω) t + φ0] σy/2 (5.1) − { }

Experimentally, the phase increments are approximated by discrete steps, similar to shaped- pulses (section 4.2.3). From slice to slice we increment the phase by an amount ∆φ. If the phase jumps are relatively small (a few degrees to about 10 degrees), then the discrete approximation is close to the continuous phase ramp. This technique is known as phase- ramping [Pat91]. This technique can be extended to excitations at multiple frequencies. The amplitude and phase in each slice during single oﬀ-resonance excitation pulses describe a vector. Given several pulses exciting at multiple frequencies, it suﬃces to add the vectors of the original pulses within each slice in order to merge the multiple pulses into a single pulse. For

example, if we were to excite at ωrf + ∆ω using an RF pulse with constant amplitude,

we apply a linear phase ramp as described above. If we were to excite at ωrf + ∆ω and

ωrf ∆ω using a single RF pulse, we would take the vector sum of the individual pulses. − The result would be a single pulse with constant phase but the amplitude would vary as cos(∆ωt). If we wanted to excite at other frequencies, we would have to adjust both the phase and amplitude from slice to slice. This method can be used to excite multiple spins using only one RF channel (see section 6.1.4 for details about RF channels). Suppose the RF channel of the spectrometer is only capable of synthesizing one fixed frequency (the rotating frame frequency) which is specified by the user but we wish to address multiple qubits at different frequencies. We can then apply the phase-ramping technique to excite at arbitrary frequencies, provided that the phase jumps between the slices is not too significant. Despite this rather simple solution, there are some peculiar aspects to this method which have to be taken into account. Since we only have one rotating frame, all of the spins that are addressed by this channel undergoz ˆ-rotations because they are off-resonance with respect to the rotating frame. Hence, any time evolution, intentional or not, results in az ˆ-rotation 128 CHAPTER 5. TOOLS OF NMRQC

which we must compensate. This can be done easily by absorbing the accumulated phase into the phase of future pulses, as described in section 4.2.2. It now becomes clear that this method requires a precise time counter which keeps track of all timing events of the spectrometer, including explicit time delays as well as implicit time delays such as internal power settling times or amplifier gating times. We use this technique to excite single and multiple spins off-resonance in many of our later experiments. This technique is also very useful to compensate for several artifacts arising from the application of simultaneous selective pulses. But to understand this effect, we need to first describe the dynamics of a spin subject to an off-resonance shaped pulse.

5.2 Transient Bloch-Siegert shifts

The RF radiation of a pulse causes a frequency shift ωBS in the Larmor frequency of spins well outside the excitation window of the pulse [EB90]. This effect is known as a transient Bloch-Siegert shift. Even though the original Bloch-Siegert shift [BS40] refers to the frequency shift induced by the counter-rotating RF field, the term has been generalized by the NMR community to refer to any frequency shift due to off-resonance radiation. This frequency shift can also be regarded as an instance of Berry’s phase [Ber84]. The 2 magnitude of ωBS is approximately ω /2(ωrf ω0) where ω1 is the RF field strength, ωrf 1 − is the frequency of the applied RF field, and ω0 is the Larmor frequency of the spin in the absence of RF fields. The frequency shift can easily reach several hundred Hertz as we show later, and always shifts away from the frequency of the RF field. This effect is only important for homonuclear spin systems. The RF pulse applied to one spin shifts the frequency of all other homonuclear spins during the application of the pulse itself. The net result is that these spins acquire a phase, which is to say they have undergone az ˆ-rotation. Since thesez ˆ-rotations are unintentional and often large, in most cases they must be compensated.

5.2.1 Calculation of transient Bloch-Siegert shifts

The calculation of the phase shift due to arbitrary pulses is similar to calculating the excitation proﬁles of selective pulses (section 4.2.3). Even though all required equations have already been shown in the previous chapter, we include them again here for completeness sake. 5.2. TRANSIENT BLOCH-SIEGERT SHIFTS 129

Shaped pulses are implemented as a sequence of time slices each consisting of a Hamil- tonian of a constant phase and amplitude. The unitary operator describing the evolution of spin a during a single slice k due to a shaped pulse at frequency ωrf is given by:

a ~ a i rf,k∆tk/ Uk = e− H (5.2)

a where tk is the length of the slice, and is the Hamiltonian in the rotating frame of Hrf,k spin a: a b b b = ~ω [cos(φ )σx/2 + sin(φ )σy/2] (5.3) Hrf,k 1,k k k b where ω1,k is the RF ﬁeld strength, and φk is the sum of two terms: one is the phase appropriate to the pulse shape (usually 0◦ or 180◦), and the other is a term linearly varying with time at a rate ωrf ω0 divided by the pulse length (in the frame of spin a, the pulse − rotates at frequency ωrf ω0. The total unitary operator for slices 1 through k is given by: −

k U a = U a (5.4) T,k Y l l=1

Thez ˆ-rotation acquired by spin a during the ﬁrst k slices is given by:

φa = arg[U a (1, 1)] arg[U a (2, 2)] (5.5) BS,k T,k − T,k

a where (1, 1) and (2, 2) denote the matrix elements of UT,k. By deﬁnition, the phase acquired a in the ﬁrst slice is φBS,0 = 0. Since each slice is of lengthtk, we can approximate the frequency shift of spin a during slice k of the pulse at ωrf :

φa φa a BS,k Bs,k 1 ωBS,k − − (5.6) ≈ tk

a We have plotted both the frequency shift ωBS,k and the accumulated phase acquired in Fig. 5.1. We have written MATLAB codes which calculate the transient Bloch-Siegert shifts. They are shown in Appendix A.

5.2.2 Correction of the transient Bloch-Siegert shift

From Fig. 5.1, we can see that the total acquired phase is 27◦. This is a substantial phase shift and must be corrected because thez ˆ-rotations are unintentional and result in erroneous 130 CHAPTER 5. TOOLS OF NMRQC

30 300

250 25

200 20

150

[Hz] 15 BS

ω 100

phase [degrees] 10 50

5 0

−50 0 0 500 1000 1500 2000 0 500 1000 1500 2000 time [µs] time [µs]

(a) (b)

Figure 5.1: (a) Plot of ωBS during the application of a Hermite 180 shaped pulse with ωrf ω0 = 5kHz and a pulse width of 2000µs. (b) Plot of φBS during the same pulse. The − total acquired phase reaches 27◦ - a substantial amount that must be corrected. evolution. The phase shift can be corrected by applying the appropriate inversez ˆ-rotation. We have discussed several different approaches to applyz ˆ-rotations earlier; however only one of them is also practical here. Correcting the phase shift by explicitly applying az ˆ-rotation throughx ˆ andy ˆ-rotations is not possible because these pulses would themselves induce a phase shift in the other homonuclear spins which would also need correction. Hence, the only practical approach is the implicit absorption of the phase into the phase of future pulses or jumping the carrier frequency as described in section 4.2.2. We have corrected the transient Bloch-Siegert shift by implicitly absorbing the phase rotations. Though the transient Bloch-Siegert shift is documented in the NMR literature, typically not much is done about it, partially because it simply does not have as serious a detrimental effect in traditional NMR experiments as it does in ours. Hence this effect is often brushed aside. However, we have approached this effect in great detail. This allowed us to gain a new perspective and invent a new technique which we describe in the next section.

5.3 Simultaneous selective pulses - artifacts and solutions

A direct consequence of the transient Bloch-Siegert shift is that simultaneous shaped pulses on homonuclear spins do not result in the desired unitary transform. Suppose we apply RF a b radiation simultaneously on two spins a and b with resonance frequencies ω0 and ω0 with b a a b ω0 > ω0 . Then the pulse at ω0 shifts the frequency of spin b to ω0 + ∆ωBS. As a result, 5.3. SIMULTANEOUS SELECTIVE PULSES - ARTIFACTS AND SOLUTIONS 131

the pulse on spin b is no longer on-resonance because it is applied at frequency ωb which is off-resonance by an amount ∆ωBS. Similarly, the pulse on spin a is now off-resonance by − an amount ∆ωBS. As we saw in section 5.2, these off-resonances can reach a few hundred Hertz, and can therefore result in spin dynamics that significantly deviate from the intended rotations. Fig.5.2a shows the simulated inversion profiles for a spin subject to two simultaneous

Hermite 180◦ pulses, separated by 3273 Hz. The centers of the inversion profiles have shifted in frequency and the inversion is incomplete, which can be seen most clearly from the substantial residualx ˆyˆ-magnetization (> 30%) over the whole region intended to be inverted. Since this is an off-resonance effect, perfect rotations cannot be achieved regardless of the tip angle.

1 1 Z Z XY XY

0.5 0.5

0 0 amplitude amplitude

−0.5 −0.5

−1 −1 −6000 −4000 −2000 0 2000 4000 6000 −6000 −4000 −2000 0 2000 4000 6000 spin frequency [Hz] spin frequency [Hz]

(a) (b)

Figure 5.2: (a) Simulation of the amplitude of thez ˆ andz ˆyˆ-component of a spin as a function of its frequency. The spin starts out along +ˆz and is subject to two simultaneous Hermitian-shaped pulses with carrier frequencies at 0 and 3273 Hz (vertical dashed lines), with a calibrated pulse length of 2650µs (ideally 180◦). (b) Same as (a) but with the frequency shift correction.

Previous work [MM93] addressing this problem suggested the use of additional soft pulses to cancel out the frequency shift of the spins. However, these pulses may have undesired eﬀects on other spins in the spectrum. Another approach [KF95] is to time-reverse the pulses, but this is not applicable for time-symmetric pulses (which are commonly used in NMR experiments), and furthermore the time-reversal induces a frequency dependent phase shift of the corresponding spin. A last method involves the brute-force optimization of the pulse shapes for simultaneous pulses [PRN91]. While this method does deliver pulse 132 CHAPTER 5. TOOLS OF NMRQC

shapes capable of rotating spins close in frequency, it is most commonly used in imaging experiments and has not found the same degree of application in the NMR community which has continued to use standard pulse shapes. In practice, simultaneous soft pulses have usually been avoided in NMR [LKF99] or the poor quality of the spin rotations was just accepted.

Frequency correction scheme

We have developed an eﬀective, intuitive, albeit simple procedure to address this problem [SVC00] as an alternative to the brute-force method1. By shifting the frequency of the simultaneous RF pulses, we can stay on-resonance with both of the spins at all times during the pulse. This is most easily done by phase-ramping techniques. The calculation of the frequency shift follows the description of section 5.2 and the frequency shift can be implemented by phase-ramping techniques, explained in section 5.1. The frequency shift is independent of the input state of the spins and hence it only needs to be computed once, namely before the experiment.

Fig. 5.2b shows the improvement of the spin rotation using the frequency shift correction technique - only very little leftover magnetization remains, in contrast with Fig. 5.2a. We simulated the inversion profiles for a variety of pulse shapes (gaussian, hermite shaped, reburp pulses) and frequency separations and found significant improvements in the spin rotations. We also verified improvements for arbitrary rotation angles. The improvements are most pronounced when the frequency window of the shaped pulse is between two and eight times the frequency separation between the spins, improving the unitary operator up to a factor of 15. This method can be easily extended to three or more simultaneous soft pulses.

Experimental results of this method are shown in Fig. 5.3, with the same parameters as described in Fig. 5.2. Clearly, using the frequency correction scheme, the resulting leftover magnetization is comparable with that of a single pulse. We use this method in the experiments of sections 6.5 and 6.6.

1This idea emerged through discussions with Lieven Vandersypen. Together we worked out and reﬁned this method. 5.4. UNWINDING J-COUPLING DURING SELECTIVE PULSES 133

Figure 5.3: The figure shows four pairs of spectra experimentally measured spectra (absolute value). For each pair, the spectrum of spin a is on the left and the spectrum of spin b is on the right. The first pair was taken after a 90◦ Hermitian pulse on a, and the second pair was taken after a Hermitian 180◦ pulse on a. These serve as reference spectra. The third pair was taken after simultaneous Hermitian 180◦ pulses on the two spins without the correction scheme. The last pair is the same but with the correction scheme. The experimental parameters are the same as those used in the simulations of the previous figure.

5.4 Unwinding J-coupling during selective pulses

The evolution of n coupled spins has been described in section 4.3 (Eq. 4.19). Coupling between spins is required for two-qubit gates, and hence is an important ingredient for quantum computation. However, this coupling is always turned on, including during the application of long, selective pulses. Thus far, we have ignored the eﬀect of coupling during selective pulses but it can result in signiﬁcant errors in the evolution of the spins. Single self- refocusing pulses have been designed to minimize coupled evolution between the selected spin and all others. However, the coupling between unselected spins is sill active, which may cause undesired evolution of the unselected spins. Furthermore, the self-refocusing behavior is disturbed during multiple simultaneous soft pulses [KF95]. We have developed a method to improve single spin rotations in the presence of scalar couplings between the spins. The basic idea is to approximate an ideal but not directly obtainable unitary evolution (single qubit rotation) by concatenating several experimentally accessible unitary evolutions (which include coupled evolution during the long, selective RF pulse). This approach relies on well-known Taylor expansion techniques, and can be given 134 CHAPTER 5. TOOLS OF NMRQC

in closed form for the case of time independent Hamiltonians. Numerical simulations tools are needed for the treatment of time-dependent Hamiltonians. Let us begin by looking at at two-qubit model.

5.4.1 Two qubit model

Suppose we wish to apply an ideal unitary evolution by switching on a Hamiltonian B for some time ∆t: iB∆t Uideal = e− (5.7)

Let us approximate Uideal by a second order Taylor expansion:

2 ∆t 2 Uideal II i∆t[B] [B] (5.8) ≈ − − 2

Suppose we can control the experimentally accessible Hamiltonian to be either H = A + B (where A is another term, which does not commute with B) or H = A (we explain the − negative sign later), but we still wish to apply Uideal. Let us concatenate the following unitary evolutions:

iAα1∆t i(A+B)∆t iAα1∆t Uexp = e e− e (5.9) where α1 times ∆t equals the time during which we want the Hamiltonian A to be active. − Setting α1 = 1/2 we can show that Uexp = Uideal to second order. In other words, despite the limitations on the experimentally accessible Hamiltonians, we can still implement the desired unitary evolution up to second order. Let us repeat a similar analysis for a two qubit system and approximate an ideal single qubit rotation to third order. We will also be more speciﬁc about what Hamiltonians are given by setting B = XI where XI is short-hand for the Kronecker product of the Pauli matrix σx and the identity matrix I. The strength of B is implicitly absorbed in ∆t by letting ∆t equal half the ﬂip angle of the rotation in radians. The ideal unitary evolution is then approximately:

∆t2 ∆t2 Uideal II(1 ) iXI∆t(1 ) (5.10) ≈ − 2 − − 6

Suppose now that in addition, there exists a Hamiltonian A arising from scalar coupling between the qubits and is given by A = βZZ where β = 2tpwJ with tpw being the pulse 5.4. UNWINDING J-COUPLING DURING SELECTIVE PULSES 135

width in seconds and J the coupling strength between the two qubits in Hertz. ZZ is the Kronecker product of two σz Pauli matrices. Let us use the same concatenation of unitary evolutions as in Eq.5.9, but this time let α1 be a variable, and perform a third order expansion to get:

2 ∆t 2 2 2 2 Uexp II[1 (4α β 4α1β + β + 1)] ≈ − 2 1 − 2 ∆t 2 3 2 + iZZ∆tβ [(2α1 1) [β (8α 12α + 6α1 1) + 6α1 1] (5.11) { − − 6 1 − 1 − − } ∆t2 iXI∆t[1 (β2 + 1)] − − 6

The term containing ZZ contributes significantly to undesired evolution. We can set this term equal to zero and solve for α1 to obtain an analytic solution, which corrects to third order, any ZZ component in the resulting unitary operator. In general, the solution α1 is a function of β and ∆t. Even though applicable to arbitrary flip angles, we here only focus on the results for specific flip angles because our experiments and NMR experiments in general as well as a significant portion of quantum computer proposals employ flip angles of only 90◦ or 180◦ radians. No real valued solution exists for a flip angle of 180◦ because the negative evolution before the inversion pulse gets canceled by the evolution following the pulse. For a flip angle of 90◦ radians, α1 varies between 0.6487 and 0.6507 over the typical range of couplings experienced, i.e. 0 β 1. Due to the small functional dependence ≤ ≤ on β we assume α1 to be constant. A similarly small functional dependence is found for arbitrary flip angles between 0◦ and 90◦ as well. We also note that there still exists an error in the XI and II components, mimicking off-resonance effects. These effects can be easily corrected for all flip angles smaller than some critical angle by slightly increasing the strength of the Hamiltonian XI, but we have not explored this in our experiments. Experimentally, the desired unitary evolution can now be achieved by pre- and appending the single qubit rotation in the presence of coupling (H = A+B) by a negative evolution

(H = A) of length α1 times the length of the single qubit rotation (pw). − In the beginning of this section we claimed to have control of whether we wish to apply H = A+B or H = A. Since the term A is always active and positive in NMR systems (J- − couplings are given by nature, and cannot be explicitly controlled), and B can be externally turned on, we can only explicitly apply H = A+B (single qubit rotations in the presence of 136 CHAPTER 5. TOOLS OF NMRQC

J-coupling) and H = A (time delays). However, there exists an implicit way to implement H = A, or ’negative’ time delays. To do logic with a NMR quantum computer, we apply − a sequence of radio-frequency pulses and time delays as described in chapter 4. Consecutive pulses on one qubit can be merged into only one by calculating the effective axis and strength of rotation. Pulses on different qubits can be applied simultaneously because they commute. Thus, it is possible to decompose any pulse sequence into a repeating string of a single (or simultaneous) pulse followed by an explicit time delay so that all pulses (except for the first and last) are surrounded by time delays. By making these time delays shorter by some amount, we can implicitly apply H = A. Specifically, we wish to make the delays − before and after the pulse shorter by α1 times the pulse width.

Of course, the scenario may arise in which α1 times the pulse width is actually longer than the time delays surrounding the pulse, which means we would have to physically implement negative time. This is impossible experimentally, but can be compensated for as well. Each two-qubit quantum logic block requires a time delay proportional to 1/2J with single qubit rotations applied throughout. If we make this total duration longer by 2/J but otherwise retain the same pulse sequence, the unitary evolution remains the same. By doing this, we have increased the time delay surrounding the single qubit rotations. We keep lengthening the total duration until α1 times the pulse width is shorter than the time delay surrounding the pulse, allowing for our correction scheme to be implemented.

5.4.2 Multiple qubit model

Extending this concatenation scheme to several coupled qubits requires additional thought. Applying the above results for 2 qubits on a 3 qubit system in which all qubits are coupled to each other could lead to signiﬁcant erroneous evolution. During the single qubit rotation the two spins that are not involved in the rotation are coupled to each other, undergoing an evolution by an amount ∆t. Along with the two negative correction Hamiltonians active for some time, α1∆t, these two qubits undergo a coupled evolution of duration (1 2α1)∆t. − When α1 = 1/2, this evolution is equal to zero, and thus the scheme can be easily extended to an arbitrary number of qubits. However, as we have seen from the analysis above,

α1 0.65, and hence it appears that our method is not extendable. ≈ We investigated the use of time dependent Hamiltonians to implement single qubit rotations in hope of obtaining diﬀerent values for α1 (speciﬁcally the value 1/2). We developed a mathematical tool capable of applying a time dependent Hamiltonian of the form 5.4. UNWINDING J-COUPLING DURING SELECTIVE PULSES 137

H(t) = A+g(t)B where A is still the term due to scalar coupling, and g(t) can be varied as a function of time. The time dependent part g(t)B of the Hamiltonian can be approximated, and in fact is also experimentally implemented as a series of constant Hamiltonians, each with a diﬀerent value of g. Thus, the overall evolution can be calculated as:

U = exp( i(A + gjB)tj) (5.12) Y − j where B = XI, tj is the duration of each slice, gj is the amplitude of B in each slice and A = πJZZ/2 with J denoting the coupling strength between the two qubits. The term

exp( i(A + gjB)tj) for each slice can be approximated to much higher orders than 3 using − standard mathematical software programs. We pre- and appended the unitary matrix U

as in Eq.5.9 by exp(iα1βZZ) to obtain Uexp. We then varied α1 so that Uexp matches the desired ideal evolution as closely as possible as determined by taking the 2-norm of the

diﬀerence between Uideal and Uexp. We chose the 2-norm to calculate the distance the between the ideal and experimentally accessible unitary matrix because it is a standard distance metric for matrices and it can be easily calculated using standard mathematical software programs. Other proper distance metrics should produce similar results. Nonetheless, there are certain peculiar aspects that arise when using the 2-norm in the context of quantum information. Consider two unitary matrices U and U. They both are physically indistinguishable, however the distance − between them is maximized according to the 2-norm. One could argue that maybe the 2- norm is not a good measure for distances between unitary matrices for quantum information. But if one considers the scenario where U and U are operations controlled by an ancilla − bit, then they are distinguishable. Therefore, it is reasonable to treat U and U as diﬀerent − unitary matrices.

The results for α1 following our minimization procedure for diﬀerent envelope shapes

of the time dependent Hamiltonian, g(t)B, and ﬂip angles (90◦ and 45◦) with very small values of β are listed in Table 5.1 (calculations were all done using standard functions in

MATLAB). Results for ﬂip angles of 180◦ are not shown because as mentioned earlier, the pulse refocuses the two negative time delays.

We note that the value of α1 for a hard 90◦ pulse diﬀers from the analytic 3rd order model by only 2%, conﬁrming that the analytical model can be used for that pulse shape,

at least for ﬂip angles less than or equal to 90◦. We also found that α1 varies by less than 138 CHAPTER 5. TOOLS OF NMRQC

shape α1[90◦] α1[45◦] hard 0.6366 0.5274 gauss 0.5759 0.5153 hermite180 0.4951 0.4988 hermite90 0.5322 0.5064 sinc 0.5342 0.5069 uburp 0.0986 0.3897

Table 5.1: Table listing the optimized values of α1 for ﬂip angles of 90◦ and 45◦ and a variety of diﬀerent pulse shapes.

5% for the typical range of β. As mentioned above, we therefore assume for the remainder of the text that α1 is independent of β. None of the shapes we tried result in α1 = 0.5 exactly but it is clear that the pulse shape has a signiﬁcant impact on the optimal amount of ’negative’ time delay. Further work may lead to a design of a pulse shape that is even more suitable for this correction method than the ones we used.

Thus far we provided a recipe for improving unitary control for single qubit rotations in the presence of strong coupling, but have not quantized how much error is still left in the quantum operation. The fidelity has been used in the past to quantify the success of a quantum operation and we shall use the gate fidelity here. The gate fidelity is given by:

Fg(Uideal,Uexp) = min F (Uideal ψ ,Uexp ψ ψ Uexp† ) (5.13) ψ | i | ih | | i where Uideal is the unitary matrix of the ideal single qubit rotation, Uexp is the corrected unitary matrix with a ’negative’ pre- and post pulse evolution and F (ψ, ρ) is the fidelity between a quantum state ψ and a density matrix ρ. The gate fidelity here then quantifies the worst-case behavior by minimizing over all possible input states.

Fig. 5.4 plots the gate fidelity for several unitary matrices as a function of β for a flip angle of 90◦ corresponding to ∆t = π/4. Clearly, the uncorrected unitary matrix U has a low gate fidelity compared to the corrected one which has a gate fidelity close to unity. Only prepending or appending (but not both) the unitary matrix U gives an improvement compared to the absence of any correction but a symmetric correction provides the best improvement in unitary control, as expected. 5.4. UNWINDING J-COUPLING DURING SELECTIVE PULSES 139

0.95

G 0.9

0.85

gate fidelity F 0.8

0.75 no correction symmetric correction un−symmetric correction 0.7 0 0.2 0.4 0.6 0.8 1 β

Figure 5.4: Gate fidelity of soft pulses as a function of β, over a range typically encountered in NMR. The solid line shows the rapid decay of the gate fidelity for stronger couplings without any correction scheme. The dashed line shows improvement when adding a ’negative’ time delay either before or after the single qubit rotation. The gate fidelity stays almost constant for the symmetric compensation by introducing ’negative’ time delays before and after the pulse.

5.4.3 Extension to multiple single qubit rotations

The discussion in the previous section dealt with just single qubit rotations, yet many experiments would beneﬁt from the application of simultaneous rotations of several qubits, especially refocusing 180◦ pulses [Fre97]. For a 2 qubit system, let us apply two Hamilto- nians, B1 = XI and B2 = IX, and assume both are of the same strength. By replacing

B in Eq.5.12 with B1 + B2, we can use the same numerical framework as before. Instead of calling this result α1, let us call it α2 to denote that it is the result when applying two simultaneous qubit rotations instead of just one. Our calculations (not included here) are summarized in Table 5.2 and suggest that α2 is (a) in general not the same as α1, (b) a very weak function of β (and for simplicity, let us approximate it as a constant for the remainder of the article as well), and (c) roughly the same for simultaneous two and three qubit rotations. Table 5.2 summarizes some values for α2, similar to the table for α1. We note that the angle of rotation resulting from the unitary matrix for ﬂip angles of

180◦ was fairly insensitive to even large changes in α2. The axis of rotation changes with 140 CHAPTER 5. TOOLS OF NMRQC

shape α2[180◦] α2[90◦] α2[45◦] hard 0.2500 0.6272 0.5566 gauss 0.3791 0.5382 0.5315 hermite180 0.4455 0.5014 0.4980 hermite90 0.4293 0.5382 0.5134 sinc 0.4444 0.5298 0.5154 uburp 0.2920 0.5031 0.3337

Table 5.2: Table listing the optimized values of α2 for ﬂip angles of 180◦ and 90◦ and a variety of diﬀerent pulse shapes.

α2 but not the angle of rotation, and this is independent of the input state. This was not the case for single pulses. We furthermore note, that for simultaneous 180◦ pulses, the time delay following the pulse is not canceled out by the delay before the pulse. Therefore, we included rotation angles of 180◦ in this table. During simultaneous pulses, strong coupling does not just manifest itself in erroneous evolution but, also in the excitation of multiple quantum coherences. These can become significant for sufficiently large values of β but were not a primary source of errors in our experiment. Future work is likely geared towards reducing multiple quantum coherence excitations. We now proceed and analyze how this scheme scales. For any number of qubits greater than three, we would have to pre- and append by three different types of negative evolution between several pairs of qubits; one between the two pulsed qubits by an amount α2 times the pulse width, another between one of the pulsed qubits and all others by an amount α1 times the pulse width, and a last one between unpulsed qubits by an amount of 0.5 times the pulse width. The order in which the negative evolutions occur does not matter since any Hamiltonians involving Kronecker products of only σz matrices and identity matrices commute. However, unless α2 = α1 = 0.5, the negative evolutions are not straight forward to implement experimentally using our previously described methods. Fortunately, when using simultaneous pulses it may not be necessary to correct all of the erroneous evolution due to cancellation effects during refocusing schemes. Consider the refocusing scheme drawn in Fig. 4.8 which selects out only one coupling of a 7 qubit quantum system where all qubits are pairwise coupled. Suppose now that the refocusing pulses require long shaped pulses. We argued in the previous subsection how these pulses need to be pre - and appended by multiple negative evolutions. If those 5.5. PULSE SEQUENCE DESIGN AND SIMPLIFICATION 141

evolutions were left out, one would expect this refocusing scheme to fail. Upon further inspection however we notice that the erroneous evolutions between all pairs of qubits except for qubits 1 and 2 cancel out during the pulse sequence. This is in fact true, regardless of the values for α1 and α2. The evolution between qubits 1 and 2 during the refocusing pulses can be easily compensated by shortening each evolution segment by the total pulse width of all refocusing pulses divided by the number of delay segments - no other adjustments are necessary for this refocusing scheme.

Together with the negative time-delay technique for single 90◦ pulses, we can now construct two qubit gates such as the cnot with a high degree of unitary control, despite using long selective pulses in the presence of strong scalar couplings. The improvements are not perfect, and nonidealities still exist, and future work may enhance current schemes even further.

5.5 Pulse sequence design and simpliﬁcation

The goal in NMR quantum computing is to eﬃciently obtain a pulse sequence by taking the quantum algorithm, designing the required quantum circuit and decomposing it into one- and two-qubit gates and ﬁnally into a sequence of RF pulses and delay times. The procedure is analogous to compiling code on a classical machine. We have already “compiled” quantum code in the description of quantum gates (sections 2.2.3 - 2.2.4) and their implementation in NMR (sections 4.2 - 4.3). But this was all done manually to create recipes to simplify quantum circuits and/or pulse sequences. The goal would be to develop a quantum compiler that does these steps for us. Good pulse sequences typically are the shortest and most robust sequences. It is extremely important that such a quantum compiler not require exponential overhead - otherwise, the speed-up by using quantum algorithms is completely negated. This section summarizes the techniques we have acquired that relate to quantum circuit and/or pulse sequence optimization. We have already mentioned some of them, but we summarize them here.

Simplifying quantum circuits

Since the compilation process first designs a quantum circuit, then decomposes it into one and two-qubit gates and finally creates a sequence of RF pulses and delay times, the simplification process occurs on each of these levels as well. 142 CHAPTER 5. TOOLS OF NMRQC

By studying a particular quantum algorithm in detail, either the general algorithm or the particular instance we are trying to solve, we can sometimes reduce the resources required. Recall that in both the Deutsch-Jozsa and Grover algorithms we transform a quantum state from x ( 0 1 )/ (2) to ( 1)f(x) x ( 0 1 )/ (2). Since the second register (a single | i | i − | i p − | i | i − | i p qubit in this case) does not change its state by this transformation, we can leave it out.

After this top-level simpliﬁcation, the next step is to look closer at the one and two- qubit gates. Here we make use of the commutation rules to move blocks about (refer to section 2.2.4 for some commutation relationships). This could lead to cancellation of two consecutive blocks, or to the simultaneous application of two blocks. Considering that some blocks can be implemented using several equivalent constructions, there is a good chance to cancel out several blocks in a quantum circuit. Finally, we can also use the fact that unitary operations acting on diagonal input states do not require proper phases (for example, Eqs. 4.21 and 4.22 lead to the same output state provided the input is diagonal).

At the lowest level, each one or two-qubit gate can again be implemented using several equivalent constructions using diﬀerent RF pulses and/or delay times. We can choose construction properly to cancel out adjacent pulses.

Besides simplifying a quantum circuit at these three levels, we must also pay attention to the robustness of the sequence. For example, J-couplings only come into play at the lowest level, but very small J-couplings may lead to long gates during which decoherence has a large impact. Hence, it is important to work around small couplings, so several iterations of the compilation scheme may be required.

The overall simplification process needs to pay attention to other non-idealities such as RF-inhomogeneities and unrefocused coupling during RF pulses. Inhomogeneities in the 2 static field, B0, can be unwound by an X pulse. But what if we have to apply a train of 180◦ pulses, as required for example in long refocusing schemes (see Fig.4.8)? Do all refocusing pulses have the same phase, or should we make some of the pulses X¯ 2? It turns out that we do need to change the phases of the pulses for a long train of refocusing pulses. For example, X2X2X¯ 2X¯ 2 performs much better than X2X¯ 2X2X¯ 2 ([LFF82]). Furthermore, during some refocusing schemes, such as the one from Fig. 4.8, two spins are never pulsed so they experience static field inhomogeneities. We thus also need to carefully design refocusing schemes, keeping in mind the phases of refocusing pulses as well as inhomogeneities. 5.6. PULSE SEQUENCE FRAMEWORKS 143

Composite pulses

Composite pulses have been used in NMR to remove certain kinds of errors, often at the cost of long durations. For example, the eﬀects of RF inhomogeneities (or pulse width miscalibrations) can be reduced by applying composite pulses. The sequence XY 2X should 2 result in a better 180◦ rotation, starting from thez ˆ-axis than X , in the presence of RF ﬁeld inhomogeneities [Fre97]. But since these composite pulses are longer, they might introduce more errors than they remove, especially if J-couplings cannot be perfectly unwound. IN section 7.4 we design a composite pulse for a Josephson phase qubit to improve the quality of single qubit rotations.

Summary

Obviously, ﬁnding the optimal sequence is a rather ambitious undertaking. It is however, very desirable, because strong error cancellations have been observed in several complex quantum computing experiments, including the ones we describe in sections 6.3 and 6.6.

5.6 Pulse sequence frameworks

Over the course of the years, we have made tremendous use of our developed tools. Not all tools were used in all experiments because we discovered a lot of the techniques along the way. The phase-ramping technique, compensation of the transient Bloch-Siegert shift, and frequency correction method were used in the two experiments described in sections 6.5 and 6.6. The experiments in section 6.6 also used the scheme to unwind J-couplings during selective pulses. Incorporating all of these tools into the experiments required the development of a software framework. As described in this chapter, when applying shaped pulses, we have to keep track of additional phase accumulations (transient Bloch-Siegert shifts) as well as the elapsed time of the pulse sequence (oﬀ-resonant excitation). We wish to apply these adjustments at a lower-level and not have to include them explicitly in our pulse sequence. In other words, we wish to simply type X(3), for example, which would execute anx ˆ-rotation on spin 3. The framework would then automatically send the correct pulse shape of the desired duration on the correct spin with the phases properly adjusted, and also keep track of all timing events and transient Bloch-Siegert shifts associated with the application of that pulse. We have written an extensive framework for the seven spin experiment (section 144 CHAPTER 5. TOOLS OF NMRQC

6.6), and we show parts of the framework pseudo-code in Appendix C. A signiﬁcant portion of my thesis work went into developing this framework.

5.7 Other advances

In addition to our own tools, several other groups have developed a library of techniques as well. However, we have not used them in our experiments because our methods have worked well and we did not require other techniques. Cory’s group [FPB+02] use very strongly modulated pulses to average out undesired effects, provided there was exact knowledge of the Hamiltonian of the system. The resulting pulses are amplitude and phase modulated and are shorter compared to the traditionally used soft-pulses. This comes at the cost of requiring numerical methods to create these pulses, and hence this method might not scale beyond a few qubits. Jones’ group [KJ01, CLJ02] has worked on designing composite pulses better suited for quantum computation when faced with off-resonance errors and pulse length miscalibrations. Glaser’s group looked at improving traditional NMR pulse sequences [KRLG02]. Specif- ically, they looked at how the sequence for coherence polarization transfer between two spins can be optimized in the presence of relaxation. They also report unexpected gains in sensitivity for the most commonly used experiments in NMR spectroscopy. It is clear that the traditional tools used in NMR experiments can be improved for use in quantum computations. Conversely, the newly developed methods may find useful applications in traditional NMR experiments. For example, the frequency correction method for simultaneous shaped pulses (section 5.3) has been incorporated by Varian into their software because it was straightforward to do so, and because they believed it to be of broad enough application. In fact, we know that this method has now been used in a few NMR experiments, based on Varian’s newsletters. It has also been included in a review article of NMR techniques for organic chemistry [HM01].

5.8 Summary

In chapter 4 we outlined how NMR techniques could serve as an excellent test bed for experimental quantum computation. Nonetheless, several artifacts arise when trying to implement 5.8. SUMMARY 145

some of these techniques, especially when using homonuclear spin systems. Speciﬁcally, we found solutions to the following artifacts:

(Transient) Bloch-Siegert shifts: This effect results in phase shifts and off reso- • nance effects when applying single and multiple shaped pulses.

J-coupling during RF pulses: This eﬀect occurs during any shaped pulse because • J-couplings are always active, and is especially pronounced with larger J-couplings.

We furthermore continued to develop tools towards simplifying quantum circuits, including state initialization procedures. We have used these methods as well as those described in detail earlier and have developed them extensively over the years. These techniques culminated in the experiment described in section 6.6. My contributions lie in the detailed investigation of the transient Bloch-Siegert shift (section 5.2), the frequency correction method (section 5.3, the method to unwind J-couplings (section 5.4), pulse sequence simpliﬁcation techniques (section 5.5), and the design of the software framework (section 5.6). However, none of these contributions were solely my own work but rather emerged together in collaboration with Lieven Vandersypen. In the next chapter, we present a series of experiments in which we explore how our methods and concepts apply to real quantum computations, in order to gain a deeper understanding of what it would take to build a scalable quantum computer. Chapter 6

Experimental realization of NMR quantum computers

Following a brief description of the experimental apparatus (section 6.1), we summarize some of the major NMR quantum computing experiments performed to date (section 6.2), not just by our group1. We then present in detail a series of 6 experiments in which we explored quantum computation in practice. These are a study of systematic errors with three qubits (section 6.3), an implementation of efficient cooling of one of three spins (section 6.4), a realization of the order-finding algorithm with five qubits (section 6.5), quantum factorization of the number 15 using seven qubits (section 6.6), a realization of a universal quantum computation using a spin-3/2 quantum system (section 6.7), and a realization of an adiabatic optimization algorithm using three qubits (section 6.8).

6.1 The NMR spectrometer

The main components of a NMR spectrometer are outlined in Fig. 6.1. A sample containing a large number of identical molecules (our quantum computers) dissolved in some liquid solution is placed in a strong, static magnetic ﬁeld. The RF pulses are generated by a radio-frequency coil placed next to the sample inside a probe. The same coil is used to detect the signal of the spins during the measurement. The experiment is controlled by a

1The experiments of sections 6.8 and 6.7 took place in the Media Lab at MIT. The remaining experiments took place at the IBM Almaden Research Center. Both instruments were almost identical, and we only provide a detailed description of the spectrometer at IBM while noting diﬀerences in footnotes.

146 6.1. THE NMR SPECTROMETER 147

workstation. We next explain each component in more detail.

sample tube

mixer

capacitor

directional RF computer coupler oscillator

RF coil amplifier B 0

static field coil

Figure 6.1: Schematic overview of a NMR spectrometer.

6.1.1 Sample

The physical qubits are represented by the spin-1/2 nuclei of a molecule. Since the signal of a single molecule is too weak to be detected with current technologies, we use on the order of 1018 molecules to enhance the signal. Each molecule serves as a quantum computer, and all 1018 of them undergo the same operations. It is important to keep in mind that the power of quantum computers does not stem from the number of identical quantum computers, but instead, the number of qubits per quantum computer. We have already discussed some of the design parameters to synthesize proper molecules (see section 4.8). These molecules are dissolved in some liquid solvent at room temperature and atmo- spheric pressure. The solvent is chosen based on the solubility of the molecules in the solvent, and on the coherence times of the qubits obtained in this solvent. The concentration strength of solved molecules is a trade-off between signal-strength and coherence times. The NMR solvents are usually deuterated because the deuterium signal is used as part of a feedback loop which keeps the magnetic field strength constant over the course of several experiments (more on field strength in section 6.1.2). 148 CHAPTER 6. EXPERIMENTAL REALIZATION

The sample solution is held in a thin-walled glass NMR sample tube with a 5mm outer diameter (4.2mm inner diameter). The sample tube is ﬁlled with our liquid solution to about 5cm from the bottom. It is critical that the walls of the tube be straight and uniform in thickness, in order to minimize ﬂuctuations of the magnetic susceptibility. We have used high quality sample tubes, mostly from Wilmad but also New Era Enterprises.

The sample is prepared carefully by removing oxygen because O2 is paramagnetic and causes rapid relaxation. This is done via several cycles of freeze-thawing the sample; repeated freezing and thawing of the sample while subjecting the sample to a vacuum after each freezing step. We also remove water (but this is only needed when the quantum computer molecules react with H2O), and any other particles suspended in the solution. Finally, the open end of the sample tube is flame sealed to prevent water, oxygen and other impurities from leaking into the solution. The filled and sealed sample tube is then placed in a sample holder called “spinner” (see Fig. 6.2). As the name suggests, the sample holder does not only hold the sample in place but it is also used to spin the sample about the axis of the static magnetic field to average outx ˆyˆ inhomogeneities in the static field.

Figure 6.2: A typical NMR sample: A sample tube containing the dissolved quantum computer molecules is held in place by a sample holder. 6.1. THE NMR SPECTROMETER 149

6.1.2 Magnet

Superconducting coil

The sample inside the sample holder is placed in the room temperature bore of a superconducting magnet built by Oxford Instruments. The magnet is made of a superconducting solenoid immersed in a bath of liquid Helium (4.2 Kelvin). A current passing through the solenoid produces the static magnetic ﬁeld. The helium vessel is surrounded by a vacuum seal, a liquid nitrogen vessel and another vacuum seal. The whole magnet is mounted on air-cushioned vibration isolation legs (see Fig. 6.3).

The MIT magnet had an active vibration isolation system by the “Technical Manu- facturing Corporation” (TMC), instead of the air cushion, but unfortunately it has never lived up to its promised expectations. Since this magnet is located on the fourth floor, it is subject to a much larger amount of vibrations than it would be in the basement. Vibra- tions typically manifest themselves in very ragged looking spectra, especially around the base of each spectral line. For very narrow lines, as is the case for most small molecules, the resulting spectra were not acceptable for high quality experiments. This was and still is the case even with the isolation system turned on. In fact, it was possible to tell when someone was walking around in the laboratory and when not! When a sudden and strong vibration occurred (for example, when somebody accidentally ran into a table), the units overloaded and actually amplified vibrations at certain frequencies. This forced us to run high precision experiments overnight where there are less vibrations. It was then determined by TMC technicians that the magnet legs were not strong enough, leading to much stronger than expected vibrations in the horizontal plane. The magnet legs were subse- quentially reinforced, and better amplifiers for the units were installed. This helped the situation a little, for example, it is now no longer possible to tell when somebody is walking around in the laboratory. But the system is not working well enough yet, especially when stronger vibrations occur. I am currently unaware of any constructive means to address this situation.

The current that passes through the superconducting solenoid is on the order of 100 A in strength, producing a field of about 11.7 Tesla at the center of the magnet (about 200.000 times as strong as the earth’s magnetic field). The field strength outside the magnet drops off dramatically, reaching only 5 Gauss about 3 meters away from the magnet (10 times 150 CHAPTER 6. EXPERIMENTAL REALIZATION

Figure 6.3: An Oxford Instruments 500MHz narrow-bore NMR magnet (the IBM magnet was wide-bore). The ﬁll ports for liquid nitrogen and helium are at the top of the magnet. The cabinet next to one of the magnet legs (visible on left side of the picture) contains transmit/receive switches, preampliﬁers and mixers. The probe is inserted at the bottom of the magnet, while the sample is inserted from the top.

the earth’s magnetic field)2. Due to the strong fields, we must make sure that no magnetic objects (eg. screwdrivers) are placed close to the magnet. Magnetic objects can be pulled into the bore, and may be impossible to take out unless we de-energize the magnet. It is possible for the superconducting solenoid to no longer be superconducting, in which case the energy in the solenoid is released to the helium which then boils off extremely rapidly. This is known as a magnet quench and shown in Fig. 6.4. A magnet quench is a rather spectacular process, though highly costly in both time and money. It mostly occurs during the initial installation, but may also occur when the magnet is subject to severe vibrations (eg. earthquakes) or sometimes even during helium refills, but this is very rare. Strong magnetic fields are advantageous for quantum computation because the chemical

2The MIT magnet was actively shielded and thus the 5 Gauss line was about 3 feet away from the magnet 6.1. THE NMR SPECTROMETER 151

Figure 6.4: A quench of the IBM 500 MHz magnet during its installation. The superconducting solenoid suddenly dissipates all of its energy by boiling oﬀ the liquid helium.

shift is linearly dependent on the field strength, and we desire chemical shifts that are as large as possible (to reduce the RF pulse widths). However, the spin coherence times may decrease for stronger fields because of chemical shift anisotropies. It is not clear yet what field strength optimally balances these opposing criteria.

Field homogeneity

Another aspect that is of crucial importance is the magnetic ﬁeld strength homogeneity because it directly aﬀects the spectral linewidths (i.e. T2∗), and hence the signal-to-noise ratio and the spatial resolution of lines within a multiplet. These inhomogeneities can in principle be refocused during a pulse sequence, but this comes at the price of incurring longer pulse sequences. Hence, we typically try to minimize inhomogeneities right from the start.

Shimming

The inhomogeneities are reduced via two additional sets of shim coils around the bore. One set consists of several (roughly 10) superconducting coils which are adjusted during the installation process of the magnet, but are typically never readjusted, unless the magnet quenches. The second set consists of about 25 conventional electromagnetic coils whose 152 CHAPTER 6. EXPERIMENTAL REALIZATION

current can be adjusted by the user. Each of these coils produces a static magnetic field which varies spatially in strength; the Z1 coil outputs a field that varies linearly along the 3 zˆ-direction, the field due to Z2 varies quadratically, and so forth, up to fifth order . The transverse (X and Y ) coils are up to forth order, and mixed coils (eg. XZ) are typically forth order as well. The optimal shim settings are strongly probe dependent because they are sensitive to the RF coil geometry. Nonetheless, there can also be substantial sensitivities to solvent susceptibility, the quality of the sample tube walls, temperature and small magnetic particles that got pulled into the vicinity of the magnet. The homogeneity of the field is correlated with the lock signal (more on the lock signal below), the rate of decay of the FID and the spectral lineshapes and linewidths. The goal is to shim (i.e. adjust the current through the shim coils) to obtain variations in the magnetic field strength that are less than 1 part in 2 x 109 over the active sample region (about 4.5 mm in diameter, and 1.5 to 2 cm in height). But what procedure do we follow to optimize the field homogeneity? Typically, shimming has been done by hand. This can take on the order of several hours, but once a good set is found, the daily reshimming takes only a few minutes. Shimming by hand is facili- tated by following some general rules of thumb (see for example [CH90]). For example, the lower order shims have a much higher impact on the homogeneity than higher order shims. Recently however, a novel automated procedure was invented, making use of magnetic gradient fields [BCF+97, CM98]. This method is becoming more and more standard nowadays, and is being continuously refined, for example when there exist strong convection currents in the sample [EMA02]. For all experiments, except for the one described in section 6.8, we shimmed the sample by hand.

Magnetic ﬁeld drift and locking

Another important aspect to consider is that the field strength of a superconducting magnet slowly drifts over time because the current flowing through the superconductor is dissipated. The drift is tiny compared to the Larmor frequency: For a good magnet, the drift is less than about 1 Hz per hour. At this rate, the field drifts 8.76 kHz per year which is less than 0.002% per year! Hence, good magnets can last several decades without substantial loss in field strength. Compared to the precise control we require in quantum computation experiments, and

3The MIT magnet had a sixth and seventh order Z shim coil. 6.1. THE NMR SPECTROMETER 153

NMR experiments in general, the drift is still large. Consider for example, that when the spins move off-resonance the rotating frame becomes progressively further out of phase with the actual rotating frame. This drift can be compensated using an additional Z0 coil which produces an additional constant magnetic field along thez ˆ-direction and which is adjusted to keep the total field strength constant over time. The current flowing through this coil is regulated via a feedback loop aimed at locking the frequency of the deuterium signal of the solvent, and hence also the field strength, to a prescribed value. The deuterium signal is monitored every few seconds by applying a short-pulse at the deuterium frequency (about 77 MHz for our magnet). The deuterium resonance frequency is far away from the resonance frequency of other nuclei of interest. The user must first set-up the lock power, lock gain, and phase of the feedback signal. From then on, the lock mechanism runs in the background. Since field drift simply results in a shift in the resonance frequency, this effect can be

refocused using additional 180◦ pulses, but the extra pulses are undesirable. One could also simply track the drifting Larmor frequencies and make the RF source follow this shift, but this is rarely done. The magnet has to be regularly refilled with liquid nitrogen and liquid helium. Our magnet required a Nitrogen fill roughly every 10 days and a Helium fill every 3 months4.A typical Nitrogen fill in our magnet uses about 80 to 90 liters whereas a Helium fill requires about 70 to 80 liters.

6.1.3 Probe

The probe is a narrow, cylindrical tube housing the RF coil(s), a tuning and matching circuit, a temperature control system, a sample spinning mechanism and sometimes gradient coils (see Fig. 6.5).

RF coils and tune/match circuit

The RF coils are mounted near the top of the probe and closely surround the sample (the coils are thus about 1.5 cm long). The region over which the sample is coupled well to the coils is called active region and is larger than the coil area (about 2 cm). The coils typically

4The magnet at MIT only needed reﬁlling every 14 days and 4 5 months for liquid Nitrogen and helium respectively. This magnet was connected to a vacuum pump during− the installation for a much longer time than the one at IBM. 154 CHAPTER 6. EXPERIMENTAL REALIZATION

Figure 6.5: A Nalorac HFX probe. The RF coils are located near the top of the probe, connected by the BNC connectors at the bottom. The cooling air inlet and connectors for the gradient coil are also located near the bottom. The tune/match capacitor rods stick out at the bottom of the probe.

consist of 1 to 3 windings and are made of low-resistivity materials such as copper, for example. Fig. 6.6 shows a picture of the RF coils of a Bruker 200 MHz probe.

Figure 6.6: An open 200 MHz Bruker probe showing the RF coils (indicated by the white arrow).

The coils are integrated into a resonant circuit with the goal of obtaining a high quality factor Q (values of about 100 to 300 are typical) and thus also a high signal-to-noise ratio. We can tune to diﬀerent resonances via a mechanically tuned variable capacitor (the tuning capacitor). Using a second, mechanically variable capacitor, the impedance of the circuit is matched to 50 Ω. The matching is needed to make sure all of the applied power also goes 6.1. THE NMR SPECTROMETER 155

to the RF coils and is not reflected. Since the same circuit is used to detect the signal, a poorly matched circuit also results in poor signal-to-noise ratios. The actual capacitors are located close to the RF coil, and can be adjusted by long rods that stick out at the bottom of the probe. Tuning and matching is done by minimizing the reflected power from the probe for a desired frequency. It is however difficult to design and build high Q resonant circuits with multiple resonances over a wide frequency region. Hence, many commercial probes actually contain two pairs of Helmholtz coils, mounted at right angles to minimize cross-talk between the coils. For a 500 MHz probe, one coil serves frequencies over 200 MHz, or the high-band nuclei (f.ex. 1H, 19F), and the other coil serves frequencies below 200 MHz, or the low-band nuclei (f.ex. 13C). The lock frequency is typically generated using the high-band coil so that the lock signal is far away from the resonance of the nuclei of interest. Probes which have the high-band coil inside the low-band coil are called “normal” probes; “inverse” probes have the low-band coil on the inside. We have used normal probes in all experiments except for the ones described in sections 6.7 and 7.5.2. We desire pulse widths that are as short as possible. We can only apply a certain amount of power until the coil windings or the capacitors arc. The shortest pulse widths (not using shaped pulses) are between 6 and 15 µs. Despite being capable of applying short pulses, saddle-shaped Helmholtz coils have rather poor field homogeneities. The envelope of the

Rabi oscillations decays about 5% per 90◦ pulse. Hence, the error of a single 90◦ pulse is roughly 5%. This error can be largely undone by using clever composite pulses (see section 5.5). But the additional pulses often increase the experimental time, and furthermore often do not address the problem of J-coupling still being active during the pulse (see section 5.4). The field homogeneity can be greatly improved by designing coils with different geometries. This is possible but comes at the price of not being able to easily change samples or sacrificing B0 homogeneity. If we had a large enough molecule with many spins, not being able to switch samples would be acceptable, but the loss in B0 homogeneity is not.

We do require the B0 homogeneity to be several orders of magnitude better than the B1 homogeneity because the number of revolutions about B0 during a pulse sequences is also many orders of magnitude larger than the number of Rabi oscillations about B1. Another option is to limit the sample region to the homogeneous region of the RF coils. However, this results in an abrupt change in the magnetic susceptibility going from the sample to the 156 CHAPTER 6. EXPERIMENTAL REALIZATION

glass or gas. Specially designed susceptibility plugs do exist that match the susceptibility of the solvent, but they are hard to use in ﬂame sealed samples and only gave modest improvements.

Additional functions of the probe

The probe does not just contain the RF coils and the tune/match circuit but also a temperature control system, a mechanism to spin the sample holder, and sometimes gradient coils. The temperature control is regulated by the flow of cool (or hot) nitrogen air inside the probe which is controlled by the user. The MIT magnet used a Kinetics temperature control unit (Model XRII1851) with dynamic temperature control from 80 to 100 , and a − ◦ ◦ stability of 0.1◦, at a flow rate of 28 liters per minute. The inlet air is normal dehumidified air with a dewpoint below 100 . Nitrogen air would be better because it is very inert. − ◦ Thermocouples accurately measure the temperature near the sample. In our experiments, we operated at room temperatures. Separate nitrogen air flows to suspend the sample holder on a pocket of nitrogen air, and spins it about thez ˆ-axis. The spinning rate is is also user controlled and ranges from 0 to 60 Hz. As mentioned earlier in this section, the probe may contain a Z gradient coil, X,Y ,Z gradient coils, or none. Theses coils produce magnetic fields that vary linearly along thex ˆ, yˆ, orz ˆ-direction. For the two experiments of sections 6.7 and 7.5.2 we used a double resonance HX probe where X means that the low-band coil is tunable over a wide range. For all other remaining experiments, we used a Nalorac HFX probe where X was tuned to 13C. Both probes are equipped with gradient coils, but we only used the Z gradient for gradient shimming in section 6.8.

6.1.4 Transmitter

The function of the transmitter is to send RF pulses to the probe. We used a custom- modiﬁed Varian UNITY INOVA spectrometer, equipped with four RF channels. Fig. 6.7 shows a picture of the spectrometer electronics cabinet. The primary reference signal is generated by a crystal oscillator, producing a 10 MHz signal which is used by four frequency sources (PTS 620 RKN2X-62/X116). The PTS sources create a continuous wave signal up to 1 Vrms in the range of 1 to 620 MHz via direct synthesis. The resolution of this signal is 0.01 Hz though we never observed the last digit 6.1. THE NMR SPECTROMETER 157

Figure 6.7: The electronics cabinet of our spectrometer. The four PTS frequency synthesizers are visible on the left. The waveform generators, attenuators, and high power ampliﬁers are on the right.

in resolution; the phase noise is 63 dBc (phase noise in dB relative to the carrier power). − The stability is as good as that of the primary oscillator. The signal that is generated is set at 20 MHz above the desired RF frequency (the signal is later mixed to obtain the correct frequency). The CW signal from the PTS synthesizers is then sent to four transmitter boards. These boards function to generate the actual RF pulse. The input signal is gated to achieve the desired pulse width. The minimum pulse length is 100 ns, with a 50 ns resolution. The phase of the pulse can be controlled with a resolution of 0.5◦. The phase adjustment is done in two steps. A coarse phase shifter sets the phase in 90◦ steps. A ﬁne phase shifter adjusts the phase in 0.5◦ steps by mixing two quadratures with adjustable amplitudes. Each board contains a linear attenuator which controls the amplitude of the pulse from zero to the full amplitude in 4096 steps. The board also contains a single-sidebanded mixer which mixes the gated signals with a 20 MHz signal so the outgoing pulses have the desired frequency5.

5The phase shifting is done on the 20 MHz signal because it is much easier to do phase shifting for one ﬁxed frequency rather than an arbitrary one. When this is mixed with the gated signal the phase is carried 158 CHAPTER 6. EXPERIMENTAL REALIZATION

The amplitude and phase control in the transmitter board can be used to create shaped pulses. Four fast memory boards (called waveform generators) are used to load the information required for several consecutive shaped pulses quickly enough onto the transmitter boards. From the transmitter board, the signals are sent to four coarse attenuators which can attenuate the signal over a range of 79 dB in steps of 1 dB. These coarse attenuators lack the fine control that the transmitter boards have, but they can attenuate over a far greater dynamic range. The coarse attenuators can be changed from one pulse to the next, but are fixed during a pulse. We have added fine stepper attenuators (1 dB range, 0.1 dB resolution) immediately before the input to the coarse attenuators to even out difference between input power for the four transmitter boards. The attenuated signal is then sent to a set of linear amplifiers which turn the low-power signal into a high power RF pulse. Two amplifiers (AMT model 3900-15) each contain a low-band (6 to 200 MHz, 300 W maximum pulse power, 30 W CW mode, 60 dB gain) and a high-band (200 to 500 MHz, 100 W maximum pulse power, 15 W CW mode, 50 dB gain). The amplifiers are designed to have extremely short rise and fall times (200 ns), and fast blanking circuits (< 2 µs on/off with a noise < 20 dB over the thermal noise). The blanking circuits are especially important to avoid noise from the amplifiers in-between the pulses. In the standard configuration the signal is automatically routed from the transmitter boards 1 and 2 to the high or low-band amplifier within the first amplifier unit (depending on whether the signals are high or low-band). The signal from transmitter boards 3 and 4 is similarly routed to the high or low-band amplifier within the second amplifier unit. All of the high-band outputs of the two amplifier units are combined with high power combiners (and so are all of the low-band outputs). The high-band and low-band signals then each pass through a PIN diode transmit/receive switch. In the transmit mode, the signal passes with very little attenuation (0.5 dB) to the probe while isolating the amplifiers from the receiver preamplifiers (see section 6.1.5). The preamplifiers are further protected via quarter wave-length cables. The signals then pass narrow-band, high-pass or low-pass filters to filter out broad-band noise from the amplifiers. Finally, the high-band and low- band signals are routed to the high-band and low-band coils of the probe. For the seven qubit experiment of section 6.6 we installed an additional frequency source, gating circuit, power amplifier and narrow-band filter in order to be able to send a CW signal by the outgoing signal. 6.1. THE NMR SPECTROMETER 159

at the 1H frequency during the pulse sequence but not during acquisition. This was needed to decouple our qubits from protons present in the molecule without sacriﬁcing one of the four transmitter channels.

6.1.5 Receiver

The function of the receivers is to record the voltage induced in the RF coil by the spins. Commercial Varian spectrometers used to only have one receive channel, but several channels are now becoming commercially available. Varian custom-built a four channel receiver system which we tested and used in our experiments. With the PIN diode in the receive mode, the NMR high-band and low-band signals are passed from the probe to high-band and low-band preamplifiers (preamps) with low loss (0.1 to 0.2 dB). The noise figure is 1.7 dB and 1.2 to 1.6 dB for the high-band and low-band preamps. This is low enough so the noise is dominated by the coils rather than the preamps. The preamp gain is about 35 dB. The amplified signals are then mixed with the outputs of the PTS frequency sources to an intermediate frequency (IF) of about 20 MHz. The resulting four signals are routed to the receiver boards in the electronics cabinets where they are mixed with an additional 20 MHz signal down to audio frequencies, separated into two quadratures and passed through audio filters (with a filter band width ranging from 100 Hz to 256 kHz). The two quadratures and then amplified to a desired level and digitized. The maximum sampling rate is 1 MHz and the maximum number of points is 524288. The digitized signal is then uploaded to a workstation. Since the digitized signal is obtained by mixing it with the same frequency source that is used in the transmitter chain, the phase of each receive channel is coherent with the phase of the transmitter channel. Hence, the same phase of the spectra is detected every time we apply the same pulse sequence.

6.1.6 Console (Workstation)

The spectrometer is operated using software from Varian (Vnmr), running on a Sun Ultra 10 workstation. All experiments can be controlled entirely by the software provided that the hardware has been properly conﬁgured for a particular experiment. The pulse sequences are written in C, and several low level commands are provided by Varian’s software. These low-level commands typically involve speciﬁc parameters, such as 160 CHAPTER 6. EXPERIMENTAL REALIZATION

the pulse width, phase, pulse shape, and channel number. For convenience we can create higher-level functions (i.e. the software framework described in section 5.6) that call the lower-level functions so we do not have to specify all the detail for each pulse. Each pulse sequence and framework must be compiled, and is then submitted to the spectrometer. The digitized signal can be Fourier transformed and displayed by the Varian software as well. Further processing, such as zeroth and first-order phase corrections or line-broadening, can also be applied. Most of our analysis is not done by the Varian software. Instead, we interface with the spectrometer via MATLAB which allows us to set up a large number of experiments to run automated. The data is also stored in some desired directory for future analysis. We can then use other MATLAB routines to (automatically) analyze the data as it is accumulated. For example, additional routines add up several spectra resulting from several temporal labeling experiments (section 4.4.6), and can derive the density matrix based on several input spectra. The flexibility in MATLAB programs thus allows easy execution of a pulse sequence and a straight-forward analysis of the collected data. Clearly, the spectrometer (NMR quantum computer) is not a self-contained single unit. This quantum computer, and in fact any other quantum computer, requires the use of a powerful classical computer, facilitating pulse sequence execution or design. This is in itself perfectly acceptable provided that the classical resources required do not scale exponentially with increasing size of the quantum computer or the problem size. Next, we provide a brief overview of quantum computing experiments, listed by several different categories.

6.2 Overview of NMRQC experiments

Nuclear magnetic resonance was invented in 1946, and turned into a powerful tool for the study of molecular structure and reaction dynamics. Since the discovery of NMR quantum computation, we learned that several pulse sequences used in traditional NMR experiments have a meaning in quantum computation. For example, the INEPT sequence for polarization transfer is now recognized as a cnot gate! Over the past few years, we have learned to implement sets of pulse sequences to study quantum computation using NMR. Several groups besides our own have and continue to actively pursue NMR quantum computation. We now provide a summary of quantum 6.2. OVERVIEW OF NMRQC EXPERIMENTS 161

information experiments using NMR, showing how our experiments ﬁt in compared with other work.

Quantum algorithms

The first quantum algorithms ever implemented were Grover’s algorithm [CGK98, JMH98] and the Deutsch-Jozsa algorithm, [CVZ+98, JM98] all for two qubits. The experiments were performed by Chuang’s group at UC Berkeley and Stanford University and Jonathan Jones’ group at Oxford University. The quantum counting algorithm was implemented soon afterwards, also by Jones’ group using two-qubits [JM99]. The two-qubit Grover search was re-implemented using a subsystem of a three qubit system [VYSC99], demonstrating logical labeling. This is the time-frame in which I joined the group - when experimental quantum computation was still in its infancy. Soon after, a three qubit Grover search was implemented, up to 28 Grover iterations, involving 280 two-qubit gates; a record which still stands today [VSS+00]. From then on, several other small scale implementations have been performed: A three- qubit Deutsch-Jozsa algorithm using transition selective pulses [LBF98], another more advanced version using swap gates to avoid small couplings [CLP00], and yet another implementation without swap gates [KLL00]. A partial implementation of a five-qubit Deutsch- Jozsa algorithm was carried out by Glaser’s group [MFM+00]. The implementation of quantum algorithms reached a new level with the full implementation of a Shor-type algorithm using five qubits (section 6.5, [VSB+00]). This work involved the use of exponentiated permutations and combining them with the quantum Fourier transform (The QFT had been implemented on its own earlier [WPF+01]). This experiment was followed by the implementation of the simplest instance of Shor’s algorithm using seven qubits to factor the number 15 into its prime factors (section 6.6; [VSB+01]). Following this work, a three-qubit adiabatic quantum optimization algorithm was implemented (section 6.8; [SvDH+03]). All of these implementations used spin-1/2 nuclei, but higher order spin systems have also been explored for use in quantum computation. The implementation of certain quantum gates has been discussed [KF00], as well as initialization schemes [Fun01]. A two-qubit continuous Grover search was partially implemented [EF02]. This work was followed by the implementation of a full two-qubit Grover search algorithm [Kot03]. 162 CHAPTER 6. EXPERIMENTAL REALIZATION

Quantum error correction

The first demonstration of quantum error correction was achieved in Cory’s group [CMP+98], together with Laflamme and Knill at Los Alamos. They implemented a three-qubit phase- error correction scheme in which gradient fields introduced artificial errors. A more complete version of the same experiment was later carried out by the same group [SCS+00]. In the meantime, Chuang’s group demonstrated a two-qubit phase error detection code [LVZ+99]. More recently, the Los Alamos group implemented a five-qubit phase and amplitude correction code for full bit flip and phase flip errors [KLMN01]. In the same year, Cory’s group also implemented a noiseless subsystem for quantum information processing [VFP+01] in which gradients artificially introduce noise.

Quantum simulations

Only little work has been done in the ﬁeld of experimentally simulating quantum mechanics. Cory’s group simulated the dynamics of truncated harmonic and anharmonic oscillators using two qubits [STH+99]. The same group later implemented a non-physical three-body interaction [TSS+99].

Other quantum protocols

The GHZ state and its derivatives (a GHZ state is a maximally entangled state of three particles) have been studied as well. First, an eﬀective-pure GHZ state was prepared [LKZ+98], and later a similar experiment was done with seven spins [KLMT00]. The claim of having created entangled states has been refuted based on the fact spins at room temperature are too mixed to be entangled [BCJ+99] (rather, a three and seven spin coherence has been observed). Nonetheless, these experiments remain the ﬁrst and and relatively simple experiments with three and seven qubits respectively. GHZ correlations have since been further studied on mixed states [NCL00]. Another interesting experiment has been done, also at Los Alamos: A teleportation protocol has been implemented using three qubits [NKL98]. The superdense coding protocol was also done [FZF+00]. An approximate quantum cloning experiment has been implemented as well (we can only approximately clone an unknown quantum state) [CLJ02]. Furthermore, the quantum Baker map was implemented by Cory’s group [WLEC02]. 6.2. OVERVIEW OF NMRQC EXPERIMENTS 163

Polarization work

We have already explained in detail that the thermal polarization of nuclear spins in liquid solution poses a severe limit on the usefulness of NMR quantum computers. Several experiments have been performed in an attempt to increase the thermal polarization. An algorithm approach was done at IBM/Stanford by implementing the basic building block of the Schulman-Vazirani cooling scheme [CVS01]. However, this scheme is impractical as long as the initial polarizations are very small. Also at IBM/Stanford, the initial polarization was increased by a factor of 10, and subsequently a two-qubit Grover search was implemented [VLV+01]. In a diﬀerent approach [HBG00] para hydrogen was transferred into a suitable molecule leading to a polarization of 10% which is much larger than the thermal 5 polarization of 10− . A quantum algorithm was subsequently performed on this molecule.

Solvent work

Liquid crystals are necessary for higher-order spins, but they can also increase the clock speed of the quantum computer when using spin-1/2 nuclei. This was ﬁrst demonstrated at IBM/Stanford [YSV+99] and further studied at IBM/Berkeley [MCK00].

Work on higher-order spins

Higher-order spin systems can offer some intriguing alternatives to spin-1/2 systems. Khitrin and Fung [KF00] were the first to demonstrate an effective-pure state and pointed out how certain operations could take less physical time compared to two-level systems. The same group followed up their initial work by demonstrating different techniques for the creation of pseudo-pure states on higher-order spin systems [Fun01, KSF01]. Sinha and Murali used these concepts to demonstrate classical logic gates [SMRK01, MSM+02]. Fung’s group then demonstrated a two-qubit continuous Grover search using a quadrupolar system [EF02]. This was followed up by a full implementation of a two-qubit search algorithm by Chuang’s group [KSJC03]. Clearly, quantum computing experiments using quadrupolar nuclei have not reached the same maturity as implementations using coupled spin-1/2 nuclei.

Comparison chart

Fig. 6.8 provides an overview of the most important quantum computing experiments performed to date, plotting the complexity of the experiment versus the number of qubits. The 164 CHAPTER 6. EXPERIMENTAL REALIZATION

complexity is determined by the number of quantum gates that were involved, the demands on the coupling network (experiments which only require nearest neighbor interactions are easier than those which need the entire or nearly the entire coupling network), and the number of qubits. The chart illustrates how our work relates to the work of other groups, both NMR and non-NMR. From the ﬁgure, it becomes clear that NMR is currently still the leading technology for implementing quantum algorithms, though over the past year, many 1-qubit implementations have been shown with non-NMR systems. It is also clear, that our experiments consistently pushed the envelope for state-of-the-art quantum computing experiments.

Figure 6.8: Overview of quantum computing experiments performed to date. The complexity of the experiment depends on the number of quantum gates involved, the demands on the coupling network, and the number of qubits. The numbers are the year published. The dotted ovals indicate experiments I was directly involved with, and are the focus of this chapter. 6.3. GROVER SEARCH ALGORITHM (3 QUBITS) 165

6.3 Grover search algorithm (3 qubits)

We now turn to the experiments performed in this thesis work, beginning with the demonstration of a 3-qubit Grover search algorithm.

6.3.1 Problem statement

In this work we studied the effect and cancellation of systematic errors in one-qubit gates. Aside from the well-understood scaling limitations due to the use of a high-temperature (almost random) system instead of a low-temperature (low entropy) polarized spin system, the crucial limitation in applying this method to implement larger quantum algorithms has been systematic errors in the quantum gates. These gates are implemented by applying pulses of radio-frequency (RF) electromagnetic fields of precise duration and phase, which are in practice highly inhomogeneous over the sample volume, causing the gate fidelity to be less than 95%. Producing a homogeneous

B1 field is difficult because of the sample geometry and the necessity of keeping the field transverse to the B0 field. If such systematic errors simply accumulated, these observations would imply that fewer than 90 gates applied to any one spin could ever be cascaded in these systems6. One technique which has been proposed for controlling errors in quantum gates is quantum error correction, but this is associated with a large overhead and is simply not practical given the current state-of-the-art of NMR techniques. Such techniques are in principle capable of controlling completely random errors, including irreversible decoherence phenomena. However, in this case the errors are systematic and might be easier to control provided a deeper understanding of their origin. We tested to what extent systematic errors can be controlled (1) by avoiding them through pulse sequence simplification techniques and (2) by letting them cancel out. The experiment consisted of the repeated application of the two main steps of Grover’s algorithm (the controlled phase flip and inversion about average, described in section 2.3.2) for three spins.

The three-qubit Grover algorithm can ﬁnd a “marked” element x0 out of an unsorted array with n = 8 elements in only two oracles queries (or functions calls), whereas a classical search would require 4.375 searches on the average. This algorithm requires a toffoli gate

6Assuming 90 gates with a ﬁdelity of 95% each, the total success rate is only 0.9590 0.01. ≈ 166 CHAPTER 6. EXPERIMENTAL REALIZATION

as well as several one-qubit gates, and the amplitude of the state x0 is expected to reach | i its ﬁrst maximum after two iterations, and oscillate as the number of iterations increases.

6.3.2 Experimental set-up

The first method to reduce the number of one- and two-qubit gates, used to realize any given n-qubit unitary operation, starts with a library of efficient implementations for often-used building blocks. For example, four equivalent realizations of az ˆ-rotation using an RF coil in the transverse plane, can be constructed (see section 4.2.2, or section 5.5). If only cnot and single-qubit gates were used for the implementation of the required toffoli gate, as described in [BBC+95], 70.5 π/2 pulses and 8 evolutions of 1/2J would be required. Using our simplification techniques and the circuit drawn in Fig. 2.6, this reduces to 19 pulses, 2 evolutions of 1/2J and 3 of 1/4J. The resulting pulse sequence is listed in Appendix B. The second method to reduce the complexity concerns the initialization of the qubits to the ground state. The thermal density matrix can be transformed into an effective-pure state via temporal labeling (see section 4.4.3). The original scheme involves the summation of 2n − 1 experiments each of which cyclically permutes all of the populations. The pulse sequence required to do this can be rather complex. Here, we reduce the number of experiments from seven to three by approximating the effective-pure state so that the expected variance of the 2n 1 population is only 7 % of the average value. Each of the three experiments is − also much simplified with respect to its complexity.

6.3.3 Experimental results

13 We selected C-labeled CHFBr2 for our experiments with JHC = 224 Hz, JHF = 50 Hz and JFC = 311 Hz. The scalar interaction with the Br nuclei is averaged out. The output − state of each spin i, 0 or 1 , can be determined from the phase of the signal induced in the | i | i RF coil after applying an Xi read-out pulse. In fact, the spectrum of the signal of any one spin suﬃces to determine the output state of all the spins given that they are in an eﬀective pure energy eigenstate and that they are all mutually coupled: each spectrum then contains only a single line, the frequency of which, combined with the knowledge of the Jij, reveals the state of the remaining spins (Fig. 6.9 (a), inset). We also reconstructed the complete output deviation (traceless) density matrix using quantum state tomography (Fig. 6.9 (a)). The agreement between experimental results and theoretical predictions is good, considering that about 100 pulses were used and that the systematic error rate exceeds 5% per 6.3. GROVER SEARCH ALGORITHM (3 QUBITS) 167

Figure 6.9: Experimental deviation density matrices ρexp for x0 = 1 0 1 , shown in magnitude with the sign of the real part (all imaginary components| i were| small),i| i| i after 2 (a) and 28 (b) Grover iterations. The diagonal elements give the population difference with respect to the average. The off-diagonal elements represent coherences between the basis states. Inset: The corresponding 13C spectra (13C was the least significant qubit). The receiver phase and read-out pulse are set such that the spectrum be absorptive and positive for a spin in 0 . | i

RF pulse (the measured signal loss due to RF ﬁeld inhomogeneity after applying Xi). This suggests that the systematic errors cancel each other out to some degree. We examined this in more detail in a series of experiments with increasingly longer pulse sequences, executing up to 28 Grover iterations (repeated executions of the two main steps of Grover’s algorithm).

6.3.4 Discussion

Theoretically, the probability of obtaining x0 oscillates as a function of the number of | i iterations k, reaching a ﬁrst maximum for k = (√N). Fig. 6.10 (a) shows that the O diagonal entry dx0 of ρexp oscillates as predicted but the oscillation is damped as a result of errors, with a time constant Td of 12.8 iterations. However, Td would have been smaller than 1.5 if the errors due to just the RF ﬁeld inhomogeneity were cumulative (Fig. 6.10

(a), solid line). Remarkably, after a considerable initial loss, dx0 decays at a rate close to 13 the C T2 decay rate (dashed line), which can be regarded as a lower bound on the overall error rate. A more complete measure to quantify the error and benchmark results is the relative error r = cρexp ρth 2 / ρth 2, where ρexp and ρth are the experimental and k − k k k 168 CHAPTER 6. EXPERIMENTAL REALIZATION

theoretical (traceless) deviation density matrices, and 2 is the 2-norm. Comparison of r k · k with c = 1 and c equal to the inverse of the signal loss (Fig. 6.10 (b)) reveals that signal loss dominates over other types of error. Furthermore, the small values of r with c > 1 suggest that x0 can be unambiguously identiﬁed, even after almost 1350 pulses. This is conﬁrmed | i by the density matrix measured after 28 iterations, which has a surprisingly good signature (Fig. 6.9 (b)). Given the error of > 5% per single π/2 rotation, all these observations demonstrate that substantial cancellation of errors took place in our experiments.

Figure 6.10: (a). Experimental (error bars) and ideal (circles) amplitude of dx0 , with ﬁts (dotted) to guide the eye. Dashed line: the signal decay for 13C due to intrinsic phase 13 randomization or decoherence (for C, T2 0.65 s). Solid line: the signal strength retained after applying a continuous RF pulse of the≈ same cumulative duration per Grover iteration as the pulses in the Grover sequence (averaged over the three spins; measured up to 4 iterations and then extrapolated). (b). The relative error r.

The error cancellation achieved was partly due to a judicious choice of the phases of the refocusing pulses, but a detailed mathematical description in terms of error operators is needed to fully exploit this effect in arbitrary pulse sequences. This difficult undertaking is made worthwhile by our observations. This conclusion is strengthened by a similar observation in another experiment [VYSC99]. Also, we believe that error cancellation behavior is not a property of the Grover iterations, because we found experimentally that the choice of implementation of the building blocks dramatically affects the cancellation effectiveness. In summary, more than 280 two-qubit quantum gates involving 1350 RF pulses were successfully applied, which far exceeds the number of gates that were applied in any previous 6.4. SCHULMAN-VAZIRANI COOLING (3 QUBITS) 169

NMR quantum computing experiment, and also appears to overcome the limitations of 90◦ pulses in NMR systems. Whereas the cancellation of systematic errors makes it possible to perform surprisingly many operations, the methods for simplifying pulse sequences reduce the number of operations needed to implement a given quantum circuit. This combination permitted the successful realization of Grover’s algorithm with 3 spins, up to 28 full iteration cycles, and brings many other interesting quantum computing experiments within reach.

6.4 Schulman-Vazirani cooling (3 qubits)

6.4.1 Problem statement

In this experiment [CVS01], we report the experimental realization of the key step of the Schulman-Vazirani scheme using liquid state NMR techniques by cooling down the spin temperature of one of three spins. The fundamental ideas of the Schulman-Vazirani cooling scheme have been introduced in section 3.1.3, but the actual scheme for three spins is shown here. Polarization enhancement experiments from a high-γ nucleus to a low-γ nucleus are common in NMR, however this experiment focuses on the polarization enhancement of three equivalent nuclei. In this case, eigenvalue conservation allows a maximum polarization enhancement factor of 3/2. The following procedure serves as a building block for the Schulman-Vazirani algo-

rithm (Fig. 6.11). Given three qubits a, b, and c with identical initial polarizations = 0,

the initial state xa xb xc , or for short xa xb xc , is one of the eight possible | i ⊗ | i ⊗ | i | i| i| i 1+0 3 1+0 2 1 0 states 0 0 0 , 0 0 1 ,..., 1 1 1 , with respective probabilities ( ) ,( ) ( − ), | i| i| i | i| i| i | i| i| i 2 2 2 1 0 3 ..., ( −2 ) . First perform a cnot operation on c conditioned on the state of b. The new state of the

three qubits is x x x = xa xb xb xc , where denotes addition modulo 2. Note | a0 i| b0 i| c0 i | i| i| ⊕ i ⊕ 20 that conditioned on xc0 = 0 , the polarization of b is now 2 (b is almost twice as cold | i | i 1+0 as before); conditioned on x = 1 , the polarization of b is 0 (b is at inﬁnite temperature). | c0 i | i However, overall, the polarization of b is still the same as before, 0. The polarization of a is of course also still 0. We then perform a NOT operation on c followed by a Fredkin gate (Table 1) with c as the control qubit. The result is that a and b are swapped if and only if x = 0 (and thus if and only if b has been cooled): x x x = x x x | c0 i | i | a00i| b00i| c00i | b0 i| a0 i| c0 i if x = 0 , and x x x = x x x otherwise. On average, a is thus colder than | c0 i | i | a00i| b00i| c00i | a0 i| b0 i| c0 i 30  before. The resulting polarization of a is = + ( ), where the higher order terms 2 O 170 CHAPTER 6. EXPERIMENTAL REALIZATION

are negligible, so the polarization of spin a is enhanced by a factor of 3/2.

Figure 6.11: A quantum circuit that implements the boosting procedure for Schulman- Vazirani’s algorithm. The controlled-swap operation has been replaced by an equivalent set of gates: two cnot’s and a toffoli.

6.4.2 Experimental set-up

The quantum circuit of Fig. 6.11, results in the unitary operation (with a the most signiﬁcant qubit) 0 1 0 0 0 0 0 0   1 0 0 0 0 0 0 0      0 0 1 0 0 0 0 0     0 0 0 0 1 0 0 0    U =   , (6.1)  0 0 0 0 0 1 0 0       0 0 0 1 0 0 0 0     0 0 0 0 0 0 1 0       0 0 0 0 0 0 0 1  which transforms the thermal density matrix as

3 1 Ia + Ib + Ic Ia + Ib IaIc IbIc . (6.2) z z z → 2 z 2 z − z z − z z where 2Iz = σz. The propagator U thus redistributes the populations in such a way that the highest populations are moved to states with a = 0 . This can be clearly seen by | i | i 6.4. SCHULMAN-VAZIRANI COOLING (3 QUBITS) 171

expressing the resulting density matrix in explicit matrix form:

1 0 0 0 0 0 0 0   0 3 0 0 0 0 0 0      0 0 1 0 0 0 0 0     0 0 0 1 0 0 0 0  1   ρf =   . (6.3) 2  0 0 0 0 -1 0 0 0       0 0 0 0 0 -1 0 0     0 0 0 0 0 0 -1 0       0 0 0 0 0 0 0 -3 

Because the density matrix remains in a diagonal state after application of each quantum gate in Fig. 6.11, the boosting procedure can actually be implemented using a simpliﬁed quantum circuit (see section 5.5): replacing each gate with a gate whose unitary matrix is correct up to phases preserves the transformation given by Eq. 6.2. Consequently, the Toﬀoli gate, for which the fastest known implementation takes on the order of 7/4J seconds

(taking all Jij to be J), can be substituted with a Toﬀoli gate correct up to phases — ≈ consisting of a 90 yˆ rotation of b when c is in 1 , followed by a 180 zˆ rotation of b when a is ◦ | i ◦ in 1 and a 90 yˆ rotation of b when c is in 1 — which takes only 1/J seconds. The actual | i − ◦ | i pulse sequence used in the experiment is given in Fig. 6.12. This sequence was designed by standard pulse sequence simpliﬁcation techniques supplemented by Bloch-sphere intuition. The resulting unitary operator is

0 1 0 0 0 0 0 0   1 0 0 0 0 0 0 0      0 0 -1 0 0 0 0 0     0 0 0 0 -1 0 0 0  ˜   U =   . (6.4)  0 0 0 0 0 1 0 0       0 0 0 -1 0 0 0 0     0 0 0 0 0 0 1 0       0 0 0 0 0 0 0 1 

A 2 mol % solution of C2F3Br in deuterated acetone was used as the homonuclear three- spin system, for its remarkable spectral properties: strong chemical shifts (0, 28.2, and 48.1 172 CHAPTER 6. EXPERIMENTAL REALIZATION

ppm, arbitrarily referenced) and large scalar couplings (Jab = 122.1 Hz, Jac = 75.0 Hz, − and Jbc = 53.8 Hz) combined with long relaxation times (T2’s 4-8 s). The experiments ≈ were conducted at 30.0◦C.

X 2 X 2 Y Z Y X 2 X 2 Z X 2 X 2 Y Z Y

X Z 1/2 X Z X Z 1/2 X Z

Y Z Y X 2 X 2 X 2 X 2 X 2 X 2 X 2

1/2J bc −1/2J ab 1/4J bc −1/2J ab 1/4J bc 1/2J ab

Figure 6.12: Pulse sequence to implement the boosting procedure. This pulse sequence is designed for molecules with Jab < 0 and Jac,Jbc > 0.

All pulses were spin-selective, and varied in duration from 1 to 3 ms. Hermite 180 and av90 [AV93] shaped pulses were employed for 180◦ and 90◦ rotations respectively, in order to minimize the eﬀect of the J couplings between the selected and non-selected spins during the pulses. Couplings between the unselected spins are irrelevant whenever those spins are along zˆ. Bloch-Siegert shifts (see section 5.2) were accounted for in the pulse sequence ± out of necessity. The duration of the entire sequence of Fig. 6.11 is about 70 ms.

6.4.3 Experimental results

The theoretical predictions for the spectrum of each spin after the boosting procedure can be derived most easily from Eq. 6.3, taking into account the sign and magnitude of the J- couplings. After a readout pulse on spin a, the four spectral lines in the spectrum of a should ideally have normalized amplitudes 1 : 2 : 1 : 2, compared to 1 : 1 : 1 : 1 for the thermal equilibrium spectrum (for spins b and c, the boosting procedure ideally results in normalized amplitudes of 0 : 1 : 0 : 1 and 1 : 0 : 0 : 1, respectively). So the prediction is that the − boosting procedure increases the signal of spin a averaged over the four spectral lines by a factor of 3/2, equal to the bound for polarization enhancement established in Section 6.4.1. The experimentally measured spectra before and after the boosting procedure are shown in Fig. 6.13. The proper operation of the boosting procedure is further validated via the experimentally measured density matrix (Fig. 6.14). 6.4. SCHULMAN-VAZIRANI COOLING (3 QUBITS) 173

2 2 2

1 1 1

0 0 0

−1 −1 −1 −100 0 100 −100 0 100 −100 0 100 2 2 2

1 1 1

0 0 0

−1 −1 −1 −100 0 100 −100 0 100 −100 0 100

Figure 6.13: Experimentally measured spectra of spin a (Left), spin b (Center) and spin c (Right), after a readout pulse on the corresponding spin, for the spin system in thermal equilibrium (Top) and after applying the boosting procedure (Bottom). The real part of the spectra is shown, and the spectra were rescaled in order to obtain unit amplitude for the thermal equilibrium spectra. Frequencies are in Hz with respect to the Larmor frequency of the respective spins.

6.4.4 Discussion

Clearly, the signal of spin a has increased on average as a result of the boosting procedure, and the relative amplitudes of the four lines are in excellent agreement with the theoretical predictions. The measured areas under the four peaks combined before and after polarization transfer have a ratio of 1.255 0.002. The spectra of spins b and c after the boosting ± procedure are also in excellent agreement with the theoretical predictions, up to a small overall reduction in the signal strength. The experimentally measured density matrices not only demonstrate that the boosting procedure exchanges the populations as intended, but also that it doesn’t signiﬁcantly a a excite any coherences. The experimentally measured Tr(ρf Iz )/Tr(ρiIz ) gives a polarization enhancement factor of 1.235 0.016, consistent with the enhancement obtained just from ± the peak integrals of spin a. The experimental implementation of the boosting procedure thus successfully increased the polarization of spin a. 174 CHAPTER 6. EXPERIMENTAL REALIZATION

3 3

0 0

−3 −3 000 000 111 111 111 000 111 000

Figure 6.14: Pictorial representation of the theoretical (left) and experimentally measured (right) density matrices, shown in magnitude with the sign of the real part (all imaginary components were very small).

Despite the excellent qualitative agreement between the measured and predicted data, the quantitative polarization enhancement of spin a is lower than ideally achievable. Given the absence of substantial coherences (Fig. 6.14), we attribute this suboptimal enhancement primarily to signal attenuation due to RF ﬁeld inhomogeneity and, to a lesser extent, due to transverse relaxation. The minor excitation of coherences is attributed mostly to incomplete removal of undesired coupled evolution during the RF pulses.

In summary, we have experimentally demonstrated the building block for the hyperpolarization procedure outlined by Schulman and Vazirani on a homonuclear three-spin system. However, the repeated boosting required in a much larger spin system would be counteracted by relaxation and other causes of signal decay, such as RF ﬁeld inhomogeneity. Also, it should be noted that when starting from thermal equilibrium at room temperature, the overhead in the number of nuclear spins required for the complete Schulman-Vazirani scheme is impractical (see section 3.1.3). However, for higher initial polarizations, this cooling scheme could be a most valuable tool. 6.5. ORDER-FINDING (5 QUBITS) 175

6.5 Order-ﬁnding (5 qubits)

6.5.1 Problem statement

This experiment [VSB+00] combines the quantum Fourier transform (QFT) with exponentiated permutations, demonstrating a quantum algorithm for order-finding. This algorithm has the same structure as Shor’s algorithm and its speed-up over classical algorithms scales exponentially. Up to this point, the quest for the experimental realization of quantum computers has culminated in the creation of specific entangled quantum states, with four qubits using trapped ions[SKK+00], and seven qubits[KLMT00] using liquid state NMR [GC97, CPH98b], and in the successful implementation of Grover’s search algorithm [CGK98, JMH98, VSS+00] and the Deutsch-Jozsa algorithm [CVZ+98, JM98, MFM+00] on two, three, and five qubit systems. However, a key step which remains yet to be taken is a computation with the structure of Shor’s factoring algorithm [Sho94a, EJ96], which appears to be common to all quantum algorithms that achieve exponential speed-up [CEMM98]. This structure involves two components: exponentiated unitary operations and the quantum Fourier transform (QFT). Implementing these components is challenging because they require not just the creation of static entangled states, but also precise dynamic quantum control over the evolution of multiple entangled qubits, over the course of tens to hundreds of quantum gates for the smallest meaningful instances of this class of algorithms. The evolution of the states is precisely where NMR quantum computers appear to have an exponential advantage over classical computers [SC99]. This experiment reports the experimental implementation of a quantum algorithm for finding the order of permutation; its structure is the same as for Shor’s factoring algorithm and it scales exponentially faster than any classical algorithm for the problem. We experimentally implemented the order-finding quantum algorithm to determine the order of a representative subset of all 4! = 24 permutations on 4 elements, including instances of each possible order. It can be proven that the best classical algorithm needs two queries of the oracle to determine r with certainty, and that using only one query of the oracle, the probability of finding r using a classical algorithm can be no more than 1/2. One optimal classical strategy is to first ask the oracle for the value of π3(y): when the result is y, r must be 1 or 3; otherwise r must be 2 or 4. In either case, the actual order can be 176 CHAPTER 6. EXPERIMENTAL REALIZATION

guessed only with probability 1/2. In contrast, the probability of success is 0.55 with only ∼ one oracle-query using the quantum algorithm on a single quantum computer. Depending on the measurement outcome, make a probabilistic guess r0 as shown also in Fig. 6.15. It is easy to verify that Pr[r = r] is 0.55, regardless of the probability distribution of r or π. 0 ∼ In fact, since in our implementation an ensemble of 1018 quantum computers contribute ∼ to the signal, our output data enables the order to be deduced with virtual certainty.

Figure 6.15: (Left) The probabilities that the measurement result m is 0, 1,..., or 7, given r (for an ideal single quantum computer). (Right) The optimal probabilities with which to make a guess r0 for r, given m.

6.5.2 Experimental set-up

We custom synthesized a molecule [GMS68] containing five 19F spins which served as the qubits (Fig. 6.16). When placed in a static magnetic field, each spin has two discrete energy eigenstates, spin-up, 0 , and spin-down, 1 , described by the Hamiltonian ~ωiIzi, | i | i where ωi is the transition frequency between the spin-up and spin-down states and Iz is thez ˆ component of the spin angular momentum operator. In this molecule, all five spins are remarkably well-separated in frequency, ωi, and are mutually coupled with a coupling

Hamiltonian of the form 2π~JijIziIzj (Fig. 6.16). The linewidths of the NMR transitions are 1 Hz, so the T2 quantum coherence times of the spins were at least 0.3 s. The T1 ∼ ∼ time constants were measured to be between 3 and 12 s. The five spins were prepared in an effective-pure state via the product operator approach (see section 4.4.6). We used 9 experiments (instead of the theoretically minimum of 7 experiments), giving a total of 45 product operator terms. The 14 extra terms were canceled out pairwise, using NOT (Ni) operations to flip the sign of selected terms. The 9 state preparation sequences were

C51C45C24N3,C14C31C53N2,C54C51N2,

C31C43C23N5,C21C52C45C34,C53C25C12N4, 6.5. ORDER-FINDING (5 QUBITS) 177

Figure 6.16: Structure of the pentaﬂuorobutadienyl cyclopentadienyldicarbonyliron complex, with a table of the relative chemical shifts of the 19F spins at 11.7 T [Hz], and the J-couplings [Hz]. A total of 76 out of the 80 lines in the 5 spectra are resolved.

C12C15C13C41,C32C13C25N4,C35C23N1. This effective pure state served as the initial state for the order-finding algorithm. The actual computation was realized via a sequence of 50 to 200 radio-frequency (RF) ∼ ∼ pulses, separated by time intervals of free evolution under the Hamiltonian, for a total duration of 50 to 500 ms, depending on π. The pulse sequences for the order-finding ∼ ∼ algorithm were designed by translating the quantum circuits of Fig. 2.11 into one- and two-qubit operations, employing the simplification methods from section 5.5. The transformation x y x πx(y) is realized by: | i| i 7→ | i| i

r = 1: P54 C35 P† C35 P34 (Pij rotates spin j by 90 aboutz ˆ if and only if spin i is • 54 ◦ 1 ). | i

r = 2: C35. •

r = 3: C32 C25 C32 C21 P14 C51 P† C51 P54 C21 P15 C41 P† C41 P45. (this sequence • 14 15 does the transformation πx(y) for y = 2 only; sequences for r = 3 that would work for any y are prohibitively long).

r = 4: C24 P34 P54 C35 P54. • Each transformation was tested independently to conﬁrm its proper operation. These pulse sequences were implemented on the four-channel Varian Unity INOVA spectrometer described in section 6.1. One channel served two qubits, using the methods from section 5.1, and its frequency was set at (ω2 +ω3)/2. The other three channels were set on the resonance of spins 1, 4 and 5. On-resonance excitation of spins 2 and 3 was achieved using phase- ramping techniques. All pulses were spin-selective and Hermite shaped. Rotations about thez ˆ-axis were implemented by adjusting the phases of the subsequent pulses. Unintended phase shifts of spins i during a pulse on spin j = i (transient Bloch-Siegert shifts) were 6 178 CHAPTER 6. EXPERIMENTAL REALIZATION

calculated and accounted for by adjusting the phase of subsequent pulses (see section 4.2.2). During simultaneous pulses, the eﬀect of these phase shifts was largely removed by shifting the frequency of the pulses via phase-ramping (section 5.3). The unintended evolution due to J-coupling during the selective pulses was compensated by slightly reducing the delay times7.

Upon completion of the pulse sequence, the states of the three spins in the ﬁrst register were measured and the order r was determined from the read-out. Since an ensemble of quantum computers rather than a single quantum computer was used, the measurement gives the bitwise average values of mi(i = 1, 2, 3), instead of a sample of m = m1m2m3 with probabilities given in Fig. 6.15. Formally, measurement of spin i returns Oi = 1 − 2 mi = 2Tr(ρIzi), where ρ is the ﬁnal density operator of the system. The Oi are obtained h i experimentally from integrating the peak areas in the spectrum of the magnetic signal of spin i after a 90◦ read-out pulse on spin i, phased such that positive spectral lines correspond to positive Oi. The theoretically predicted values of Oi (i = 1, 2, 3) for each value of r follow directly from the probabilities for m in Fig. 6.15. For reference, we also include the values of O4 and O5 (for y = 0; if y = 0, O4 and O5 can be negative): for the case r = 1 the Oi are 6 1, 1, 1, 1, 1; for r = 2 they are 1, 1, 0, 1, 0; and for r = 4 they are 1, 0, 0, 0, 0. For r = 3, the

Oi (i = 1, 2, 3) are 0, 1/4, 5/16, and O4 and O5 can be 0, 1/4 or 1/2, depending on y. ± ± The value of r can thus be unambiguously determined from the spectra of the three spins in the ﬁrst register. This was conﬁrmed experimentally by taking spectra for these three spins, which were in excellent agreement with the theoretical expectations.

In fact, the complete spectrum of any one of the ﬁrst three spins uniquely characterizes r by virtue of extra information contained in the splitting of the lines. For the spectrum of spin 1 the values of Oi given above indicate that for r = 1, only the 0000 line (see Fig. 6.17) is visible since spins 2 5 are all in 0 . Furthermore, this line should be absorptive and − | i positive since spin 1 is also in 0 . Similarly, for r = 2 the 0000, 0001, 0100 and 0101 lines | i are expected to be positive, and for r = 4 all 16 lines should be positive. Finally, for r = 3, the net area under the lines of spin 1 should be zero since O1 = 0, although most individual lines are expected to be non-zero and partly dispersive.

7The method from section 5.4 was anticipated but not fully developed at this time, and hence we adjusted the delay times experimentally. 6.5. ORDER-FINDING (5 QUBITS) 179

6.5.3 Experimental results

The experimentally obtained thermal equilibrium spectrum for spin 1 is shown in Fig. 6.17a. After the state preparation, only the 0000 line should be visible in the spectrum of each spin. The resulting data are remarkably clean, as illustrated for spin 1 in Fig. 6.17b.

a b − − − − − − − − − − − − − − − − 10 1010 1011 1110 1111 0010 0011 0110 0111 8 80 1000 1001 1100 1101 0000 0001 0100 0101 6 4 40 2 0 0 −50 0 50 −50 0 50

Figure 6.17: (a) The spectrum of spin 1 in equilibrium. The 16 lines are due to shifts in the transition frequency ω1 by J1j/2, depending on whether spin j is in 0 or 1 . In equilibrium, all the 32 states are± nearly equally populated, hence the 16 lines| i in each| i spectrum have virtually the same intensity. Taking into account the sign and magnitude of the J1,j, the 16 lines in the spectrum of spin 1 can be labeled as shown. (b) The same spectrum when the spins are in an effective pure state. Only the line labeled 0000 is present. All spectra shown here and in the following figure display the real part of the spectrum in the same arbitrary units, and were obtained without phase cycling or signal-averaging. A 0.1 Hz filter was applied. Frequencies are in units of Hz with respect to ω1.

Results for the four representative permutations are presented in Fig. 6.18. In all cases, the spectrum is in good agreement with the predictions, both in terms of the number of lines, and their position, sign and amplitude. Slight deviations from the ideally expected spectra are attributed mostly to incomplete refocusing of undesired coupled evolutions and to decoherence. 6.5.4 Discussion

The success of the order-finding experiment required the synthesis of a molecule with un- usual NMR properties and the development of several new methods to meet the increasing demands for control over the spin dynamics. The major difficulty was to address and control the qubits sufficiently well to remove undesired couplings while leaving select couplings active (we used a significant portion of the tools from section 5). Furthermore, the pulse 180 CHAPTER 6. EXPERIMENTAL REALIZATION

a − − − − − − − − b − − − − − − − − y π(y) 40 y π(y) 25 0 0 0 1 20 1 3 1 0 2 2 15 2 3 20 3 2 3 1 10 − − − − − − − −

− − − − − − − − 5 0 0 −50 0 50 −50 0 50

d − − − − − − − − − − − − − − − −

c − − − − − − − − − − − − − − − − y π(y) y π(y) 8 0 0 0 3 1 3 6 1 0 5 2 1 2 1 3 2 4 3 2 0 2 0 −5 −50 0 50 −50 0 50

Figure 6.18: Spectra of spin 1 acquired after executing the order-ﬁnding algorithm. The respective permutations are shown in inset, with the input element highlighted. The 16 marks on top of each spectrum indicate the position of the 16 lines in the thermal equilibrium spectrum. The spectra obtained for the r = 3 case was averaged 16 times to obtain better signal-to-noise. sequence had to be executed within the coherence time. Clearly, the same challenges need to be faced in moving beyond liquid state NMR, and we anticipate that solutions such as those reported here are useful in future quantum computer implementations, in particular in those involving spins, such as solid state NMR [Kan98], electron spins in quantum dots [LD98] and ion traps [SKK+00].

6.6 Shor’s factoring algorithm (7 qubits)

6.6.1 Problem statement

Shor’s factoring algorithm has been one of the key driving factors for studying quantum computing both theoretically and experimentally. An experimental demonstration of prime factorization has been considered the “Holy Grail” of small scale quantum computation: The 6.6. SHOR’S FACTORING ALGORITHM (7 QUBITS) 181

fundamental interest in this ﬁeld as well as the unprecedented complexity of the experiment would make an experimental demonstration of quantum factoring a new milestone. We have achieved this goal by factoring the number N = 15 into its prime factors [VSB+01]. Note that since this experiment nicely combines all tools that we have developed over the years, we go into detail describing this experiment.

Quantum circuit for N = 15

In section 2.3.3 we outlined Shor’s algorithm and pointed out that it fails when N is even or a prime power. Hence, the smallest number for which Shor’s algorithm can be meaningfully implemented is N = 15. For this choice, and following arguments from section 2.3.3, a total of 12 qubits would be required; m = 4 qubits for the second register and n = 2m = 8 for the ﬁrst. Valid choices for a are 2, 4, 7, 8, 11, 13 or 14. The general quantum circuit using the modular exponentiation method from section 2.3.3 is shown in Fig. 6.19.

x 7 H x H 6 x 5 M H x H 4 E 0 x 3 QFT H x A H 2 x 1 H x S. H 0 1 a a2 a4 a8 a16 a32 a64 a128

Figure 6.19: Outline of the quantum circuit to factor the number N = 15. The first register consists of n = 8 qubits, initialized to 0 . The second register consists of m = 4 and is initialized to 1 . We then apply Hadamard| i gates to all qubits in the first register, followed by the multiplication| i of ax mod 15. Next the Quantum Fourier Transform is applied, and finally the first register is measured.

n 1 As outlined in section 2.3.3, the values of a2 − are found by repeated squaring. Tab. 6.1 displays ax mod 15 for the valid choices of a as well as the resulting period and prime factors. From this table, we can see that the valid choices for a can be grouped into two categories:

If a = 4, 11 or 14, the numbers by which we need to multiply are equal to unity • (a2k mod 15 = 1) for all 1 k n. ≤ ≤ 182 CHAPTER 6. EXPERIMENTAL REALIZATION

x r ar/2 1 gcd 0 1 2 3 4 5 6 7 ± 2 1 2 4 8 1 2 ... 4 24/2 1 = 3, 5 3,5 ± 4 1 4 1 4 ... 4 42/2 1 = 3, 5 3,5 ± 7 1 7 4 13 1 7 ... 4 74/2 1 = 48, 50 3,5 ± a 8 1 8 4 2 1 8 ... 4 84/2 1 = 63, 65 3,5 ± 11 1 11 1 11 ... 2 112/2 1 = 10, 15 5,3 ± 13 1 13 4 7 1 13 ... 4 134/2 1 = 168, 170 3,5 ± 14 1 14 1 14 ... 2 142/2 1 = 13, 15 -,5 ±

Table 6.1: The table lists f(x) = ax mod 15 for all valid values of a. From f(x) we can determine the period r for each value of a which is used to determine ar/2 1. Finally, we calculate the greatest common denominator of ar/2 1 with 15 to determine± the prime factors of N = 15. ±

For a = 2, 7, 8 or 13, a2k mod 15 = 1 for all 2 k n. • ≤ ≤

Since multiplications by unity are trivial and can be left out two qubits, x0 and x1, in the first register would be sufficient to implement Shor’s algorithm independent of the choice of a. If we were to retain qubits xk for all 2 k n the final outcome of the ≤ ≤ first register would not change, in theory. However, experimentally these qubits decohere and may even be coupled to other qubits, introducing errors. These and other errors could create higher order periods which may be measured by the QFT. Thus, retaining x2 such that n = 3 constitutes a more meaningful test of the modular exponentiation and the QFT. Note however, in general for large N, it is unlikely that a2k mod N = 1 for any k less than 2n. Therefore, the dramatic reductions shown here are generally less likely, even though the procedure shown does present a standard methodology to reduce quantum circuits. Together with the m = 4 qubit in the second register, we used a total of seven qubits. The resulting simplified quantum circuit is outlined in Fig. 6.20. We chose one representative parameter a from each category: a = 11 (the “easy” case) and a = 7 (the “hard” case). The two controlled multiplications are done as follows: The first step of the modular exponentiation is to multiply 1 by a mod 15 controlled by x0 (qubit 3) which can be achieved by controlled-addition of (a 1) mod 15. Considering the least significant bit in the second − register is qubit 7, the controlled addition for a = 11 is done by CN34CN36 and similarly, for a = 7 by CN35CN36 (gates A and B of Fig. 6.21a and b). 6.6. SHOR’S FACTORING ALGORITHM (7 QUBITS) 183

x H 2 x 1 0 H x QFT H 0 1 a a2

Figure 6.20: Simpliﬁed quantum circuit for factorizing the number N = 15.

The second step is multiplication of y by a2 mod 15. For a = 11, a2 mod 15 = 1. This trivial multiplication has been left out as explained above. For a = 7, we multiply y by 2 a mod 15 = 4. This corresponds to shifting the bits of y by two positions, i.e. y3y2y1y0 7→ y3y2y1y000. In order to take the remainder of y3y2y1y000 divided by 15, we note that the

weight of y2 is now 16 and the weight of y3 is now 32. Furthermore, 16 mod 15 = 1 and

32 mod 15 = 2. Therefore, y3y2y1y000 mod 15 = y1y0y3y2. In words, multiplying y3y2y1y0

by 4 mod 15 boils down to swapping y3 with y1 and y2 with y0. Both swap operations must

be controlled by x1. Thus the resulting gates are fredkin gates which can be achieved by gates C,D,E and F, G, H of Fig. 6.21b (see section 2.2.5 for the construction of fredkin gates).

a b 1: 1: 2: 2: 3: 3: 4: 4: 5: 5: 6: 6: 7: 7: A B A B C D E F G H

Figure 6.21: (a) Detailed quantum circuit for the case N = 15 and a = 11. (b) Detailed quantum circuit for the case N = 15 and a = 7. The gates shown in dotted lines can be left out and the gates shown in dashed lines can be replaced by simpler gates (see following section). 184 CHAPTER 6. EXPERIMENTAL REALIZATION

Quantum circuit simpliﬁcation

The reductions described thus far lead to a reduction from 12 to 7 qubits and are rather speciﬁc to the choice N = 15. We now extend earlier techniques to simplify quantum circuits to describe a technique to reduce any quantum circuit using a quantum analogue to peephole optimization [ASU86]. Classical peephole optimization works by taking a small window and moving through compiled code while trying to optimize the existing code. Unreachable or redundant code is eliminated this way. Quantum circuits can be simpliﬁed in a similar manner as we demonstrate using our circuit:

1. The control qubit of gate C is 0 , so C is ’unreachable’ and can thus be left out; | i

2. Similarly, the control of F is 1 so it is always executed, and therefore replaced by a | i NOT on qubit 5;

3. We only measure the first register and trace over all other qubits. Therefore, any unitary evolution within the second register that leaves the first register in the same state commutes with the QFT thus plays no role in the final measurement of the first register and can consequently be left out. Gates H and E commute with the QFT and can be left out.

4. The targets of D and G are in a basis state, so the doubly controlled gates can be 2 2 implemented as CY24† CZ64CY24 and CY27† CZ57CY27 (CZij stands for a 90◦ zˆ rotation of j if and only if i is in 1 ), reducing their total length by almost 50%, assuming all | i coupling strengths are the same.

5. The refocusing schemes were kept as simple as possible. To this end, A was carried out after E.

We emphasize that none of the simplifications made use of prior knowledge of the outcome, i.e. the period r. Based on the dramatic reductions achieved here, we believe continued effort towards building quantum compilers benefits future quantum computer implementations immensely. 6.6. SHOR’S FACTORING ALGORITHM (7 QUBITS) 185

6.6.2 Experimental set-up

Molecule

We chose to work with the same ﬁve-qubit molecule but with the inner two carbon atoms 13 99% C enriched to obtain two additional qubits. The J-couplings, chemical shifts T1 and

T2 relaxation times for this molecule are shown in Fig. 6.22. Based on the J13C19F we believe the iron to be located as shown, different from the results in [GMS68] which were based on infrared spectroscopy. The assignment of qubits to spins involved a trade-off between sensitivity (the first register is measured and hence we want qubits 1, 2, and 3 to be 19F) and the demands of the coupling network (to execute the algorithm as fast as possible). We chose the following qubit assignment: Qubit 1 spin 1; qubit 2 spin 3; qubit 3 spin 7→ 7→ 7→ 5; qubit 4 spin 2; qubit 5 spin 4; qubit 6 spin 7; qubit 7 spin 6. The protons in 7→ 7→ 7→ 7→ the molecule broaden the lines of the nearest 13C spin, and thus were decoupled during the pulse sequence.

i ωi / 2π T1,i T 2,i J7i J6i J5i J4i J3i J2i 1 −22052.0 5.0 1.3 −114.3 25.2 6.6 14.5 −221 37.7 2 −4918.7 10.0 1.7 79.9 2.1 12.9 54.1 −5.7 3 489.5 13.7 1.8 12.5 3.9 18.6 −3.9 1 3 F 6 4 15186.6 2.8 1.8 41.6 19.4 59.5 5 F 7 C 5 25088.3 3.0 2.5 1.0 −13.5 F C C 6 4244.3 31.6 2.0 68.9 C 2 F Fe 7 −4519.1 45.4 2.0 F 4 CO C 5 H 5 CO

Figure 6.22: The seven spin molecule, along with its J-coupling constants, T1 and T2 relaxation times (in seconds), and chemical shifts (in Hertz) at 11.7 Tesla.

The synthesis of this molecule was quite complex, and is summarized in Fig. 6.23. Ethyl (2-13C)bromoacetate (Cambridge Isotope Laboratories, Inc) was converted to ethyl 2-ﬂuoroacetate by heating with AgF followed by hydrolysis to sodium ﬂuoroacetate using

NaOH in MeOH. This salt was converted to 1,1,1,2-tetrafluoroethane using MoF6 and was subsequently treated with two equivalents of n-butyl lithium followed by I2 to provide triflu- oroiodoethene. Half of the ethene was converted to the zinc salt which was recombined with 13 the remaining ethene and coupled using Pd(Ph3P)4 to give (2,3- C)hexafluorobutadiene. The end product was obtained by reacting this butadiene with the anion obtained from 186 CHAPTER 6. EXPERIMENTAL REALIZATION

treating [(p-C5H5)Fe(CO)2]2 with sodium amalgam [GMS68]. The product was purified with column chromatography, giving a total yield of about 5%. The sample, at 0.88 ± 0.004 mol% in perdeuterated diethyl ether was dried using 3-A molecular sieves, filtered through a 0.45 micron syringe filter, and flame-sealed in the NMR sample tube using three freeze-thaw vacuum degassing cycles.

H H H O AgF O NaOH O H 13C C H 13C C H 13C C - Br OEt F OEt F O Na+

H F BuLi/ H F BuLi/ - F MoF6 hexane hexane H 13C C F 13C C 13C C THF F F F F F F bp = -26C bp = -78C bp = -51C

F F 1) Zn o I F 13 I2 2) Pd(Ph P) C C 3 4 F 13C C F distill distill 13C C F F bp = 30C bp = 3-6C F F

F F Na(Hg) C 13C [Fe(Cp)(CO) 2] 2 F F 13 THF C C Fe F Cp CO CO Figure 6.23: Diagram of the synthesis of the 7 qubit molecule.

Temporal labeling

The effective pure state was created via 2 stage extension of the scheme of section 6.5. During the first stage the five 19F spins were made effective pure using the summation of nine experiments each consisting of a different series of Ni and Cij. These nine experiments 13 are repeated 4 times, each time with additional Cij and Ni involving the C spins such they also are made progressively pure. Hence, in the second set of nine experiments the first carbon (spin 6) is turned from I to Z. In the third set, the second carbon is turned 6.6. SHOR’S FACTORING ALGORITHM (7 QUBITS) 187

from I to Z. Finally, in the fourth set, the ﬁrst and second carbon are turned from II to ZZ. Table 6.2 summarizes the four sets of the labeling scheme.

set 1 set 2 C32C13C51C41 C32C13C51C41N6N7C61 C54C25C12N3 C54C25C12N3N6N7C61 C25C12C31C51N4 C25C12C31C51N4N6N7C61 C23C12C41C31N4N1 C23C12C41C31N4N1N6N7C15C61C15 C54C25C32C54 C54C25C32C54N6N7C61 C15C14C31C13 C15C14C31C13N6N7C62C63 C23C52C45C51 C23C52C45C51N6N7C61C62 C23C52C45C23C52N1 C23C52C45C23C52N1N6N7C64 C54C52C41N3N5N2 C54C52C41N3N5N2N6N7C61C62 set 3 set 4 C32C13C51C41N6N7C71 C32C13C51C41N7C71C67 C54C25C12N3N6N7C71 C54C25C12N3N7C71C67 C25C12C31C51N4N6C71 C25C12C31C51N4C71C67 C23C12C41C31N4N1N6C71C75 C23C12C41C31N4N1C71C75C67 C54C25C32C54C76N7C72 C54C25C32C54C72C67 C15C14C31C13N6C72C73 C15C14C31C13C62C63C76 C23C52C45C51N6C71C72 C23C52C45C51C71C72C67 C23C52C45C23C52N1C74 C23C52C45C23C52N1C74C67 C54C52C41N3N5N2C71C72 C54C52C41N3N5N2C71C72C67

Table 6.2: Summary of the 36 temporal labeling experiments on the spins, grouped into four sets. Time goes from left to right; Cij denotes a cnot operation ﬂipping spin j if spin i is in the state 1 ; Ni is a NOT and simply ﬂips spin i. | i

Summation of all the resulting 36 experiments and the application of a 180◦ pulse on qubit 7 results in the required effective pure state (first register initialized to 0 , and the | i second to 1 ). All of the CNij were chosen to minimize the time for the pulse sequence to | i minimize decoherence effects. On average, each series of CNij gates lasted 200ms in our experiments. The pulse sequence for each cnot gate is given by:

2 2 2 2 ¯ for Jij > 0 Xj1/4JijXi Xj 1/4JijXi Xj Yj (6.5) 2 2 2 2 for Jij < 0 Xj1/4 Jij X X 1/4 Jij X X Yj (6.6) | | i j | | i j which takes advantage of the fact that the thermal equilibrium state as well as the state 188 CHAPTER 6. EXPERIMENTAL REALIZATION

after each Cij is diagonal.

Spectrometer

Similar to the experiment described in section 6.5, we have more qubits to address than available RF channels. Hence, we used the same technique of exciting spins oﬀ-resonance, and keeping track of all timing events of the spectrometer. The frequency source of channel

1 was set to the resonance frequency of spin 1, ω1. Source 2 was set to (ω2 + ω3)/2, source

3 to (ω4 + ω5)/2 and source 4 to (ω6 + ω7)/2. We installed an additional frequency source, power ampliﬁer and power combiner for the proton decoupling.

We used Hermite 180 and Gaussian 90 pulse shapes for the 180◦ and 90◦ pulses respectively. Transient Bloch-Siegert shifts were compensated just as in the previous experiment of section 6.5. We also applied the frequency correction scheme from section 5.3. The resulting phase ramped pulses are precalculated and stored as separate pulse shapes. Coupled evolution during the pulses was unwound as explained in section 5.4. Gates A and G use a full seven qubit refocusing scheme which also refocuses T2∗ (see section 4.3.3). The other gates use simpliﬁed refocusing schemes, depending on which spins need to be refocused (we still tried to minimize T2∗ eﬀects however).

The resulting pulse sequence contains about 300 180◦ pulses and about 30 90◦ pulses, and is outlined in Fig. 6.24. The duration of the temporal labeling part is on the order of 200 ms, the modular exponentiation takes about 400 ms, and the QFT lasts about 120 ms - for a total of about 720 ms.

Read-out

Upon completion of this pulse sequence, the reduced density matrix of the three qubits in the ﬁrst register is expected to be:

ρ = wl lN/r lN/r (6.7) X | ih | l

We deduce the period r from the output spectra which is particularly simple here because in both cases (a = 11 and a = 7), the period is a power of 2. From this period, we obtain the factors of N = 15 by calculating the greatest common denominator of ar/2 1 with 15. ± 6.6. SHOR’S FACTORING ALGORITHM (7 QUBITS) 189

Figure 6.24: Pulse sequence for implementing Shor’s algorithm to factor N = 15 with a = 7, using seven qubits. The tall lines represent 90◦ pulses selectively acting on one of the seven qubits (horizontal lines) about positivex ˆ (no cross), negativex ˆ (lower cross) and positivey ˆ (top cross). Note how single 90◦ pulses correspond to Hadamard gates and pairs of such pulses separated by delay times correspond to two-qubit gates. The smaller lines denote 180◦ selective pulses used for refocusing, about positive (darker shade) and negative xˆ (lighter shade). Rotations aboutz ˆ are denoted by even smaller and thicker lines and were implemented with frame-rotations. Time delays are not drawn to scale. The vertical dashed black lines visually separate the steps of the algorithm; step (0) shows one of the 36 temporal averaging sequences. 190 CHAPTER 6. EXPERIMENTAL REALIZATION

Qubit 1 (Spin 1) Qubit 2 (Spin 3) Qubit 3 (Spin 5)

−200 −100 0 100 200 −50 0 50 −50 0 50

Qubit 4 (Spin 2) Qubit 5 (Spin 4)

−100 0 100 −50 0 50 Figure 6.25: Experimentally obtained thermal equilibrium spectra of the ﬁve ﬂuorine spins. The real part of the spectrum is shown, in arbitrary units. The frequencies are with respect to ωi/2π, in Hertz.

6.6.3 Experimental results

Fig. 6.25 show the thermal equilibrium spectra of the five 19F spins, corresponding to the five fluorine spins of the molecule of Fig. 6.22. Fig. 6.26 shows the thermal equilibrium spectra of the carbon spins. Each multiplet contains up to 26 = 64 spectral lines because each spin is coupled to all other spins. For spin 1, all lines are resolved8. For all other spins, some of the lines fall on top of each other, but this is of no consequence for our implementation of Shor’s algorithm because we are only interested in the integral of the spectrum: positive, negative, or zero. The thermal equilibrium was transformed into an effective-pure state via the temporal labeling experiments shown earlier. The resulting spectra for qubits 1, 2, and 3 are shown in Fig. 6.27. These spectra are obtained by summing 36 spectra, each obtained after

8NMR spectroscopists are continually amazed to see a spectrum with well-resolved 64 lines. 6.6. SHOR’S FACTORING ALGORITHM (7 QUBITS) 191

Qubit 6 (Spin 7) Qubit 7 (Spin 6)

−75 0 75 −150 0 150 Figure 6.26: Experimentally obtained thermal equilibrium spectra of the two carbon spins. The real part of the spectrum is shown, in arbitrary units. The frequencies are with respect to ωi/2π, in Hertz. performing one of the temporal labeling experiments. As expected only one line is visible in the spectrum indicating we have indeed obtained an eﬀective-pure state.

Qubit 1 (Spin 1) Qubit 2 (Spin 3) Qubit 3 (Spin 5)

−200 −100 0 100 200 −50 0 50 −50 0 50 Figure 6.27: Experimentally measured spectra of the ﬁrst register, similar to the thermal equilibrium spectra, after putting the seven spins into an eﬀective-pure state.

We then applied the pulse sequence for the two choices a = 11 and a = 7. The measured spectra upon completion of the “easy” case (a = 11) are shown in Fig. 6.28. Clearly, the lines of qubits 1 and 2 are up, and hence a positive integral is obtained. Since by convention a positive integral means the spin was in the state 0 we infer that both qubits 1 and 2 are | i in 0 . Qubit 3 has as many lines point up as down, and hence the integral of the spectrum | i is zero. This means, that this spin is in an equal mixture of 0 and 1 . Since qubit 3 is | i | i now the most signiﬁcant bit [Cop94], we conclude that the state of the ﬁrst register is in an equal mixture of 000 and 100 , or 0 and 4 in decimal notation. The periodicity is thus | i | i | i | i 4. Since this equals the inverted period 8/r we known that r = 8/4 = 2. We now obtain 192 CHAPTER 6. EXPERIMENTAL REALIZATION

Qubit 1 (Spin 1) Qubit 2 (Spin 3) Qubit 3 (Spin 5)

−200 −100 0 100 200 −50 0 50 −50 0 50 Figure 6.28: Experimentally measured spectra of the spins in the ﬁrst register (Bottom) and the ideally expected spectra (Top), after the completion of the “easy” case of Shor’s algorithm (a = 11). Positive and negative integrals of the spectra denote the qubit state 0 and 1 respectively. | i | i Qubit 1 (Spin 1) Qubit 2 (Spin 3) Qubit 3 (Spin 5)

−200 −100 0 100 200 −50 0 50 −50 0 50 Figure 6.29: Similar to the previous figure but for the “difficult” case (a = 7). gcd(112/2 11, 15) = 3, 5 - thus our prime factors are 3 and 5, as expected. ± Similar spectra obtained after the completion of the “hard” case are shown in Fig. 6.29. From these spectra we conclude that spin 1 is in the state 0 , and spins 2 and 3 are in an | i equal mixture of 0 and 1 . Hence, the first register is in an equal mixture of 000 , 010 , | i | i | i | i 100 , and 110 , or 0 , 2 , 4 , and 6 in decimal notation. This has a periodicity of 2, and | i | i | i | i | i | i hence r = 8/2 = 4. We now obtained gcd(74/2 1, 15) = 3, 5 which are again the correct ± prime factors of 15. Even after the application of the complex pulse sequence shown in Fig. 6.24, we have unambiguously determined the correct prime factors of the number N = 15. Nonetheless, there are discrepancies between the ideally expected spectra and the experimentally obtained ones, especially for qubit 3 in the “difficult” case. In fact, also the effective-pure state spectra already show small deviations from the expected data. We have 6.6. SHOR’S FACTORING ALGORITHM (7 QUBITS) 193

worked extremely hard to improve on these results, learned a lot in the process and made substantial progress since the first initial results. However, we were unable to improve on the shown results and we started to believe that something more fundamental was prevent- ing us from obtaining better data: decoherence. We tested this hypothesis by running a few extra experiments and based on those results we strongly believed that decoherence could be a substantial contributing factor to the quality of the data. We thus decided to attempt to model the effect of decoherence. We believe that being able to model the effect of decoherence could be an invaluable tool in estimating the feasibility of future experiments.

Decoherence model

The decoherence of coupled spin systems has been studied in NMR since the 60’s, beginning with a microscopic semi-classical description by Bloch, Wangness and Redfield [Red65, WB53]. These initial ideas have been worked out by many others later [VV78, Jee82, EBW87] using a so-called superoperator formalism. One can simulate decoherence based on knowledge of internuclear distances, chemical shift anisotropy tensors, and so forth, and of correlations of these mechanisms in time and space. Such calculations are in principle possible, but the required parameters are not always available. Furthermore, the superoperator matrix for seven coupled spins would be of size 47 x 47 and thus involve 268435456 degrees of freedom. This is more than most current computers can store in memory. Hence, simulating decoherence using this approach appears intractable. Instead, we wanted to derive the simplest and predictive decoherence model. The most straightforward assumption that one can make, and which allows us to use the phenomelog- ical Bloch equations to describe relaxation in terms of T1 and T2, is that the spins undergo independent relaxation. This is to say that there are no correlations between the relaxation of one spin and all other spins - each spin experiences random fluctuations of the local magnetic field which are not correlated with the fluctuations of the magnetic fields experienced by other nuclei. It is not clear a priori that this should be the case in our molecule, but at least such a model could serve as a good first idea of the impact of decoherence. Compar- ing the simulated spectra with the obtained spectra reveals whether the model carries any predictive value. We have already explained relaxation for single qubits in section 2.4 using the operator 194 CHAPTER 6. EXPERIMENTAL REALIZATION

sum representation. Our goal is to extend this model to multiple qubits9. Assuming independent relaxation, we have made the following observations that helped us develop such a model:

Generalized amplitude damping (GAD) error operators acting on diﬀerent spins com- • mute (by assumption) .

Phase damping (PD) error operators acting on diﬀerent spins commute (by assump- • tion).

GAD and PD error operators commute with each other. The error operators Ek (see • section 2.4) for GAD commute with the Ek for PD when applied to σx, σy, σz and σI and thus also when they act on any arbitrary density matrix ρ.

i t PD commutes with the ideal unitary time evolution e where = 0 + J (see • − H H H H section 4.3.1).

As a result, we need not simulate the GAD and PD processes simultaneously. Instead, we can simply simulate each one after the other. This allows us to ignore any cross-products of the Ek operators and hence makes the simulation much more manageable. Nonetheless, there are a few non-commuting processes:

GAD does not commute with the ideal unitary evolution under . • H GAD and PD do not commute with the ideal unitary evolution during the application • of RF pulses.

In order to retain a workable model, we approximated these processes as if they did commute, at least on short time scales10. Speciﬁcally, we we modeled a delay time of i t/~ duration t by ﬁrst applying the evolution e− H , followed by GAD acting on spin 1 for time t, followed by GAD acting on spin 2 for time t and so forth, followed by PD acting consecutively on the spins for a duration t. If the duration t is short enough such that only little GAD and PD acts on the spins, then this simulation should approximate the real evolution reasonably well. In our experiments, each two-qubit gate in the circuit consisted

9Lieven Vandersypen developed this model for seven qubits, and I wrote the code which simulates the decoherence effects during Shor’s algorithm. 10For example, we can model the evolution due to two non-commuting Hamiltonians to first order by simulating the effect of the two Hamiltonians one after the other. 6.6. SHOR’S FACTORING ALGORITHM (7 QUBITS) 195

Qubit 1 (Spin 1) Qubit 2 (Spin 3) Qubit 3 (Spin 5)

−200 −100 0 100 200 −50 0 50 −50 0 50 Figure 6.30: Comparison between our experimentally obtained spectra (Bottom) and the simulated spectra based on the decoherence model (Top), for the “easy” case of Shor’s algorithm (a = 11).

of several smaller delay segments due to the refocusing schemes. Each delay segment was

short compared with the T1 and T2 relaxation times so that we believe our approximation to be reasonably accurate. Similarly, an RF pulse of duration pw was modeled via an ideal single spin rotation, followed by GAD and PD acting no the spins consecutively. Here as well, the pulse width of the RF pulses was short compared to the relaxation times of the spins.

Due to these approximations, the complete pulse sequence of Shor’s algorithm (including the 36 state preparation sequences) was modeled in only a few minutes on four IBM

power3-II processors. Our model only required the measured T1 and T2 relaxation times. Hence, this model has no free parameters.

The simulated spectra obtained after running the simulation as well as the experimentally obtained spectra are shown in Figs. 6.30 and 6.31. In both cases, the simulation reproduced the main observed non-idealities rather convincingly, especially for qubit 3 in the “hard” case.

The good agreement between the simulated and experimentally obtained spectra lead us to believe that the assumptions underlying our simulations are largely valid. We attribute the remaining deviations between the data and the simulations to the other approximations

of our model, as well as other experimental imperfections such as RF and B0 inhomogeneity, imperfect calibrations, and uncomplete unwinding of the J-couplings during the RF pulses. 196 CHAPTER 6. EXPERIMENTAL REALIZATION

Qubit 1 (Spin 1) Qubit 2 (Spin 3) Qubit 3 (Spin 5)

−200 −100 0 100 200 −50 0 50 −50 0 50 Figure 6.31: Similar to the previous ﬁgure but for the “diﬃcult” case (a = 7).

6.6.4 Discussion

We have successfully implemented the simplest instance of Shor’s algorithm to factor the number N = 15 into its prime factors, demonstrating the proper operation of the algorithm. This is in itself already a remarkable achievement. In addition we developed a predictive decoherence model which simulates most of the observed non-idealities in the spectra. This is the ﬁrst NMRQC experiment, in which decoherence errors were dominant over other errors. Clearly, the demands of Shor’s algorithm are pushing the limits of the current state-of-the-art. Our developed decoherence model may be an valuable tool to assess the feasibility of future NMRQC experiments. Because of the good agreement between our data and the simulation, we can infer that the degree of unitary control was very high (i.e. our developed methods from chapter 7 worked well). This bodes well for related proposed implementations of quantum computers [Kan98, LD98].

6.7 Grover search using higher-order spins (2 qubits)

6.7.1 Problem statement

Almost all of the experimental demonstrations of quantum algorithms to date employed single or coupled two-level quantum systems. Higher order spin systems however could oﬀer a variety of interesting alternatives for implementing quantum algorithms. These range from allowing faster implementations of certain quantum operations [Fun01], over the possibility of using high polarization methods [VLV+01], to performing quantum computation without 6.7. GROVER SEARCH USING HIGHER-ORDER SPINS (2 QUBITS) 197

static magnetic fields [FG02]. Because of these intriguing applications, these systems have been studied in much detail. This includes the demonstration of classical logic [SMRK01, MSM+02] and the creation of pseudopure states for quantum computing [KF00, KSF01, Fun01]. These implementations have mostly been classical and the next step in testing higher- order spins consists of implementing quantum algorithms. However, higher-order spin systems are subject to the transient Bloch-Siegert shift because the different transitions are close in frequency (see section 5.2). This phase shift could lead to a significant source of errors in quantum algorithms if not corrected, and in fact it has already been necessary to correct for it in spin-1/2 implementations of quantum algorithms as we have seen in section 6.5 and 6.6. This Bloch-Siegert shift has been extensively studied for spin-1/2 particles, but not so for higher-order spins. Hence, in order to implement quantum algorithms using higher-oder spins, it is critical to correct the Bloch-Siegert shift for such systems. Here, we explicitly show how to generalize the techniques to calculate and compensate the Bloch-Siegert to multi-level systems. We then test our methods by implementing all four cases of the two-qubit grover search algorithm on a four level subsystem of a spin- 7/2 particle. This work constitutes, to our knowledge, the first full test of a quantum algorithm using higher-order spins. Besides generalizing our methods from spin-1/2 to multi-level systems, this work also required the development of two other key components. First, we designed an easily accessible universal set of discrete unitary gates for spin-3/2 systems. Second, to verify the correct operation of Grover’s search algorithm we need to reconstruct the resulting density matrix which fully describes the quantum system following the computation. We have explained the Bloch-Siegert shift in details in section 5.2 for spin-1/2 systems. The analysis for a spin-3/2 system is very similar. To recall, quadrupolar systems will serve as our multi-level system with the quadrupole Hamiltonian, including the Zeeman term, taking the form [Abr83, Sli96]:

2 e qQ 2 H = ~ωIz + (3I I(I + 1)) (6.8) 4I(2I 1) z − − where eq is the electric ﬁeld gradient and eQ the quadrupole moment. From Eq.6.8 and taking I = 3/2 we can see that there are four energy levels which are equally split due to the quadrupole coupling. The allowed transitions in such a system correspond to ∆m = 1 such ± 198 CHAPTER 6. EXPERIMENTAL REALIZATION

that three distinct transitions are allowed. These transitions can be selectively addressed at frequencies ω12, ω13, and ω23, as shown in Fig.4.11. Similar to the spin-1/2 case[VP77], the Hamiltonians associated with the transition at 12 12 frequency ω12 are given by H = 00 01 + 01 00 and H = i 01 00 i 00 01 . The x | ih | | ih | y | ih | − | ih | Hamiltonians associated with the other two transitions take a similar form. We can now calculate the Hamiltonian of a spin-3/2 system due to a shaped pulse applied at ωrf by modifying Eq. 5.3:

12 12 23 23 rf,k = ~ω1,k[cos(φk1)H /2 + sin(φk1)H /2 + cos(φk2)H /2 + sin(φk2)H /2 H x y x y 34 34 + cos(φk3)Hx /2 + sin(φk3)Hy /2] (6.9) where ω1,k is the RF field strength as before. The phase φk1 is the sum of the phase of the pulse and the linearly varying term with a rate ωrf ω12 divided by the pulse length. The − phases φk2 and φk3 can be calculated in similar fashion. Based on Eq. 6.9 we can calculate the resulting unitary matrix and thus the phase or Bloch-Siegert shifts between successive energy levels. We note that a pulse on-resonance with one of the transitions will induce a phase shift on all other transitions. But how do we compensate for these undesired phase shifts? This is an important question since phase shifts can result in significant errors if not compensated. Theoretically, the phase shifts can be compensated by applying transition selectivez ˆ-rotations. Experimen- tally, these have to be implemented viax ˆ andy ˆ-rotations which in turn induce Bloch-Siegert shifts on the other transitions. Hence this approach is not very practical. A better approach is to absorb the phase shifts into the phase of future pulses, similar to the approach used in previous experiments using spin-1/2 systems [VSB+01] and as described in section 5.2. This method works because changing the phase of subsequent pulses effectively changes the phase of the rotating frame which is equivalent to implementingz ˆ-rotations. However, for multi-level systems one may have to adjust the phase of subsequent pulses by different amounts for the different transition selective pulses, as we discuss next. Suppose our goal is to apply a selectivez ˆ-rotation on the first transition, resulting in iα/2 iα/2 the unitary transform with diagonal entries [e− , e , 1, 1]. In order to achieve this transform, we have to change not just the phase of subsequent pulses on the first transition by α but also the phase of subsequent pulses on the second transition by α/2. All transition − selectivez ˆ-rotations can be implemented in this manner and hence we can compensate all 6.7. GROVER SEARCH USING HIGHER-ORDER SPINS (2 QUBITS) 199

Bloch-Siegert shifts. The shown calculation and compensation methods are not limited to two or four level systems. In general, the methods can be extended to a spin-n/2 system or multi-level system in a straightforward manner by including the extra oﬀ-resonant terms into Eq. 6.9. In our system, we use the shown calculations for a spin-7/2 system and demonstrate the methods by implementing the two-qubit Grover search algorithm in this system.

6.7.2 Experimental set-up

In order to implement Grover’s algorithm, however, we need to ﬁrst construct a set of universal gates for a four level system. We have shown this explicitly in section 4.7.2, and Fig. 4.12. Recall that the controlled Z-gate as shown in Fig. 4.12 requires virtually no time because it simply involves transition selectivez ˆ-rotations which can be done implicitly. This in contrast to its implementation in a two-level system which requires a time equal 1/(2J) where J is the coupling strength between the two qubits. It is precisely this feature which may permit us to implement certain algorithms in less time than is possible using two level systems. Besides applying several of these unitary gates to implement a quantum computation and verifying the compensation of the Bloch-Siegert shift, we also want to verify the performance of the gates by reconstructing the density matrices following the computation. Recall the observables of a spin-3/2 system, as given in Eq. 4.80. From these, we can see that ρ21, ρ32, and ρ43 are directly obtainable by measuring the observables. The remaining entries are reconstructed using the appropriate unitary rotations listed in table 6.7.2. A similar list of operations can be designed for I > 3/2. We test all of our methods by incorporating them into the two-qubit Grover search algorithm. Grover’s algorithm is interesting as it requires polynomially fewer numbers of search iterations than classical computers [Gro97]. To recall, it can be easily described as follows: given some unknown function f(x) with the property f(x0) = 1 for one unique − x0 and f(x) = 1 otherwise, the task is to determine the special element x0. The Grover quantum circuit [VSS+00] can be translated into a sequence of transition selective pulses, using the scheme drawn in Fig. 4.12. The resulting steps are summarized in Fig. 6.32, illustrating how to obtain the pulse sequence from the two-qubit quantum circuit. Recall thatz ˆ-rotations require virtually no experimental time, making controlledz ˆ-rotations very fast. This comes at the cost of increased time for single qubitx ˆ ory ˆ-rotations, as evident 200 CHAPTER 6. EXPERIMENTAL REALIZATION

operation O1 O2 O3 I ρ21 ρ32 ρ43 iH12xπ/4 e− 0.5(ρ22 ρ11) n/a n/a iH23xπ/4 − e− n/a 0.5(ρ33 ρ22) n/a iH34xπ/4 − e− n/a n/a 0.5(ρ44 ρ33) iH23xπ/2 − e− iρ31 n/a iρ42 iH23yπ/2 iH34xπ/2 − e− e− iρ41 n/a n/a Table 6.3: Summary of the density matrix reconstruction procedure. The shown entries only reﬂect which part of the observable is used for the density matrix reconstruction. The entry n/a means that we do not use this measurement for our density matrix reconstruction. From these measurements, all terms in the density matrix can be reconstructed.

from the implementation of a hadamard gate.

6.7.3 Experimental results

All the experiments were performed at room temperature. In a high ﬁeld, high temperature approximation the distribution in the populations of the spins is approximately linear with energy, and hence the population distribution can be written as [KF00], P11 = A + ∆,

P10 = A + 2∆, P01 = A + 3∆, andP00 = A + 4∆ where A = N/4( 5Nhω/16πkT ), − ∆ = Nhω/8πkT ), N is the total number of spins, ω is the Larmor frequency. Therefore, the diﬀerence between two neighboring populations is a constant. This population distribution is not a suitable input state for quantum computation but can be transformed into something that is via temporal labeling [GC97].

Temporal labeling works by permuting through all populations except for P00. This is achieved by performing three experiments and summing their results: (1) Identity. (2)

Swap populations P11 and P10, followed by swapping P10 and P01. (3) Perform the same procedure as in (2) but in reverse order. Since each swap operation does not need to be a perfect swap (we can ignore phases because the input state is diagonal), we only need to apply one transition selective pulse. Therefore, this sequence is signiﬁcantly shorter when compared to the I = 1/2 case[GC97, CFH97]. We implemented the sequence of transition selective pulses at MIT using a Varian Unity Inova 11.7 Tesla spectrometer with a Varian H-X probe. The quantum system is a I = 7/2 cesium nucleus oriented in a nematic liquid crystal phase. The sample was prepared by mixing 50% by weight each of cesium pentadecaﬂuorooctanoate[BJS93] and D2O. Cesium 6.7. GROVER SEARCH USING HIGHER-ORDER SPINS (2 QUBITS) 201

(a)

H H H Qubit 1 f(x) f(0)

H H H Qubit 2

A B C

(b)

Transition −3/2 −> −1/2 Z Z 3 1

Transition 1/2 −> −1/2 H Z H Z H −2 −2 Transition 3/2 −> 1/2 Z Z 1 3

A B C

(c)

Z Y Z Y 2 1 2 1

H Y Y −2 2

Z Y Z Y 2 −1 2 1

H1 H2

Figure 6.32: Procedure to obtain the pulse sequence for a spin-3/2 system implementing a two-qubit Grover algorithm. Step A initializes the system into an equal superposition. Step B implements the oracle function call. Step C implements the inversion about average where f(0) ﬂips the phase of the 00 state. The letters in the boxes denote the axis of rotation, and the subscript denotes| thei angle of rotation in units of π/2. (a) Two-qubit quantum circuit diagram. (b) Outline of the pulse sequence for a spin-3/2 system, implementing the circuit shown in part (a). (c) Detailed pulse sequence implementing the hadamard gate.

has eight energy levels and a Larmor frequency of about 65MHz at the given ﬁeld strength.

We use the central four energy levels Iz = 3/2, 1/2, 1/2, 3/2, forming a spin-3/2 subsys- − − tem (see Fig.7.28a). We chose these energy levels as they have the longest coherence times.

The experiments were performed at a temperature of 27◦C representing the best trade-oﬀ between lineshapes, decoherence times and energy level splittings. The linewidths ranged from 3 to 11 Hz for the three transitions with the central transition being the narrowest. The transverse coherence times of these transitions were measured using the inversion recovery method, and are T1 = 60 70ms for transition 1 and 3, and T1 = 120 ms for transition − 2. The longitudinal coherence times T2 were roughly equal to the measured T1 times. The 202 CHAPTER 6. EXPERIMENTAL REALIZATION

1 x = 3 1 x = 2 o o | | ρ ρ | |

0 0 00 00 01 11 01 11 10 10 10 10 01 01 11 00 11 00

1 x = 1 1 x = 0 o o | | ρ ρ | |

0 0 00 00 01 11 01 11 10 10 10 10 01 01 11 00 11 00

Figure 6.33: Plot of the experimentally obtained absolute value of the deviation density matrices for x0 = 3, x0 = 2, x0 = 1 and x0 = 0. For visual clarity, each plot has been adjusted such that the minimum diagonal value equals to zero. splittings between the energy levels is about 7.5kHz. The three transitions were selected by using Gaussian shaped pulses of a 620 µs duration. For this length, the Bloch-Siegert shifts on the different transitions can be as large as tens of degrees which makes it obvious that they need to be corrected. Upon completion of the experiments, we reconstructed the full deviation density matrices, shown in Fig. 6.33. Ideally we expect the density matrix to be equal to ρ = x0 x0 . | ih | We can clearly identify x0 from the plots for all four cases, and observe that off-diagonal elements are small. Nonetheless, there is a substantial reduction of the expected signal intensity by about 30%, as evident by the x0 x0 being smaller than unity. The loss in | ih | signal strength leads to a large error of about 35%, defined as the 2-norm ρesp ρideal 2 || − || with ρideal being equal to zero except for the x0 x0 element, which is equal to unity, and | ih | ρexp is the experimentally obtained density matrix, adjusted by an identity portion such that no element on the diagonal is smaller than zero. We wanted to verify that most of the error (comprised of the reduced signal strength, 6.7. GROVER SEARCH USING HIGHER-ORDER SPINS (2 QUBITS) 203

and remaining off-diagonal elements) is not due to incorrect compensation of the Bloch- Siegert shift but instead other sources. Theoretically, omitting the correction for the Bloch- Siegert shift should dramatically increase the amount of off-diagonal elements, as indicated in Fig. 6.34. We repeated the experiments, but omitting the compensation for the Bloch- Siegert shift. The results are plotted in Fig. 6.35. The quality of the results is clearly reduced compared with Fig. 6.33, and with the error definition from above, the error is now larger by about 45%. From this we estimate that Bloch-Siegert shifts, if not corrected, have about as much impact on the quality of the results than all other sources of errors in our experiments.

1 x = 3 1 x = 2 0 0 | | ρ ρ | |

0 0 00 00 01 11 01 11 10 10 10 10 01 01 11 00 11 00

x = 1 1 0 1 x = 0 0 | | ρ ρ | |

0 0 00 00 01 11 01 11 10 10 10 10 01 01 11 00 11 00

Figure 6.34: Plot of the expected absolute value of the deviation density matrices for x0 = 3, x0 = 2, x0 = 1 and x0 = 0, without any compensation of the Bloch-Siegert shifts.

In conclusion, we have extended known methods to calculate and compensate for transient Bloch-Siegert shifts in higher-order spin systems by implementing the two-qubit Grover search algorithm on a spin-3/2 system. Similar to spin-1/2 implementations of quantum algorithms, we ﬁnd it necessary to compensate these phase shifts in higher-order spins systems as well. This work also presents the ﬁrst full test of a quantum algorithm using higher-order spins, requiring the design of a set of easily applicable universal unitary gates 204 CHAPTER 6. EXPERIMENTAL REALIZATION

1 x = 3 1 x = 2 o o | | ρ ρ | |

0 0 00 00 01 11 01 11 10 10 10 10 01 01 11 00 11 00

1 x = 1 1 x = 0 o o | | ρ ρ | |

0 0 00 00 01 11 01 11 10 10 10 10 01 01 11 00 11 00

Figure 6.35: Plot of the experimentally obtained absolute value of the deviation density matrices for x0 = 3, x0 = 2, x0 = 1 and x0 = 0, without any compensation of the Bloch- Siegert shifts. For visual clarity, each plot has been adjusted such that the minimum diagonal value equals to zero. and the reconstruction of the full deviation density matrices upon completion of the computation. Higher order spin system are an attractive test-bed for a wide variety of interesting experiments in the field of quantum information, and are also useful in simulating atomic physics experiments. We believe this experiment is just the first step towards a manifold of useful investigations of quantum effects using higher order spin systems.

A (yet) unexplained eﬀect

There is one effect that we have observed during our experiments on higher-order spins, which we briefly mention here. We are currently working on explaining this effect, but have not developed a useful simulation yet. The effect is not spectrometer related because we found the same effects using a Bruker AMS 500 MHz spectrometer, at the Whitehead Institute at MIT11.

11Thanks to Pete Carr for reserving the spectrometer time. 6.7. GROVER SEARCH USING HIGHER-ORDER SPINS (2 QUBITS) 205

Typically, when we calibrate our pulses, we set a particular power level and array the pulse width until we observe no signal. At this point we know that we are applying a

180◦ pulse. When sending two of such pulses back-to-back we should still observe no signal

because the pulses eﬀectively implement a 360◦ rotation. We have veriﬁed and tested this in all of our spin-1/2 molecules. However, when we started using our higher-order spin-7/2

system, this observation was no longer true. Even after careful calibration of a 180◦ pulse, we still observed a large signal (about 10% of the full signal strength) when applying two

180◦ pulses consecutively. This came as a surprise to us since we never observed this for spin-1/2 systems, and after further investigation, we observed the following:

Typical pulse widths were 600 µs (though we found similar results using pulses that • were as long as 2 ms) for a 180◦ pulse, and we used gaussian shaped pulses.

We focused on the central transition as it is the narrowest, giving rise to the best • SNR.

Applying two 180 pulses of the same phase gives rise to a signal that is about 10% • ◦ of the maximum intensity even after careful calibration of a single 180◦ pulse.

The same signal intensity is observed when applying the calibrated pulse of the same • duration but twice the power.

The same signal intensity is observed when applying the calibrated pulse of the same • power but twice the duration.

Applying two 180 pulses but of opposite phase leads to no signal, as we would usually • ◦ expect.

Applying two 180 pulses with the second being 90 degrees out of phase with respect • ◦ ◦ to the ﬁrst leads to half the signal strength we would get when both pulses are of the same phase. Considering this and the previous observation we plotted the resulting signal strength as a function of the phase of the second pulse and found an interesting relationship (see Fig. 6.36).

When both pulses are of the same phase, the second one has to be applied at less • power (about 10% less) in order to observe no signal. If we applied a third pulse, it would have to be at the same strength as the ﬁrst to get no signal. If we applied 206 CHAPTER 6. EXPERIMENTAL REALIZATION

fourth pulse, it would too have to be about 10% less in power than the ﬁrst (i.e. the same strength as the second), and so forth.

Applying two composite 180 pulses (each with the sequence XY 2X; see section 5.5) • ◦ did not signiﬁcantly improve the results, indicating that rf inhomogeneity is not a likely cause.

We found similar effects independent on which of the seven transitions we are using. • We found similar effects using different shaped pulses exciting the specific transition. • Even after characterizing the effects in detail, we have been unable to model this effect yet. We currently believe that it may be related to the orientation of the nuclei with respect to the magnetic field. For example, the nutation profile of a spin I = 1 system in a powder with no static magnetic field is not sinusoidal but rather follows a Bessel function [Pet75]. The Bessel function is obtained after integrating over all possible orientations of the spin. In this case, we do have a static magnetic field and the liquid crystal imposes restrictions on the orientation of the nuclei, but we believe there may be some residual effects. In either case, in our experiment we simply calibrated the 180◦ pulses as usual, and assumed the 90◦ pulses to simply require half the power.

0.12

0.1

0.08

0.06

0.04

0.02

signal strength of max. intensity 0

−0.02 0 1 2 3 4 phase of second pulse [π/2]

Figure 6.36: Plot of the signal strength as a function of the phase of the second pulse when applying two 180◦ pulses on one transition of a spin-7/2 system. Assuming the usual spin dynamics, we would expect the plot to be a horizontal line through the 0 point. 6.8. ADIABATIC QUANTUM OPTIMIZATION ALGORITHM (3 QUBITS) 207

6.7.4 Discussion

Higher-order spin systems could prove useful for a wide variety of interesting experiments in the field of quantum information, and are also useful in simulating atomic physics experiments (see section 7.5). We have provided the first full test of a quantum algorithm applied to such a higher-order spin system by designing a set of easily applicable universal unitary gates and reconstructed the full deviation density matrices upon completion of the computation. We believe this experiment is just the first step towards a manifold of useful investigations of quantum effects using higher order spin systems.

6.8 Adiabatic quantum optimization algorithm (3 qubits)

6.8.1 Problem statement

Ever since the discovery of Shor’s and Grover’s algorithms, the quest of finding new quantum algorithms has proved to be a formidable challenge. In fact, the challenge has been extended to include a more general goal: Instead of searching for algorithms that reduce the computation time on a quantum computer, how can quantum resources be used in general to solve hard and relevant problems? In other words, the requirement that the algorithm must outperform classical algorithms has been relaxed. One answer was recently provided by Farhi [FG00] and Hogg [Hog00] using adiabatic evolution. The scaling behavior of this optimization algorithm is unknown at this point and could well be dependent on the problem. If it does scale polynomially for at least one NP-complete problem (see [NC00] for more details on the complexity classes of algorithms), then all classical NP-complete problems could be solved in polynomial time. This is because we can convert one NP-complete problem into another efficiently classically, so we can convert only to the problem which has a polynomial solution on a quantum computer. Disregarding the potentially enormous applications or even uncertainty in its scaling behavior, this algorithm is still remarkable because it offers new insight into the usefulness of quantum resources, and could lead to the invention of other algorithms. Despite the numerous implementations of quantum algorithms, a demonstration of an adiabatic quantum algorithm thus far has remained beyond reach. The goal of this experiment [SvDH+03] was to provide the first experimental implementation of an adiabatic quantum optimization algorithm. NMR techniques are especially 208 CHAPTER 6. EXPERIMENTAL REALIZATION

attractive because several tens of qubits may be accessible, which is precisely the range that could be crucial in determining the scaling behavior of adiabatic quantum algorithms. Compared to earlier implementations of search problems [JMH98, VSS+00], this experiment is a full implementation of a true optimization problem, which does not require a black box function or ancilla bits. This experiment was made possible by overcoming two experimental challenges. First, an adiabatic evolution requires a smoothly varying Hamiltonian over time, but the terms of the available Hamiltonian in our system cannot be smoothly varied and may even have ﬁxed values. We developed a method to approximately smoothly vary a Hamiltonian despite the given restrictions by extending NMR average Hamiltonian techniques [RPW68, HW68]. Second, general instances of the optimization algorithm may require the application of Hamiltonians that are not easily accessible. We developed methods to implement general instances of a well known classical NP-complete optimization problem given a ﬁxed natural system Hamiltonian. We provide a concrete procedure detailing these methods. We then apply the results to our optimization problem which is known as Maximum Cut or maxcut [GJS76].

6.8.2 Experimental set-up

We have already outlined how the adiabatic algorithm works and how to discretize the continuous adiabatic algorithm in section 2.3.4. Let us repeat the discretization procedure here again for completeness sake. Ideally, we wish to slowly and smoothly vary the Hamiltonian from H(0) at time t = 0 to H(T ) at time t = T . Assume that this is not possible and instead we have to discretize the procedure. Suppose that the discrete time Hamiltonian H[m] be a linear interpolation from some beginning Hamiltonian H[0] = Hb to some ﬁnal problem Hamiltonian H[M] = Hp such that H[m] = (m/M)Hp+(1 m/M)Hb. The unitary evolution of the discrete algorithm − can be written as: U = U = e i((1 m/M)Hb+(m/M)Hp)∆t (6.10) Y m Y − − m m where ∆t = T/(M +1), and M +1 is the total number of discretization steps. The adiabatic limit is achieved when both T,M and ∆t 0. → ∞ → Full control over the strength of Hb and Hp is needed to implement Eq. 6.10. However, this may not necessarily be a realistic experimental assumption. We next show how the 6.8. ADIABATIC QUANTUM OPTIMIZATION ALGORITHM (3 QUBITS) 209

discrete time adiabatic algorithm can still be implemented when Hb and Hp cannot both be applied simultaneously and when they are both ﬁxed in strength.

When both Hb and Hp are ﬁxed, we can approximate Um to second order by using the Trotter formula exp((A+B)∆t) = exp(A∆t/2)exp(B∆t)exp(A∆t/2)+ (∆t2) [Tro59]. O Higher order approximations can be constructed if more accuracy is required.

Now suppose Hb and Hp are both constant. Since any unitary matrix is generated by an action iH∆t, we can increase the eﬀect of a constant Hamiltonian H by lengthening − the time ∆t. Thus, we can implicitly increase the strength of Hb and Hp, even when they are constant, by simply increasing the time during which they are applied. This technique also allows cases when the accessible Hamiltonians are not of the required

strength, for example when we are given Hb0 = gHb and Hp0 = hHp but still wish to

implement Hb and Hp. Using all of the described techniques, we can now write Um as:

iH [(1 m/M)∆t/2g] iH [(m/M)∆t/h] Um e− b0 − e− p0 (6.11) ≈ ◦ where A B = ABA. Each discretization step is of length (1 m/M)∆t/g + (m/M)∆t/h, ◦ − which is not constant when g = h. As an illustration consider Fig. 6.37. 6 m = 0 m = 1 m = 2 m = M g∆ t

strength h∆ t

time Figure 6.37: Illustration of Eq. (6.8). The shaded and clear boxes denote the strength and duration of the Hamiltonians Hb and Hp respectively.

In this experiment we choose ∆t = T/(M + 1) to be constant as we vary the number of discretization steps M + 1. This way, the total run time T increases with M + 1, allowing us to test the behavior of the algorithm when approaching one of the conditions for the adiabatic limit. Even when the discrete approximation is not close to the adiabatic limit, the implemented algorithm can often ﬁnd solutions using relatively few steps but lacks the guaranteed performance of the adiabatic theorem [Hog02]. Adiabatic evolution has been proposed to solve general optimization problems, including NP-complete ones. In this general setting, the algorithm can depend on the existence of a black box function or the usage of large amounts of workspace. Our goal here is to optimize 210 CHAPTER 6. EXPERIMENTAL REALIZATION

a hard natural problem in a way that avoids these difficulties. We first describe which problem we chose and later on explain why it does not require ancilla qubits. We found the maxcut problem to be a well-suited problem to demonstrate an adiabatic quantum algorithm because it allows a variety of interesting test cases. It also has applications in the study of spin glasses [Bar85] and VLSI design [CRS98], among others. The decision variant of the maxcut problem is part of the core NP-complete problems [GJS76], and even the approximation within a factor of 1.0624 of the perfect solution is NP-complete [AKMSP99]. The maxcut problem can be understood as follows. A cut is defined as the partitioning of an undirected n-node graph with edge weights into two sets. We define the payoff as the sum of weights of edges crossing the cut. The maximum cut is a cut that maximizes this payoff. By assigning either si = 0 or si = 1 to each node i, depending on its location with respect to the cut, the maxcut problem can be restated as finding the n bit number s that maximizes the payoff. An extension of the maxcut problem is to let the nodes themselves carry weights, which can be regarded as the nodes having a preference on their location. As an illustration consider a graph with three nodes as drawn in Fig. 6.38.

w2 w 2 23

w12 3 0 w3 1 1 w13 w1 Figure 6.38: Illustration of a graph consisting of 3 nodes and 3 edges. The edges carry weights w12, w13, and w23. When min(wij) = w23 as indicated by the length of the edges, the maxcut corresponds to the drawn cut. The solution is therefore s = 100 and also s = 011 due to symmetry. This symmetry can be broken by assigning the weights w1, w2, and w3 to the nodes.

The payoﬀ as a function of the cut deﬁned by s is given by

P (s) = wisi + si(1 sj)wij (6.12) X X − i i,j where wij are the edge weights, wi denotes the preference of the nodes to be on the 1 side n of the cut, and si is the value of the i-th bit of s, for 0 s 2 1. ≤ ≤ − The smallest meaningful test case of the maxcut problem requires 3 nodes and admits 6.8. ADIABATIC QUANTUM OPTIMIZATION ALGORITHM (3 QUBITS) 211

a variety of interesting cases by varying wi and wij. We aimed at two goals when choosing a representative set of weights. First, we wanted the minimum energy gap gmin to be smaller than the one for a 3-qubit adiabatic Grover search. Second, we wanted a resulting energy landscape with both a global and local maximum such that a greedy classical search would incorrectly find the local maximum half the time12. These goals are met by the choice w1 = w2 = w3 = 2, w12 = 2, w13 = 1, w23 = 3. The payoff function for this set of weights is P (s) = [0 6 7 7 5 9 8 6] where s = [000 001 010 011 100 101 110 111]. The global maximum lies at s = 101 so the answer on the quantum computer following measurement should be 101 , and not at the local maximum s = 110. | i In the quantum setting, this payoff function P (s) can be encoded into the Hamiltonian

Hp by rewriting Eq. 6.12 using Pauli matrices:

Hp = wi(I σzi)/2 + wij(I σziσzj)/2 (6.13) X − X − i i

n n where I is the 2 x2 identity matrix and σzi is the Pauli Z matrix on spin i. The identity matrices in the equation above only lead to an overall phase which cannot be observed, and hence they can be ignored. The diagonal values of Eq. 6.13 are equal to P (s). Because of

the direct encoding of P (s) into Hp no black box function or ancilla qubits are required, which makes this a full implementation of an optimization problem. Similar to Eq.(6.13), recall that the natural NMR Hamiltonian is given by:

= ωiσzi/2 + πJijσziσzj/2 + env (6.14) H − X X H i i

Despite the similarities, the spin-spin couplings of Eq.(6.14) are generally diﬀerent from a randomly chosen set of weights. Therefore, we require a procedure to turn the ﬁxed Jij

into any specified weights wij. This is achieved using traditional NMR refocusing schemes (see section 4.3.3). We have modified a refocusing scheme to effectively change the couplings to any arbitrary value. Consider the pulse sequence drawn in Fig. 6.39. Based on this scheme, we can derive the under-constrained system (α + β γ δ)J12 = w12,(α β γ + δ)J13 = w13, and − − − − (α β +γ δ)J23 = w23, which can be solved for positive α, β, γ, and δ such that Jij wij. − − → 12A greedy search is done by first choosing a random node configuration s, and then repeatedly moving to a new configuration s0 which differs from the previous configuration by only one node value si0 and which also has the highest payoff, until the payoff is maximized. 212 CHAPTER 6. EXPERIMENTAL REALIZATION

+ + + + 1 + + − − 2 + − − + 3 α β γ δ

Figure 6.39: Refocusing scheme to effectively change Jij into wij. The delay segments are of length α, β, γ, and δ. When all segments are of equal length, all of the couplings are effectively turned off.

The single weights wi are implemented by introducing a reference frame for each spin i which rotates about B0 at frequency (ωi wi)/2. In other words, we go oﬀ-resonance by an − − amount wi compared to the Larmor frequency ωi of the spin. In order to apply the single qubit rotations of our refocusing scheme on resonance, we apply the reference frequency shift only during the delay segment α, which we can always choose to be a positive value.

Thus, Hp is implemented by applying the refocusing scheme from Fig. 6.39 while going oﬀ-resonance during the delay segment α.

A full implementation of an adiabatic algorithm also requires a proper choice of Hb.

We choose Hb = σxi for several reasons. First, its highest two excited states are non- Pi degenerate. Second, it can be easily generated using single qubit rotations, and third, its highest excited state is created from a pure state with all qubits in the 0 state by applying | i a Hadamard gate on all qubits. We require the initial state to be the highest excited state of Hb because we are optimizing for the maximum value of Hp. A minimization problem would require an initialization into the ground state. The full adiabatic quantum algorithm is now implemented by ﬁrst creating the highest excited state of Hb. We then apply M + 1 unitary matrices as given by Eq. 6.11 and illustrated by Fig.6.37. Accordingly, from slice to slice, we decrease the time during which

Hb is active while increasing the time during which Hp is active. The resulting pulse sequence is illustrated in Fig. 6.40. Finally, we measure the quantum system and read-out the answer. 13 We selected C-labeled CHFBr2 for our experiments- the same molecule that we have already used in the experiments described in section 6.3.3. Similarly, we created an eﬀective- pure state by summing over three temporal labeling experiments.

In our experiments, we actually implemented 0.5Hp and 0.5887Hb instead of Hp and

Hb. This ensures that the error due to the 2nd order Trotter approximation is bounded 2 by i < 0.0356 where i is the contribution of the i-th undesired Pauli matrix. We pPi | | 6.8. ADIABATIC QUANTUM OPTIMIZATION ALGORITHM (3 QUBITS) 213 ¡ ¡ ¡ ¡ ¦ ¦ § § & ' * + 0 1 ¡ ¡ ¦ ¦ § § & ' * + 0 1 ¡ ¡ ¦ ¦ § § & ' * + 0 1 ¡ ¡ ¦ ¦ § § & ' * + 0 1 ¢ ¢ £ £ ¢ ¢ £ £ ¨ ¨ © © ! $ % , - 2 3 ¢ ¢ £ £ ¨ ¨ © © ! $ % , - 2 3 ¢ ¢ £ £ ¨ ¨ © © ! $ % , - 2 3 ¢ ¢ £ £ ¨ ¨ © © ! $ % , - 2 3 ¤ ¤ ¥ ¥ ¤ ¤ ¥ ¥ " # ( ) . / 4 5 ¤ ¤ ¥ ¥ " # ( ) . / 4 5 ¤ ¤ ¥ ¥ " # ( ) . / 4 5 ¤ ¤ ¥ ¥ " # ( ) . / 4 5 m=1 m=2 m=3 m=4 Figure 6.40: Illustration of the pulse sequence which implements our discrete adiabatic optimization algorithm with a total of four steps. The horizontal lines indicate the three spins, and time goes from left to right. The empty rectangles denote the Hamiltonian Hb[m]/2 while the black rectangles denote the 180◦ refocusing pulses. The arrows indicate the delay segments during which we go oﬀ-resonance with respect of the Larmor frequencies of the spins. The ﬁrst set of rectangles denote the initialization into the highest excited state of Hb.

also choose g so the applied RF ﬁeld does not heat the sample, and g h so Jij can be ignored when applying Hb. All of these choices result in a total experimental time that is

within the shortest T2 decoherence time. We reconstructed the traceless deviation density matrices upon completion of the experiments using quantum state tomography.

6.8.3 Experimental results

We executed this algorithm for several M (with wi and wij as listed above Eq.(6.13)). Since we chose ∆t to be constant, this meant increasing the run-time T of the algorithm. The reconstructed deviation density matrices are shown in Fig. 6.41. The plots clearly display the expected pure state 101 . The local maximum at s = 110 has a decreasingly small | i probability of being measured for increasing M.

6.8.4 Discussion

Simulations using Eq. 6.11 show that this optimization algorithm performs better for increasing M. From Fig. 6.41 this appears to be true, however, we wanted to verify whether this. For this purpose, we estimate the error of our obtained deviation density matrices compared with the ideal case of M = . Fig. 6.42a plots the trace distance as a function of ∞ M, using the same arbitrary scale as in Fig. 6.41. From the plot, we observe there exists an optimal run-time of the algorithm, corresponding to 0.226 seconds in our experiment. This optimal run time is in good agreement with the prediction of the simple decoherence model 214 CHAPTER 6. EXPERIMENTAL REALIZATION

1 M = 100 1 M = 60 0.75 0.75 | |

ρ ρ | | 0.25 0.25 0 0 000 000

111 111

111 000 111 000

1 M = 30 1 M = 15 0.75 0.75 | |

ρ ρ | | 0.25 0.25 0 0 000 000

111 111

111 000 111 000 Figure 6.41: Plot of the absolute value of the deviation density matrix for M = 100 (T = 374 ms), M = 60 (T = 226 ms), M = 30 (T = 115 ms), and M = 15 (T = 59.2 ms), adjusted by an identity portion such that the minimum diagonal value equals zero. The scale is arbitrary.

that we used in the previous section. Predicting the impact of decoherence has already provided invaluable insight into estimating errors in previous experiments [VSB+01], and we believe continued eﬀort towards understanding decoherence greatly beneﬁts experimental investigations of quantum systems.

In summary, we have provided the ﬁrst experimental demonstration of an adiabatic quantum optimization algorithm. We show a concrete procedure turning a continuous time adiabatic quantum algorithm into a discrete time version, even when certain restrictions apply to the accessible Hamiltonians. Our results indicate that there exists an optimal run- time of the algorithm which can be roughly predicted using a simple decoherence model. We believe this implementation opens the door to a variety of interesting experimental demonstrations and investigations of adiabatic quantum algorithms. 6.9. SUMMARY 215

1 1.4 ideal 0.8 exp 1.2 0.6 dec

101 1 ρ error 0.4 0.2 0.8 0.6 20 40 60 80 100 20 40 60 80 100 run−time [M] run−time [M]

Figure 6.42: Experimental performance of the adiabatic algorithm. Plot of the error as a function of M (Left). The error measure is the trace distance D(ρ, σ) = ρ σ /2 where σ is the traceless deviation density matrix for M = 400, approximating M| − | , and ρ equals the ideal expected (circles), the experimentally obtained (crosses), or→ the ∞ ideal expected traceless deviation density matrix with decoherence eﬀects (diamonds). The minimum error occurs at about M = 60 indicating an optimal run-time of the algorithm. A similar observation can be made when plotting the element 101 101 of the traceless density matrices from the previous ﬁgure as a function of M (Right).| ih |

6.9 Summary

This section clearly illustrates how nuclear spins in liquid solution present an excellent test bed to explore quantum computing experimentally. In fact, the series of experiments have clearly shown the proof-of-principle of quantum computing, answering one of my overall research goals. Speciﬁcally we have shown the following:

The demonstration of an eﬀective-pure state, simulating the dynamics of nuclear spins • at zero temperature.

The observation of a surprisingly large degree of error cancellation by applying a • record of 280 two-qubit gates.

The demonstration of coherent control of 7 coupled spins (the largest quantum com- • puter built to date) by implementing the simplest instance of Shor’s algorithm.

The demonstration of a predicative and workable model of decoherence for 7 coupled • spins.

The demonstration of a full quantum algorithm using a higher-order spin system. • 216 CHAPTER 6. EXPERIMENTAL REALIZATION

The demonstration of an adiabatic quantum optimization algorithm, paving the way • towards answering whether such algorithms scale eﬃciently.

I have played a central role in each of the experiments described in this section, except for the one of section 6.4 in which I played only a smaller supporting role. The main challenges of these experiments have been addressed via the developed tools from chapter 5 and by the careful choice of molecules. These techniques are beginning to ﬁnd applications in traditional NMR experiments, but it would be somewhat of a disappointment if none of the NMR techniques can be transfered to other implementations of quantum computers. In the next chapter we show several explicit examples of how else quantum computing and NMR quantum computing techniques could apply to other, perhaps more scalable implementations of quantum computers. Chapter 7

Applications of NMRQC

In this chapter we show how some of the techniques from NMR as well as the language of quantum computation can be used and transferred to other areas of physics and quantum computer implementations. To facilitate this discussion we begin by providing a brief description of superconducting loop (Section 7.1), and ion trap quantum computers (section 7.2). From this description, the potential use of NMR techniques in these systems becomes clear. We then explicitly show how the language of quantum computing can be used to design a sequence of pulses, which implement single-qubit rotations in an optical lattice quantum computer (section 7.3). This is followed by a section which describes how the tools of NMR quantum computation can be used to design better pulse shapes and composite pulses for a Josephson phase qubit (section 7.4). We also use the operator-sum approach to estimate the feasibility of our methods in the presence of tunneling this system. In the last section we begin to connect the ideas behind quantum optical phenomena with quantum computation, and explicitly demonstrate the electromagnetically induced transparency (EIT) eﬀect in NMR (section 7.5).

7.1 Superconducting qubits

We brieﬂy describe here how superconducting loops can be used for implementing quantum computers. The discussion explains how the ﬁve DiVincenzo criteria are met and follows closely the description of [MSS01] and the thesis of Audrey Cottet [Cot02]1.

1Almost all ﬁgures in this section were drawn by the author, based on the ﬁgures in [MSS01].

217 218 CHAPTER 7. APPLICATIONS OF NMRQC

Before commencing the physical description of the qubits, it is instructive to touch upon the Josephson eﬀect [Jos62] since it plays a critical role in the design of the qubits.

The Josephson eﬀect

It is well established that superconductivity constitutes a macroscopic quantum phenomenon. In a superconducting system, all electrons (or rather Cooper pairs) are all in a single quantum state described by a single-valued wavefunction:

ψ = √neiφ (7.1) with n denoting the number of Cooper pairs. Because ψ is single-valued, any change in φ over a closed circuit in the bulk semiconductor must change by integer multiples of 2π. Due to this discretization, the magnetic ﬂux Φ through a superconducting loop has to be quantized which plays a crucial role for Superconducting QUantum Interference Devices (SQUIDs), and the ﬂux qubit (described later).

Now suppose we have two pieces of superconducting material, separated by a thin layer of insulating material. This circuit is called a Josephson Junction. Assuming the wavefunction to be of the form of Eq. 7.1, and that the two pieces of superconducting material have diﬀerent phases, φ1 and φ2, it can be shown that a current ﬂows across the junction, given by

J = J0sin(φ) (7.2) where φ = φ2 φ1 and J0 denotes the maximum current density, related to the tunneling − rate from one layer to the other and the number of Cooper pairs. It is precisely this tunneling current which is used in the design for building a qubit using a superconducting loop consisting of multiple Josephson junctions. In another design for building a qubit, a single Josephson junction can also be used (see section 7.1.3).

There exits three different designs for implementing quantum computers using superconducting loops: the Josephson charge, the Josephson flux, and the Josephson phase qubits. We describe the charge qubit first. 7.1. SUPERCONDUCTING QUBITS 219

7.1.1 The Cooper pair box

Consider the circuit model of a Cooper pair box2 drawn in Fig. 7.1. The Cooper pair box consists of a Josephson junction with energy EJ , and capacitance CJ , connected to a superconducting island which can hold the Cooper pair charge Q, and a voltage source

denoted by Vg with a capacitance Cg.

¡ ¡ ¡ E j , C j ¡ ¡ ¡ Q

C g

V g

Figure 7.1: Circuit representation of a Cooper pair box. The Josephson Junction with Josephson energy EJ and a capacitance of CJ is connected to a superconducting island which in turn is connected to a gate voltage Vg with a capacitance Cg.

In addition to EJ , the box has another energy, namely the Coulomb energy of a Cooper pair: (2e)2 EC = (7.3) 2(Cg + CJ ) with e being the electronic charge. The Cooper pair box is our physical qubit, but at this point it is not clear what the physical form of the logical states 0 and 1 is. To facilitate this discussion, let us write | i | i the Hamiltonian of this system, which can be done using the charge representation.

Charge representation

We work in the basis n , which are the eigenstates of the operator nˆ associated with the | i number of Cooper pairs on the island in excess from the neutral charge state (usually simply called “the excess number of Cooper pairs”):

nˆ n = n n (7.4) | i | i 2After Fig. 1 in [MSS01]. 220 CHAPTER 7. APPLICATIONS OF NMRQC

where n Z. At energies lower than the BCS (Bardeen, Cooper, and Schrieﬀer) gap ∈ [BCS57], there are no quasiparticles present and n forms a complete basis for the states of | i the box. Hence, the total Hamiltonian of the box can be written using this basis. The total Hamiltonian (ignoring decoherence eﬀects) includes an electrostatic term and a Josephson term. The electrostatic term can be written as:

2 Hel = EC (nˆ ng) (7.5) − where ng = CgV g/2e is the reduced gate charge.

The second term of the total Hamiltonian takes into account the tunneling of a Cooper pair through the junction. Because nˆ relates to the number of excess Cooper pairs, the Josephson term takes the form:

E H = J n n + 1 + n + 1 n ! (7.6) J 2 X − n | ih | | ih |

Obviously, this term couples two consecutive charge states. The total Hamiltonian of the system is then just the sum of Hel and HJ :

E H(n ) = E ( n n n )2! J n n + 1 + n + 1 n ! (7.7) g C X g 2 X n | ih | − − n | ih | | ih |

The spectrum of this Hamiltonian is discrete and periodic in ng. We can solve for the eigenstates k of this system: | i H(ng) k = Ek k (7.8) | i | i where Ek are the energy eigenvalues of H(ng) and k is an integer. The eigenvalues of

H(ng) can be found numerically by diagonalizing H(ng) and restricting it to a subspace of several (about 10) charge states. The results are shown in Fig. 7.2 for EC = 10EJ . In the limit where the charging energy dominates the Josephson energy, EC EJ , and near the 1 degenerate point of ng = 2 mod 1, the system can be treated as a two-level quantum system because the higher energy levels are far away. 7.1. SUPERCONDUCTING QUBITS 221

1.5

1 C 0.5 E/E

0 k=3 k=2 k=1 k=0 −0.5 0 0.5 1 1.5 2 2.5 3 n g

Figure 7.2: The energy eigenvalues of a Cooper pair box for EC = 10EJ and working only with the lowest 10 charge states. If EJ was equal to zero, there would exits degeneracies at 1 ng = 2 mod 1. These are lifted even for small values of EJ , and hence the lowest two energy levels at such a degenerate point eﬀectively reduce to a 2-state quantum system.

The charge qubit

We have just shown how the Cooper pair box can be used to synthesize an eﬀective two- level quantum system in which only the n = 0 and n = 1 charge states contribute3. Let us rewrite this eﬀective Hamiltonian in a spin-1/2 notation:

1 1 H(ng) = Bz(ng)σz Bxσx (7.9) −2 − 2 with σx and σz denoting the Pauli matrices. The charge states n = 0 and n = 1 correspond 1 0 to the spin states = and = . In this notation, the charging energy EC | ↑i 0 | ↓i 1 (controlled by the gate voltage Vg) corresponds to an eﬀective magnetic ﬁeld about the zˆ-axis:

Bz(ng) = EC (1 2ng) (7.10) − The Josephson energy corresponds to an eﬀective magnetic ﬁeld about thex ˆ-axis:

Bx = EJ (7.11)

3The following discussion closely follows the arguments from [MSS01]. 222 CHAPTER 7. APPLICATIONS OF NMRQC

Let us rewrite the Hamiltonian from Eq. 7.9 yet again, for later convenience:

1 H(η) = ∆E(η)(cos(η)σz + sin(η)σx) (7.12) −2

1 where we define the mixing angle η = tan− (Bx/Bz). It determines the angle of the effective magnetic field in thex ˆzˆ-plane. The energy splitting between the energy eigenstates 2 2 is ∆E(η) = B + B = EJ /sin(η). The logical states 0 and 1 are defined as the p x z | i | i eigenstates of our effective spin-1/2 system:

η η 0 = cos + sin (7.13) | i 2| ↑i 2| ↓i η η 1 = sin + cos (7.14) | i − 2| ↑i 2| ↓i

Using this notation, the eigenstates of the qubit in the charge state basis are vectors that are located in thex ˆzˆ-plane. Clearly, the logical qubit states here are not simply the n = 0( ) and n = 1( ) | ↑i | ↓i charge states of the system but instead contain components from both charge states. In fact, at the degeneracy point the logical states are equal superpositions of the n = 0 and n = 1 charge states.

We can rewrite the eigenstates yet again, using a diﬀerent set of Pauli matrices, σ0, in the 0 and 1 basis: | i | i 1 H = ∆E(η)σ0 (7.15) −2 z This representation is useful when visualizing one particular method for implementing single-qubit rotations.

Single-qubit rotations and initialization

The initialization process is straightforward. We simply operate at low temperatures so the thermal energy is less than the energy splitting between the two energy levels, and let the system equilibrate into its ground state. Arbitrary single-qubit rotations can be implemented, using two diﬀerent methods, (1) by switching the gate voltage Vg, and (2) by applying an AC voltage to the gate. Method (1) has been used successfully by Nakamura [NPT99]. The qubit basis here is the charge basis and . Recall that the gate voltage Vg determines the mixing | ↑i | ↓i angle η. By setting the gate voltage far from the degeneracy point, the logical states are 7.1. SUPERCONDUCTING QUBITS 223

almost identical to the n = 0 and n = 1 charge states. After letting the system equilibrate, the system should be in the ground state, corresponding to . Then, we instantaneously | ↑i vary the gate voltage to the degeneracy point, so that we are applying H σx, leading to ∝ xˆ-rotations of the qubit, in the charge basis states and . Finally, we measure the | ↑i | ↓i excess charge on the island. Since we are free to vary the Hamiltonian to any point after equilibration (i.e. not just the degeneracy point), we can in fact apply any rotation in the xˆzˆ-plane, generating arbitrary single-qubit operations. Method (2) is rather similar to the NMR techniques from chapter 4). This method induces transitions in the 0 , 1 basis. Note that since the Hilbert space spanned by 0 | i | i | i and 1) depends on ng, the amplitude of the oscillating field should be small enough so | i that we can use the DC-value for ng to define a constant Hilbert space during the applied field. Interestingly, despite being a completely different physical qubit, the dynamics of single- qubit rotations using a charge qubit are very similar to those of nuclear spins in liquid solution.

The modiﬁed Cooper pair box

Despite being capable of producing single-qubit rotations, the ability to control the Joseph- son energy EJ would be beneﬁcial. For example, it is diﬃcult using the current Hamiltonian (Eq. 7.12) to implement a no-operation, requiring H = 0. We next discuss a design which allows the control over both terms of Eq. 7.12 independently. ¢ £ ¡ E j1 ¢ £ Φx ¡ E j2

C g

V g

Figure 7.3: Circuit representation of a split Cooper pair box. The two Josephson Junctions both have Josephson energies EJ /2 and EJ /2 and capacitances CJ1 and CJ2. The junctions are connected to a superconducting island which in turn is connected to a gate voltage Vg with a capacitance Cg. The ﬂux Φx induces a phase diﬀerence Φ0δ across the two junctions. 224 CHAPTER 7. APPLICATIONS OF NMRQC

The so-called split Cooper pair box4 is drawn in Fig. 7.3, and is identical to a SQUID, if the two junctions are identical. The electrostatic Hamiltonian is the same as that of a basic

Cooper pair box but instead with a total capacitance of Cg + CJ1 + CJ2. The Josephson Hamiltonian is now the sum of the two Josephson terms of the two junctions:

HJ = EJ cos(δ1)/2 EJ cos(δ2)/2 (7.16) − − b b This does not look at at all similar to Eq. 7.6. The reason for this is because we chose to work with the phase representation here instead of the charge representation of Eq. 7.6. The phase δ is simply the conjugate variable to n given by n = i∂/∂δ, much like x (space) − and p (momentum)b are quantum conjugate variables.b Hence, we couldb have writtenb Eq. 7.6 iδ as HbJ = EJ cos(δ) and translated this into the form of Eq. 7.6 via e = n + 1 n and − b | ih | iδ b e− = n n + 1 . For convenience, let us work in the phase representation and translate b | ih | back into the charge representation later. By using trigonometric identities, we can show that Eq. 7.16 changes to:

δ HJ = EJ cos( )cos(θ) (7.17) − 2b b where θ = (δ1 δ2)/2, and δ = δ1+δ2. The operator θ is associated with the superconducting − phase onb theb island,b whichb is relatedb b to the excess number of Cooper pairs n on the island. The resistance seen by the junctions is low compared with a quantum of resistance, and hence we can treat δ as a classical variable δ [IG99]. In the charge representation, the resulting Josephson Hamiltonianb takes the form:

H = E n n + 1 + n + 1 n (7.18) J J∗ X − n | ih | | ih |

EJ∗ HJ = σx (7.19) − 2 where EJ∗ = EJ cos(δ/2). Note that Eq. 7.19 emerges when making the restriction to two levels. This is the same form as Eq. 7.6, but now with EJ∗. Since δ = Φx/Φ0, we can tune

EJ∗ to any value. Note that Eq. 7.19 is not valid when both junctions do not have the same Josephson energies, but practically, it is possible to make both junctions very similar.

Furthermore note, that in general cos(θ) can be associated with σx, and similarly sin(θ) can

4After Fig. 4 in [MSS01]. 7.1. SUPERCONDUCTING QUBITS 225

be associated with σy:

cos(θ) σx (7.20) ↔ sin(θ) σy (7.21) ↔

The total Hamiltonian of the system is now equivalent to that of a basic Cooper pair box with the difference that we can tune EJ∗. Using this setup, we can tune Bz and Bx independently from each other, making it possible to obtain H = 0. This allows us great flexibility in implementing single-qubit gates. For example, we could tune the Hamiltonian to turn on only σz, and let the system equilibrate into the ground state 0 . Rotations about | i xˆ are then implemented by tuning the gate voltage and external flux Φx to turn on only

σx. If we wish to apply a ’no-operation’, we simply turn oﬀ the Hamiltonian.

Two-qubit gates

Incidentally, the split Cooper pair box also facilitates two-qubit gates compared with the regular Cooper pair box, as we discuss here. One possibility would be to connect two Cooper pair boxes directly, via a capacitor. This would result in a charge-charge interaction which can be described by an Ising-type coupling term σz σz. The advantage would be the fact that cnot gates can be implemented in a ⊗ straightforward manner (see f.ex. section 4.3.2). However, this method also suffers from a severe drawback. Since we wish to be able to control the coupling, one needs to be able to switch it on or off. This can be achieved via an external switch connected to the system, but this would introduce dephasing effects which can be substantial. Another proposal reducing the impact of decoherence [MSS99] involves connecting the

Cooper pair boxes with superconducting inductors, giving rise to σy σy couplings. The ⊗ drawback here is that we would have no active control over this coupling, and would be required to refocus undesired evolution, similar to NMR systems (see section 4.3.3). To circumvent the decoherence problem yet maintaining control over the couplings, an alternative design was proposed by Makhlin, et. al., [MSS99], shown in Fig. 7.4. In this set-up, all N qubits are connected in parallel to a common LC-oscillator mode. Without going into any detail, we provide a hand-waving argument for the nature of the coupling5. In Eq. 7.2 we have shown that the current ﬂowing through a junction is proportional to

5For a full treatment of the coupling terms, we refer to Fig. 5 and section II.C. of [MSS01]. 226 CHAPTER 7. APPLICATIONS OF NMRQC ¢ £ ¡ ¦ § ¤ ¥

¢ £ Φx1 ¡ ¦ § Φx2 ¤ ¥ L

V g1 V g2

Figure 7.4: A register of many charge qubits coupled by the oscillator modes in the LC- circuit.

EJ sin(φ) where φ is the superconducting phase across the junction. The LC-Hamiltonian can be understood as the magnetic energy of the inductor, LI2/2, where I is the sum- − mation of the current ﬂowing through all Cooper pair boxes. Hence the magnetic energy 2 should be proportional to [ EJ (Φxisin(φi))] . Let us also tune Φxi such that only two Pi junctions, i and j, produce nonzero EJ terms. Finally, from Eq. 7.21, we can identify sin(φ) i j as σy so that the overall coupling term should be proportional to σyσy. The requirement for this to be true is that the frequency of the oscillator has to be much larger than the typical frequencies of the qubits’ dynamics. In this regime, the coupling Hamiltonian takes the form: EJ (ΦxiEJ (Φxj) i j Hcoup = σ σ (7.22) − X E y y i

EJi i EJj j EJiEJj i j H = σx σx σyσy (7.23) − 2 − 2 − EL

Similarly, we cannot apply simultaneous single-qubit rotations on diﬀerent qubits. Nonethe- less, this form of coupling together with single-qubit rotations allows the implementations of arbitrary two-qubit gates, including the cnot gate.

Let us make a few remarks on this particular coupling technique: The frequency of the 7.1. SUPERCONDUCTING QUBITS 227

oscillator scales as 1/√N where N is the number qubits. Hence, the requirement that the frequency not drop below eigenenergies of the system limits the total number of qubits which can be coupled by a single inductor. Finally, the time required for a two-qubit operation is at least as long as the time for a single-qubit rotation. There exist a few other techniques to increase the coupling strength between qubits, as well as a method for adiabatic charge manipulation [MSS01] but we do not describe them here.

Measurement

Finally, we need to discuss the measurement procedure, because otherwise the computation is meaningless. The measurement of a Cooper pair box is done via a Single Electron Transistor (SET) capacitively coupled to the qubit6, as shown in Fig. 7.5.

The quantum computation is performed while setting the transport voltage, Vtr, equal to

zero, and choosing the gate voltage of the SET, Vx, to tune the transistor island away from degeneracy points. At low temperatures, Coulomb blockade effects suppress a dissipative current flow in the system. The transistor merely modifies the capacitances of the system.

A measurement is performed by setting Vx close to a degeneracy point, and applying a small transport voltage. Different charge states induce different voltages on the island of the SET and hence the current through the transistor depends on the charge configuration of the qubit. The state 0 induces a different current than the state 1 . The statistics of the resulting | i | i current can be analyzed by calculating the probability that m1 electrons have passed through the SET by time t1, and m2 electrons by time t2. From this we can calculate the probability of observing a given value of current, averaged over a time interval, ∆t. Interestingly, the measurement basis is not the same as the 0 and 1 basis of the qubit, and hence the | i | i measurement mixes the eigenstates of the qubit. Thus the time ∆t has to be smaller than

tmix. At the same time, ∆t has to be larger than tφ, the dephasing time during which phase coherence between the two energy eigenstates is lost, because the device is too sensitive

otherwise. Given that tφ < ∆t < tmix, and for a general superposition a 0 + b 1 , we | i | i 2 2 measure a current I0 with probability a , and with probability b we measure a current | | | | I1. For a rigorous derivation of the measurement dynamics, we refer to [MSS01]. We stress that this is not a unique measurement procedure and that other clever ideas

6After Fig. 14 in [MSS01]. 228 CHAPTER 7. APPLICATIONS OF NMRQC

V tr

m ¡ ¡ ¡ ¡ ¡ ¡ ¦ § ¤ ¥ ¡ ¡ ¡ ¦ § Φx ¤ ¥ n N £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ V x V g

Figure 7.5: The circuit consisting of the qubit and the SET used as measurement device. exist, for example in [VAC+02]. However, we do not describe them here and simply refer to the literature.

7.1.2 Flux qubits

The rf-SQUID

The charge qubit is operated in the regime EC EJ .A flux qubit can be designed for 7 the condition EJ EC . The simplest design of this system is an rf-SQUID , as shown in Fig. 7.6a, which is formed by one loop with one junction. The phase difference φ across the junction is related to the total flux Φ in the loop by φ/2π = Φ/Φ0 + integer where Φ0 = h/2e is the flux quantum (the flux through a superconducting loop plus the phase difference across a Josephson junction within this loop has to equal integer units).

Furthermore, let us apply an external ﬂux Φx to the system. The Hamiltonian of this system is the sum of the Josephson junction energy (as was the case for the charge qubit), the charging energy, and magnetic contributions:

2 Φ (Φ Φx) 2 H = EJ cos 2π ! + − + EC n (7.24) − Φb0 b 2L

The first term is the Josephson energy in the phase representation. The second term follows from the energy of the inductor LI2/2 where I is the current in the loop. This current − 2 results in a flux through the loop given by I = (Φ Φx) /L). Note that Φx is externally − 2 applied and hence only the flux (Φ Φx) is created by the current I. The last term, EC n − is similar to the charging energy of the charge qubit, but here the term has been simplified

7After Fig. 9 in [MSS01]. 7.1. SUPERCONDUCTING QUBITS 229

and in the end does not contribute to the system dynamics. The gate charge ng does not play a role here because it can be screened by charge transfer through the loop. Hence the only term that is left is n. However, in this setup it no longer has to be integer values due to screening eﬀects and thus becomes a classical variable.

¤ ¤ ¥ ¥ ¡ ~ ¢ £ ¤ ¤ ¥ ¥ ¡ ¢ £ Φx ¤ ¤ ¥ ¥ Φx ¡ Φx ¢ £

(a) (b)

Figure 7.6: Two flux qubit designs. (a) The rf-SQUID, consisting of a superconducting loop and one Josephson junction, form the simplest flux qubit. (b) An improved design in which the flux Φ˜x controls the effective Josephson coupling constant of the rf-SQUID.

2 2 When the self-inductance of the loop is high, i.e. when βL = EJ /(Φ0/4π L) is larger

than but close to 1, and when the externally applied ﬂux Φx is near Φ0/2, the ﬁrst two terms in the Hamiltonian form a double-well potential. Fig. 7.7 plots the double-well potential for

two values of Φx. At low temperatures, only the lowest two states in the wells contribute, and it is these states which form the basis states of the qubit.

Suppose we tune Φx away from Φ0. In this case, the double-well is tilted, as indicated by Fig. 7.7. There are two energy levels, one whose wavefunction is localized on the left-hand side, and another with higher energy whose wavefunction is localized on the right-hand side. The two energy levels correspond to different flux values Φ, which in turn correspond to different current directions - one going clockwise and the other counterclockwise. We let the logical states 0 and 1 correspond to clockwise and counterclockwise current flows | i | i through the system (similar to a nuclear spin pointing up or down in an static magnetic field). As paradoxical as it seems, an equal superposition of 0 and 1 then corresponds to | i | i current flow in both directions at the same time!

Suppose we now tune Φx = Φ0/2. In this case, we still have two diﬀerent energy levels in the system because of the ﬁnite tunneling probability between the wells, giving rise to two 230 CHAPTER 7. APPLICATIONS OF NMRQC

0.11 Φ /Φ =0.5 x 0 Φ /Φ =0.45 0.1 x 0

0.09

0.08

0.07

0.06

0.05

Potential energy 0.04

0.03

0.02

0.01 0 0.2 0.4 0.6 0.8 1 φ

Figure 7.7: Plot of the potential energy of a ﬂux qubit for two values of Φx. The lowest two energy levels of the wells form the basis of the qubit. wavefunctions that both have the property of being equally localized in both wells. In fact both energy levels correspond to an equal superposition of 0 and 1 . Fig. 7.8 illustrates | i | i how the eigenstates change as a function of ﬂux Φx.

−+

Φx

Figure 7.8: Sketch of the energy eigenvalues as a function of Φx. Far away from the avoided level crossing, the eigenstates correspond to current ﬂowing either clockwise or anti-clockwise. At the avoided level crossing, the eigenstates are equal superpositions of the two current ﬂows.

From the discussion in the previous two paragraphs, we can now intuitively explain the

Hamiltonian of the system in the Pauli matrix notation. If we are at the point Φx = Φ0/2, the system is in an equal superposition of both states, and hence the Hamiltonian can be 1 written in the form H = 2 Bxσx where Bx describes the tunneling amplitude between the 7.1. SUPERCONDUCTING QUBITS 231

wells. It depends on the height of the barrier and thus on EJ . If we are at a point away from Φx = Φ0/2, but still maintain a double-well potential, the system is either in the state where current ﬂows clockwise or counterclockwise, and hence the Hamiltonian takes the 1 8 form H = 2 Bzσz where Bz relates to the asymmetry of the well .

The modiﬁed rf-SQUID

Similar to the charge qubit, the rf-SQUID conﬁguration does not permit individual tuning

of Bx and Bz, and hence the Hamiltonian can never be turned off. This problem can be circumvented using the design shown in Fig. 7.6b. This modified set-up is virtually the same as for the modified Cooper pair box: The single junction is replaced by a loop with

two junctions (SQUID) and an additional externally control ﬂux Φx threading the SQUID. Hence the same arguments as before apply in deriving the Hamiltonianf of the Josephson

junctions in this set-up, leading to the result that Φx now controls the eﬀective Josephson f energy EJ . This control parameter lets us tune Bx independently from Bz, similar to the split Cooper pair box model.

Single-qubit rotations and initialization

The initialization process and single-qubit rotations are virtually identical to the methods discussed for charge qubits. The initialization is done by letting the system equilibrate into

its ground state. Single-qubit rotations can be achieved by tuning the strengths of Bx and

Bz, or by NMR-type techniques in which we let the applied ﬂux Φx oscillate at frequencies corresponding to the energy diﬀerence between the ground and excited states.

The 3-junction loop

One of the criteria for the design of a flux qubit was a high a self-inductance of the loop, which is only possible in large loops, making the system very susceptible to noise (stray magnetic fluxes are picked up by the loop). To overcome this difficulty, [MOL+] and [OMT+99] proposed to use a smaller superconducting loop with three or four junctions, respectively. Let us begin by discussion the design from Fig. 7.9a9. Since these circuits have a very low inductance, the current in the loop generates only a small flux so that the total flux remains

close to the externally applied value, Φ = Φx. Due to the phase quantization requirement, 8There still exists a small tunneling probability but we have ignored this here. 9After Fig. 10 a and b in [MSS01]. 232 CHAPTER 7. APPLICATIONS OF NMRQC

¢ ¢ £ 1 £ 2 ¢ ¢ £ £

¢ ¢ £ £ ~ Φ ~ ¦ § ¦ § EJ,φ ¦ § ¦ § 3 ~ ~ EJ (Φ)

(a) (b)

Figure 7.9: Two 3-junction ﬂux qubit designs. (a) The 3-junction design by Mooij reduces the inductance of the system and hence lowers the coupling of the circuit to the environment. (b) A multi-junction design by Orlando where EJ is controlled by the application of an external ﬂux Φ. f e

the three phases across the junctions are constrained by φ1 +φ2 +φ3 = 2πΦ/Φ0. This leaves

φ1 and φ2 as independent dynamical variables. Thus, the Josephson energies lead to the following energy potential:

U(φ1, φ2) = EJ cosφ1 EJ cosφ2 E˜J cos(2πΦ/Φ0 φ1 φ2) (7.25) − − − − −

For EJ /EJ > 0.5, a double-well potential is formed within each 2π x 2π cell in the phase f plane. The optimal value of E˜J /EJ 0.7 0.8 leads to high barriers between different ≈ − cells, but tunneling is still possible within each cell. For these values of E˜J , the potential along the direction where φ1 = φ2 forms a double well. The potential along the direction from one minimum to the nearest-neighbor minimum in a different unit cell also forms a double well, but with a higher barrier. Thus, tunneling is prevented between different unit cells. These directions are indicated in the contour plot of Fig. 7.10 by the full and dashed lines, respectively. The two lowest states in the wells form a two-state quantum system.

Similar to the basic rf-SQUID, the external flux Φ affects the asymmetry of the well, and thus the Bz term in the effective two-level Hamiltonian. However, E˜J cannot be manipulated in a straightforward manner. We can circumvent this problem by replacing the third junction with a SQUID (Fig. 7.9b), just as has been done for the charge qubit, and the first design of the flux qubit. 7.1. SUPERCONDUCTING QUBITS 233

3.5

2.5 π

/2 2 2 φ

1.5

0.5

0.5 1 1.5 2 2.5 3 3.5 4 φ /2π 1

Figure 7.10: Contour plot of the potential energy U(φ1, φ2) for Φ/Φ0 = 0.5 and E˜J /EJ = 0.7. The nearly circular shapes enclose maxima while the hourglass-shaped contours enclose two minima. The solid line indicated the direction along which the potential is double-well- shaped. The dashed line indicates the direction from one minimum to the next nearest- neighbor minimum in a diﬀerent unit cell. 234 CHAPTER 7. APPLICATIONS OF NMRQC

Two-qubit gates

Several different proposals have been made to inductively couple flux qubits. One proposal is indicated by the dashed line of Fig. 7.11 which inductively couples the fluxes and currents in the lower portion of the qubit10. Essentially, the flux through the loop in qubit 1 depends on whether the flux of qubit 2 is pointing “up” or “down”. Since the fluxes through these loops control the barrier height of the potential wells, this gives rise to the coupling of the 1 2 1 2 form σxσx. We can also place the coupling loop around the top loops, giving rise to σz σz couplings. The loops from Fig. 7.9a and b can also be inductively coupled. We mentioned earlier that the currents in the loop induce a negligible flux, but as it turns out, it is this small amount that couples the qubits when placed in an external coupling loop. Here again, we can couple different loops of the two qubits, and produce couplings of different forms. If we couple the main loops (the loop that is threaded by Φ and not Φ),˜ we can create a coupling 1 2 of the form σz σz because flux changes affect Bz. If we coupled the main loop of one qubit 1 2 with the SQUID loop of the other we can obtain couplings of the form σxσz . The fluxes can be “transported” much better by building an extra superconducting loop, which can be either always turned on, or externally controlled by adding Josephson junctions into the coupling loop [MOL+].

L osc

C osc ¡ ¡ ¤ ¥ ¤ ¥ ¡ ¡ ¤ ¥ ¤ ¥ ¢ ¢ £ £ ¦ § ¦ § ¢ ¢ £ £ ¦ § ¦ §

Figure 7.11: Coupling ﬂux qubit in two diﬀerent ways. The qubits can be inductively coupled via an external superconducting loop (dashed line) or the coupling is provided by an external LC-circuit (solid line), similar to the coupling for charge qubits.

The coupling design indicated by the solid line11 in Fig. 7.11 was proposed by [MSS00] and operates on a similar basis as the coupling method used for charge qubits (Fig. 7.4).

10This loop consists of superconducting material and can be switched on or oﬀ at will, as shown in [MOL+]. 11After Fig. 11 in [MSS01]. 7.1. SUPERCONDUCTING QUBITS 235

1 2 The resulting coupling is of the form σyσy and is proportional to the tunneling amplitude

Bx within a unit cell of both qubits, and on the separation between the minima.

Measurement

The measurement process is similar to the qubit coupling method: The qubit is inductively coupled to a SQUID magnetometer12 (a shunted dc-SQUID), as shown in Fig. 7.12. ¤ ¤ ¥ ¥ ¢ ¢ £ £ ¤ ¤ ¥ ¥ ¢ ¢ £ £ ¤ ¤ ¥ ¥ ¢ ¢ £ £ R V ¡ ¡ ¡ ¡ ¡ ¡

Figure 7.12: Set-up for measuring a ﬂux qubit. The qubit (on the left) is inductively coupled to the meter (on the right).

Two operating regimes of this measurement device exist, one where R (EJ /EC )Rk or where the dc-SQUID is unshunted, and another where R < (EJ /EC )Rk (where EJ and EC are the Josephson and charging energies of the SQUID). For higher shunt resistances, the noise of the SQUID in the superconducting regime is small, i.e. without driving a current through the SQUID. Hence, despite the presence of the SQUID, dephasing effects should remain small. A read-out is performed by ramping the current I through the SQUID. Phase differences across the two junctions adapt themselves to accommodate the driving current if it is below some critical current13. However, once a critical current is reached, measurable voltages appear. The critical current depends on the flux through the SQUID, and hence the point at which the SQUID switches provides information about the external flux. The measurement of the qubit thus consists of ramping the current until a voltage is measured across the SQUID. The current value at which the SQUID switches provides information about the state of the qubit. The drawback of this measurement technique is that the switching process is stochastic

12After Fig. 20 in [MSS01]. 13This is because of the requirement that the sum of phases and the externally applied ﬂux have to be quantized. 236 CHAPTER 7. APPLICATIONS OF NMRQC

[vtW+00] even when the applied external flux is constant. For the currently available systems, the spread in measured switching currents is larger than the difference in switching currents corresponding to the two states of the qubit. Hence only weak repeated measurements are possible in this regime. In the other operating regime, we apply a current that is greater than the critical current, and voltages develop across the resistor. These voltages depend on the external flux in the SQUID and hence on the qubit states. The main disadvantage of this method is that the SQUID induces dephasing during periods where we only wish to apply coherence manipulations of the qubit. It is not clear at this point whether an operating regime exists which is a compromise of the two extremes.

7.1.3 Josephson phase qubits

The previous two subsections discussed the principles of “charge” and “flux” qubits. This subsection introduces a third design [MNAU02] in which the charging energy is very small, similar to the flux qubit, but with the added advantage that the decoherence from flux noise should be small. This qubit design does not include a superconducting loop but instead just a single junction.

The system dynamics

The system consists of a single Josephson junction as shown in Fig. 7.13. As mentioned in section 7.1.2, when we drive a current I through a Josephson junction that exceeds the critical current I0, voltages appear across the junction. In this case, the phase δ across the junction begins to evolve with time according to the relation δ˙ = 2πV/Φ0 where Φ0 = h/2e is the quantum flux. The total current I that flows through the junction has to equal the sum of the currents flowing through the three elements drawn on the right in Fig. 7.13. Summing over the three current elements and eliminate terms in V in favor of δ˙, we obtain the following classical motion for the phase difference:

Φ0 1 Φ0 ˙ ∂U(δ) Φ0 C + δ + = IN (t) (7.26) 2π R 2π ∂δ 2π 7.1. SUPERCONDUCTING QUBITS 237

I I ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

¡ ¡ ¡ V C I 0 R V

Figure 7.13: Schematic representation of a Josephson junction (left), and circuit representation (right).

The termIN (t) represents the Nyquist current noise, and

U(δ) = (IΦ0/2π)[cos δ + (I/I0)δ] (7.27) −

The washboard potential and the qubit

2 We can view Eq. 7.26 as the motion of a classical particle of mass C(Φ0/2π) , moving in

the potential U(δ). When I < I0 the particle can be trapped in one of the relative minima. But this is only a classical picture, and we thus need a quantum mechanical picture to understand how this system can serve as a qubit. The details of a Josephson phase qubit is described elsewhere [MNAL03], and we only review the basics here. The Hamiltonian of the current-biased Josephson junction with bias

source I, critical junction current I0, and junction capacitance C is

1 I Φ IΦ H = Qˆ2 0 0 cosδˆ 0 δˆ (7.28) 2C − 2π − 2π

where Φ0 = h/2e is the superconducting ﬂux quantum. The operators Qˆ and δˆ correspond to the charge and the superconducting phase diﬀerence across the junction respectively, and have a commutation relationship [δ,ˆ Qˆ] = 2ei. Quantum mechanical behavior can be 2 observed for large area junctions in which I0Φ0/2π = EJ EC = e /2C and when the bias current is slightly smaller than the critical current I < I0. In this regime the last two terms in H can be accurately approximated by a cubic potential∼ U(δ) parametrized by a barrier 238 CHAPTER 7. APPLICATIONS OF NMRQC

3/2 height ∆U(I) = (2√2I0Φ0/3π)[1 I/I0] and a quadratic curvature at the bottom of the − 1/4 1/2 1/4 well that gives a classical oscillation frequency ωp(I) = 2 (2πI0/Φ0C) [1 I/I0] . − The commutation relation leads to quantized energy levels in the cubic potential. The quantized energy levels in this potential can be visualized as indicated by Fig. 7.14a.

Microwaves induce transitions between levels at a frequency ωmn = Emn/~ = (Em En)/~, − where En is the energy of state n . The two lowest transitions have frequencies | i

5 ~ωp ω10 ωp 1 , and (7.29) ' − 36 ∆U 10 ~ωp ω21 ωp 1 (7.30) ' − 36 ∆U

These two frequencies must be diﬀerent for the qubit to behave as a two-state system. The ratio ∆U/~ωp parameterizes the anharmonicity of the cubic potential with regard to the qubit states, and gives a rough estimate of the number of states in the well.

Figure 7.14: (a) Schematic drawing of the energy levels of the particle in the tilted washboard potential U(δ) with I < I0. When the anharmonic term is suﬃciently large only 3 energy levels exist whose energy separations diﬀer by a few percent. In this case, the bottom two levels can be used as the qubit states 0 and 1 . (b) Energy level diagram of the three-level system. | i | i

Single qubit rotations are achieved by applying an AC-current to the system, tuned to the resonance frequency of the qubit transition. This has been demonstrated [MNAU02] experimentally. Since the second transition is close in frequency, the qubit is not perfectly isolated from the third energy level, leading to faulty operations. We address this problem speciﬁcally in section 7.4. The initialization of the qubit is done by letting the system equilibrate with the environment at low temperatures. 7.1. SUPERCONDUCTING QUBITS 239

Two-qubit interactions in this system are envisioned by capacitively coupling two Joseph- son phase qubits, leading to a σy σy interaction. As a first step, it is proposed to leave this ⊗ coupling turned on at all times, and to use refocusing schemes to unwind undesired evolution. It would be beneficial however, to design system in which such refocusing schemes are not required. The measurement is done by applying a 180 pulse on the 1 2 transition. If the ◦ | i ↔ | i qubit was in the state 1 , it now is in 2 . The barrier thickness of the potential well at this | i | i energy level is small, and hence the state can easily tunnel through the barrier. When this happens, the phase particle is no longer trapped and continues to roll down the washboard potential. This leads to a voltage across the Josephson junction, which can be measured. If the qubit was in the state 0 , it is not affected by the measurement pulse, i.e. no voltage | i appears across the junction. Hence, the measurement consists of applying a 180◦ pulse on the 1 2 transition followed by measuring the voltage across the junction. | i ↔ | i Note that the high tunneling rate from 2 could potentially have undesirable effects when | i this state becomes transiently populated during single-qubit operations, and we investigate these effects in section 7.4.

7.1.4 Decoherence and recent experiments

Decoherence of superconducting qubits

Thus far we have only discussed the operation and measurements of the charge, ﬂux, and phase qubits. In order to perform coherent computations, we have to ensure that the coherence times of these systems are much greater than the typical operation times. The result of several calculations [MSS01] show that in principle, 106 coherence operations should be possible. For typical choices of parameters (EJ , T , Cg, etc), the time scale of single-qubit 10 operations can be as short as τop 10 s. Two-qubit operations are estimated to be on ∼ − the order of 102 times slower than single-qubit gates. Dephasing times are estimated to be 4 on the order of τφ 10 s. While initial experimental observations showed signiﬁcantly ∼ − worse decoherence times than predicted, the progress towards improving the experiments has met with large success. In fact, in a recent experiment, the dephasing time is estimated to be 104 times larger than the single-qubit operation [VAC+02] which lets us look ahead with cautious optimism. 240 CHAPTER 7. APPLICATIONS OF NMRQC

Recent experiments and perspectives

Nakamura presented the earliest convincing demonstration of Rabi oscillations using a Josephson junction qubit [NPT99]. This experiment was recently followed up with several other impressive demonstrations of Rabi oscillations and Ramsey fringes in superconducting qubits [VAC+02, YHC+02, MNAU02]. In addition, the coherence times for such systems are being continuously enhanced through better fabrication or circuit designs. Recently, the first observation of quantum oscillations in two coupled charge qubits was reported [PYA+02] - the first step towards demonstrating a two qubit algorithm. Superconducting loops do have several appealing characteristics that make them ideal candidates for quantum bits. One and foremost, fabricating several thousands of such qubits on a chip is well within reach. If we can properly design the qubit and achieve sufficiently long coherence times, it is believed that these systems are scalable in a rather straightforward manner. Furthermore, the integration of such qubits with classical circuits using solid-state techniques may be an additional advantage. Hence, one of the most challenging aspects for the design of such qubits is the need for long coherence times. Recent experiments report dephasing times which permit the implementation of 8000 Rabi oscillations [VAC+02]. Whether similarly encouraging numbers can be reproduced for two coupled or several coupled qubits will probably be answered with in the next few years. Superconducting loops are continuously growing more mature, and it is likely that they provide the next demonstration of a quantum algorithm, following the footsteps of NMR [CVZ+98] and ion traps [GRL+03]. Considering that several of the proposals for superconducting qubits employ NMR techniques to perform single-qubit rotations, we firmly believe that several NMR concepts and methods are of significant use once these systems reach a more mature stage. Using a specific example of the potential use, we show in section 7.4 how NMR techniques can improve single qubit rotations for a Josephson phase qubit [MNAU02].

7.2 Ion trap quantum computers

In this section (following the outline of [Sac01]) we discuss the ion-trap quantum information processor. The ion trap approach essentially consists of using trapped charged atoms (ions) as quantum bits. The Coulombic interaction between these ions gives rise to coupling between qubits. The quantum states of ions are manipulated, and read-out using precisely 7.2. ION TRAP QUANTUM COMPUTERS 241

tuned laser pulses. This method allows very high degree of coherence and control to be maintained throughout an experiment, while the low temperature of operation and good isolation makes it possible to minimize the decoherence eﬀects from the environment. The initialization and measurement procedures have also been demonstrated with high ﬁdelities.

7.2.1 The qubit states and single-qubit rotations

Typical ion species used for quantum computation have only one valence electron, so that the internal electronic states of such a system are very similar to those of neutral hydrogen or alkali metal atom. A large number of internal states are available, of which we must choose appropriate ones to serve as the qubit. From a decoherence point of view, the best possible option is to choose the two sublevels of the electronic ground state, which is very stable. Typical spontaneous emission rates 6 1 are on the order of 10− yr− . There exist other, more typical causes for decoherence like fluctuating magnetic fields, but still, coherence times in the range of several hundred seconds have been observed (for example in [BMB+98]). The ground state sublevels are typically separated by microwave or radio frequencies. Transitions between these levels ( 0 and 1 ) | i | i can be driven by the application of microwave or rf fields. This method is virtually identical to NMR methods, providing a means for NMR techniques to potentially apply to ion trap quantum computers. The two sublevels can also be driven using a stimulated Raman transition, in which two laser beams are applied that differ in frequency by an amount equal to the sublevel splitting. The disadvantage is that decoherence effects are worse than with the direct excitation method, but it allows coupling between the qubit states and the motion of the ions, which makes two-qubit gates possible. Another option would be to use an electronic excited state of the ion. In this case, the transition frequencies are typically in the optical range, and the internal and motional states can be coupled directly by a laser. The drawback of this method is that very stable lasers are required (frequency stabilities below 1 Hz).

7.2.2 Measurement and initialization

Besides the qubit levels, two other internal states are required. This is shown in Fig 7.15. Here, the 0 and 1 denote the qubit levels. The state p is an unstable excited state which | i | i | i 242 CHAPTER 7. APPLICATIONS OF NMRQC

is radiatively connected to both qubit states, and it is used to initially prepare the ion in a known state. This is done by optical pumping [Hap72], where only one of the transitions, say 1 p , is driven until all the population has decayed by spontaneous emission into | i ↔ | i 0 . Optical pumping has shown state preparation ﬁdelities of 99 percent or greater. | i

Figure 7.15: Schematic of the energy levels of an ion isolated from the rest of the energy levels for its use as a qubit.

The second excited state, d , is radiatively connected to one of the qubit state, for | i example 0 in the figure. This state serves as a means of measurement. Suppose after we | i have completed our quantum computation and want to measure the state of our qubits, we drive the 0 d transition. This transition causes fluorescence if the ion is in the state | i ↔ | i 0 . In contrast, if the ion is in the 1 state, then there is no emission of photons in the | i | i measurement process. The number of photons emitted in this process of measurement is of the order of 105, a number that is large enough to efficiently distinguish the two qubit states.

7.2.3 Trapping and motion of the ions

Trapping a single ion

The previous subsection described operations on a single qubit and the measurement and initialization process. We now describe the physical means of coupling two qubits (or ions). Before we do that, we must understand more about the motion of ions in the trap. Most of the ion-trap techniques developed to date either use time-dependent electric fields in radio-frequency Paul traps, or use a combination of electric and magnetic fields in Penning traps14. Quantum computation experiments use Paul traps in favor of the Penning traps as it is difficult to stably localize ions in a Penning trap. The electric field in a rf trap rapidly

14It is impossible to trap a charged particle in free space using just static electric ﬁelds 7.2. ION TRAP QUANTUM COMPUTERS 243

oscillates when compared to the frequencies of the ions’ overall motion. The time-averaged eﬀect on the ions can be described by a pondermotive pseudopotential

2 q 2 Up(r) = 2 E(r ) (7.31) 2MΩT h i

Here, q is the ion’s charge, M is its mass, ΩT is the oscillation frequency of the field, and E2 denotes the time average. A quadrupole field, E(r) r, leads to a harmonic h i ∼ confining force. The ion is not stationary in such a time-averaged confining potential. It still undergoes a slow motion, called the secular motion. In addition, the ion also undergoes a much faster micromotion at the applied field frequency whose amplitude is much smaller than the amplitude of the secular motion. It is the secular motion that provides the basis for two-qubit gates, whereas the micromotion typically leads to undesired effects. We return to the significance of motion shortly. There are various geometries of the electric field suitable to trap ions of which the linear quadrupole field is the most suitable for ion-trap quantum computation. This field can be described as,

Elinear(r) axx ayy (7.32) ∝ − were r = x + y + z, z is along the vertical direction (see Fig. 7.16), and zx and ay give the strength of the electric field in the x and y direction. This type of field provides a confinement in the radial direction only. Hence a static quadrupole field is required for axial confinement and can be obtained by the static field geometry created by the electrodes. The resulting configuration is shown in Fig. 7.1615. The harmonic oscillation frequencies (of the secular motion) are on the order of 0.1 to 10 MHz with rf frequencies about 10 times higher.

Trapping multiple ions

If several ions are trapped, then most ions also undergo substantial micromotion which can have undesirable eﬀects (see for example [CGB+94]). It can be avoided in the linear geometry by making the axial conﬁnement weak compared to the radial. However since the 0.9 ratio of the axial to radial oscillation frequencies scales as N − for a large number of N

15This ﬁgure has been drawn by the author, after Fig. 1 in [Ste97]. 244 CHAPTER 7. APPLICATIONS OF NMRQC

ions, the slow (axial) motion frequencies limit the number of ions that can be used.

Figure 7.16: A linear ion trap consisting of four parallel rods. An oscillating electric field is generated by applying an rf voltage to the electrodes, providing radial confinement. The axial confinement is achieved by applying different dc voltages to different segments of the rods.

We have previously described how ions can be cooled nearly to the ground state. Several such cooled ions interact strongly via Coulomb interaction, and hence the ground state is a crystalline arrangement. Excitations from the ground state are described by normal modes of the ion crystal. There are in general 3N modes whose frequencies can be calculated classically. However, the simplest modes are the center-of-mass (COM) modes, in which the ion crystal moves rigidly in the trap. The COM mode frequencies are identical the oscillation frequencies of a single ion, and this is independent of N. The COM has three degrees of freedom, each of which behaves as a quantum mechanical oscillator mode. Let us label the eigenstates of the simplest COM modes as the number states n with energies n~ω. Naturally, there exist higher modes, each with three degrees | i of freedom, but we ignore them here. The combined eﬀect of the internal states and the vibrational states of a two-level systems is shown in Fig. 7.17. From this ﬁgure we can see that we can excite transitions from 0, n to 1, n = n m . | i | 0 ± i However the Rabi frequency of these transitions depends on n, i.e. which motional state the ion is in. If the motional state of the ion is unknown, then the Rabi frequency is also 7.2. ION TRAP QUANTUM COMPUTERS 245

2 1 1 0

2 1 0 0

Figure 7.17: Schematic of the energy levels of an ion isolated from the rest of the energy levels for its use as a qubit. unknown, making it impossible to apply precise transformations. Therefore, cooling to the motional ground state, or at least a known state, is required. The transition for n = n is called the carrier. The transition for n = n m is termed 0 0 ± the m-th blue (+) or red ( ) sideband, and has a frequency that is diﬀerent from the − carrier frequency. Cooling to the motional ground state is done using sideband transitions and can be understood with reference to Fig. 7.15. When the Lamb-Dicke parameter16 is small, then spontaneous decay from d to 0 tends to conserve n. By tuning a laser to the | i | i 0, n d, m 1 transition, one quantum of energy is removed from the motional mode | i ↔ | − i with each spontaneous emission. Hence the system cools to the motional ground state.

7.2.4 Two-qubit gates

We have described that single-qubit gates can be achieved similar to NMR techniques. This is possible if each ion can be separately addressed by a laser. It is still possible to perform single-qubit rotations even if a single laser addresses all ions in the trap at the same time. This can be done by applying composite pulses, similar to the description of a related example shown in section 7.3. Two-qubit gates are achieved by using the motional states of the ions. The Hamiltonian describing the interaction of the laser with the internal states and the COM quantum state can be described by the following Hamiltonian,

~ηΩ iφ iφ Hcoupl = [e− 1 0 a + e 0 1 a†] (7.33) 2√N | ih | | ih |

16This parameter characterizes the coupling strength of a motional transition 246 CHAPTER 7. APPLICATIONS OF NMRQC

where, a and a† are the annihilation and creation operator for quanta of the COM motion and N is the number of ions, such that

a g = 0, and0, a† g = e (7.34) | i | i | i ha, a†i = 1 (7.35)

Here, g and e are the ground and ﬁrst excited states of the COM of the system. The | i | i Hamiltonian Hcoupl is useful for two-qubit operations, and its action can be written as,

iH coupl t ηΩt iφ ηΩt e− ~ 0 e = cos 0 e ie sin 1 g (7.36) | ij| i 2√N | ij| i − 2√N | ij| i and iH coupl t ηΩt iφ ηΩt e− ~ 1 g = cos 1 g ie sin 0 e . (7.37) | ij| i 2√N | ij| i − 2√N | ij| i We now explain how these transformations together with the help of an auxiliary level aux | i can be used to implement a controlled-phase shift operation. The Hamiltonian corresponding to this kind of interaction can be written as,

~ηΩ iφ iφ Haux = [e− aux 1 a + e 1 aux a†] (7.38) 2√N | ih | | ih |

Let us denote the unitary transforms Uaux(θ, φ) as the evolution due to the Hamiltonian

Haux acting on the excited state of ion 2 and the auxiliary state, with the motional states.

Similarly, Ucoupl(θ, φ) be the evolution due to Hamiltonian Hcoupl acting on ion 1 and the motional states. To obtain the controlled-phase shift operation we apply the following sequence of pulses,

CPS = Ucoupl(π, π)Uaux(2π, 0)Ucoupl(π, 0) (7.39)

The internal and motional states then evolve as follows:

0 0 g 0 0 g 0 0 g 0 0 g (7.40) | i1| i2| i 7→ | i1| i2| i 7→ | i1| i2| i 7→ | i1| i2| i 0 1 g 0 1 g 0 1 g 0 1 g (7.41) | i1| i2| i 7→ | i1| i2| i 7→ | i1| i2| i 7→ | i1| i2| i 1 0 g i 0 0 e i 0 0 e 1 0 g (7.42) | i1| i2| i 7→ − | i1| i2| i 7→ − | i1| i2| i 7→ | i1| i2| i 1 1 g i 0 1 e i 0 1 e 1 1 g (7.43) | i1| i2| i 7→ − | i1| i2| i 7→ | i1| i2| i 7→ −| i1| i2| i

From this transformation we can obtain a cnot operation [CZ95a], similar to the sequence 7.2. ION TRAP QUANTUM COMPUTERS 247

of operations shown in Eq. 4.22. Together with the initialization procedure, the single-qubit operations and the measurement, we can implement a quantum computer.

7.2.5 Decoherence, summary, and perspectives

Extremely long coherence times (on the order of hundreds of seconds) have been observed for the internal electronic states of a single ion. Together with gate operation times on the order of microseconds, and high fidelity measurements, ion traps appear to be an excellent candidate for implementing a quantum computer. Nonetheless, scaling of this method to several tens of trapped ions is difficult due to the lack of easy individual addressing techniques. Decoherence channels include dissipation due to electrical resistance in the electrodes used to trap the ions, spontaneous transitions in the vibrational modes of the system, thermal radiation that drives the internal rf transitions of the ions, and experimental instabilities in the laser beam power, rf voltages, mechanical vibrations, and fluctuating external magnetic fields. Furthermore, the number of motional modes grows quickly with more trapped ions and hence mode cross-couplings become difficult to avoid. Finally, the reduced motional frequencies for an increasing number of trapped ions lead to longer and longer two-qubit gate times. Currently, it appears unlikely to work with more than about 10 trapped ions, at least in a single, linear trap. Hence, a scalable ion trap quantum computer appears to require multiple traps. Progress towards this direction has been made recently by transporting ions between two traps with high fidelities [RBKD+02]. Ion trap quantum computation has made some spectacular progress however via a demonstration of a four particle entanglement [SKK+00], and recently in the implementation of a two-qubit Deutsch-Jozsa algorithm [GRL+03]. It appears likely that ion trap systems will be the next system, following NMR quantum computers, to implement a factoring algorithm. Besides the descriptions of sections 7.1 and 7.2, there exist several other proposals for implementing quantum computers, but they are out of the scope of this thesis. For a more detailed description of a few other proposals, please refer, for example, to [MPZ00] or [NC00]. Through the descriptions of sections 7.1 and 7.2 it becomes clear that NMR techniques could potentially play a significant role in the design for pulse sequences, composite pulses, among others, in such systems. In the next section we discuss how the language of quantum 248 CHAPTER 7. APPLICATIONS OF NMRQC

computation is useful for designing an appropriate composite pulse in an optical lattice quantum computer. Such methods could also be useful for the design of composite pulses in an ion trap quantum computer.

7.3 Design of atomic physics pulse sequences

The rules for designing quantum circuits and simplifying them were very useful for solving the following problem in an optical lattice quantum computer17. Cesium atoms are used, and the hyperfine states of cesium in a magnetic field provide long lived states ideal for representing a qubit. However, it is not straightforward to perform single qubit operations using RF pulses, because of the degeneracies in energy level differences. Specifically, we have the magnetic sublevels, as drawn in Fig. 7.18.

H a H b

Figure 7.18: Schematic diagram of the magnetic sublevels of a cesium atom.

The 0 and 1 are the desired states for the qubit. Applying a RF ﬁeld at that transition | i | i unfortunately also causes an undesired transition between the levels 2 and 3 (note these | i | i excitations require a two-photon transition because ∆m = 1). Suppose, however, that we 6 ± can control an additional RF transition between 0 and 2 such that the two Hamiltonians | i | i are: 0 α 0 0 0 β 0 0     α∗ 0 0 0 β∗ 0 0 0 Ha(α) =   , and Hb(β) =   (7.44)  0 0 0 α   0 0 0 0       0 0 α∗ 0   0 0 0 0  17This problem is taken from problem set 3 from Isaac Chuang’s quantum information class, MAS 865, taught in 2002. 7.3. DESIGN OF ATOMIC PHYSICS PULSE SEQUENCES 249

where Ha couples both 0 , 1 and 2 , 3 , and Hb couples 0 , 2 . Note that α and β are | i | i | i | i | i | i both complex numbers.

The task is to ﬁnd a sequence of pulses by turning on Ha and Hb consecutively, with speciﬁed values of α and β and pulse durations, to perform an arbitrary rotation in the 0 , 1) manifold while leaving the 2 , 3 manifold untouched. | i | i | i | i One could try and solve this problem by brute-force by simply applying a sequence of pulses and optimizing the parameters α, and β, and the pulse duration for each pulse. However, there exists an elegant way using the language of quantum circuits.

Let us take the four energy levels as two qubits, with the following notation: 0 00 , | i 7→ | i 1 01 , 2 10 , and 3 11 . From Eq. 7.44 it becomes clear that Ha is applying | i 7→ | i | i 7→ | i | i 7→ | i a rotation on qubit 2 while Hb acts like a zero controlled operation (with qubit 2 being the control and qubit 1 being the target). The goal is to implement a zero controlled operation of qubit 1 on qubit 2. How do we obtain this operation from the set of allowed operations?

Fig. 7.19 shows how we start from a desired operation, say a zero controlledx ˆ-rotation about some arbitrary angle α and decompose it into operations that are experimentally accessible. Let us deﬁne

iαπ/4 e− 0 Zα = (7.45) 0 1 cos(πα/4) isin(πα/4) Xα = − (7.46) isin(πα/4) cos(πα/4) − A controlledx ˆ-rotation can be decomposed into two hadamard gates and a controlled zˆ-rotation. The hadamard gates can be simpliﬁed into ay ˆ and -ˆy-rotation. The control and target of any controlledz ˆ-rotation are interchangeable, so let us make qubit 2 the control. Similar to the implementation of az ˆ-rotation (see section 4.2.2), a controlled zˆ- rotation can be implemented using three controlled rotations. The extra singlez ˆ-rotation is needed to adjust some phases in the unitary matrix. The resulting quantum circuit can be implemented using only the Hamiltonians Ha and Hb. The controlled rotations are done by applying Hb with the correct phases. Similarly, the single qubit rotations are implemented by applying Ha with appropriate phases. The last two rotations can in principle be absorbed into one, so that a total of only ﬁve composite pulses are needed to apply the intended rotation. 250 CHAPTER 7. APPLICATIONS OF NMRQC

The following rotations are achieved in Fig. 7.19, in matrix notation:

cos(απ/4) i sin(απ/4) 0 0  −  i sin(απ/4) cos(απ/4) 0 0  −  (7.47)  0 0 exp( iαπ/4) 0   −   0 0 0 exp( iαπ/4)  − This implements ax ˆ-rotation on the desired qubit levels only. Ay ˆ-rotation can be implemented by changing the ﬁrst and last pulse in the sequence to X¯ and X.

Z 2α

X α H Z 2α H Y Y

X Y α X

Y Zα/2 Y

Figure 7.19: Outline of the derivation of the quantum circuit to implement the desired rotation, given the Hamiltonians Ha and Hb. The two wires correspond to the two qubits. The goal is to implement a zero-controlled operation with qubit 1 being the control, and qubit 2 the target. This corresponds to a rotation in the desired Hilbert space. The individual steps are explained in the text.

This small example serves as a beautiful illustration of how the language of quantum computing can be directly applied to other relevant experiments. In another example, with the aid of NMRQC techniques, the Deutsch-Jozsa algorithm was implemented using a single ion in an ion trap quantum computer [GRL+03]. It is becoming clear that techniques specifically developed for NMR purposes are beginning to find useful applications in other fields as well. In yet another example, we show how our developed tools are useful for the design of RF pulses in a superconducting loop quantum computer. 7.4. DESIGN OF PULSES FOR SUPERCONDUCTING LOOP QUANTUM COMPUTERS251

7.4 Design of pulses for superconducting loop quantum computers

This work was done in collaboration with John Martinis, and the resulting paper is in preparation for submission. Superconducting loops have been proposed as candidates for scalable qubits. Several diﬀerent proposals exist, two of which we described in detail earlier (section 7.1.1 and 7.1.2). In this section we discuss how some of our techniques are helpful in designing suitable pulse shapes and composite pulses for a Josephson phase qubit [MNAU02]. We believe that the shown methods are also applicable to other systems with similar energy level conﬁgurations. We have discussed the nature of the qubit in section 7.1.3, and thus do not repeat it here.

7.4.1 Problem statement

The energy levels of a superconducting Josephson Junction do not easily lend themselves to forming a qubit, which is a closed two-dimensional Hilbert space. Instead, one typically has three or more levels, with energy differences between successive levels differing by only a few percent. The challenge lies in successfully isolating the two energy levels from the rest of Hilbert space and to perform qubit operations on them. In other words, how is it possible to operate as quickly and with as little error as possible on a qubit subspace of Hilbert space in a Josephson phase qubit? This is especially important when the coherence times of the system are short. A Josephson phase qubit can be described by three energy levels 0 , 1 , and 2 , with | i | i | i energies E0, E1, and E2, as sketched in Fig. 7.14b. The qubit space is formed by 0 and 1 , | i | i and hence we wish to operate only within this subspace. Clearly, the higher-order transition can be avoided by exciting at the ω10 frequency for a sufficiently long duration. Because one wants to maximize the number of logic operations within a fixed coherence time, there is a need to excite the 0 1 transition as quickly as possible without populating other | i ↔ | i states. In this section we use techniques from NMR quantum computing (chapters 4 and 5) to present how single-qubit operations can be improved compared with hard pulses by using shaped and composite pulses. Though other proposals for implementing single-qubit rotations within a subspace of Hilbert space are known [WL03, TL00, PK02], we estimates the feasibility of our methods for typical parameters of a Josephson phase qubit, including 252 CHAPTER 7. APPLICATIONS OF NMRQC

the effects of tunneling out of higher lying energy levels. We show that these tunneling effects can be a significant source of decoherence if not taken into account properly. The physics of the Josephson phase qubit has already been described in section 7.1.3. From this description we provide a concrete procedure detailing our methods to simulate the effect of amplitude modulation. Our results indicate that gaussian shaped modulation [BFF+84] provides the best selectivity. We then describe composite pulses, which perform better than hard pulses, but not quite as well as gaussian shaped pulses. We proceed to show how to include tunneling effects out of the higher energy levels to estimate the feasibility of these techniques in a real Josephson junction qubit system. Our results indicate that tunneling plays a significant role, leading to the conclusion that one may wish to use at least four energy levels instead of the previously used three [MNAU02] levels to reduce tunneling effects. Though our methods have been developed in the context of Josephson phase qubits, we believe they could also be fruitful in other systems where one wishes to restrict the operation of the system only to a subspace of Hilbert space. We also take first steps towards transferring ideas from NMR quantum computing to other proposals for implementing a quantum computer which we believe to be a very rewarding approach (as we have already indicated by the previous example). Furthermore, the section on the effects of tunneling uses ideas from quantum computing to help us understand and model the physics of a real Josephson junction qubit. The state of the qubit can be controlled with a time-varying bias current at frequency

ω = ω10, given by

I(t) = Idc + ∆I(t) (7.48)

= Idc + Iµw(t)cos(ωt + φ) (7.49)

In general, the Hamiltonian for the ﬁrst three states is

E0 0 0   H = 0 E1 0 (7.50)    0 0 E2  0 δˆ 0 0 δˆ 1 0 δˆ 2 Φ  h | | i h | | i h | | i  + 0 ∆I 1 δˆ 0 1 δˆ 1 1 δˆ 2 . (7.51) 2π  h | | i h | | i h | | i   2 δˆ 0 2 δˆ 1 2 δˆ 2  h | | i h | | i h | | i 7.4. NMR TOOLS FOR JOSEPHSON PHASE QUBITS 253

ˆ The matrix elements m δ n are calculated as follows. When I = Idc and for Idc I0, h | | i → the Hamiltonian has a potential U(δˆ) that is cubic. We then solve for the energies of the eigenstates 0 , 1 , and 2 of this Hamiltonian from which the matrix elements m δˆ n can | i | i | i h | | i be simply computed.

It can be shown that to ﬁrst approximation

i(ωt+φ) E0 g(t)e 0  i(ωt+φ) i(ωt+φ)  H g(t)e E1 √2g(t)e + Hnr (7.52) ≈ −  i(ωt+φ)   0 √2g(t)e− E2  where the basis states are 0 , 1 , and 2 , from left to right, ω is the frequency of the applied | i | i | i time-varying current, and g(t) = 1.014I (t) ~/2ω C/2 represents the amplitude of the µw p 01 radiation. The Hamiltonian Hnr contains additional diagonal and off-diagonal elements, but they are all far off-resonance from ω10 and ω21 such that Hnr only has negligible effects.

It is furthermore convenient to move into a doubly rotating frame, deﬁned by the unitary operator 1 0 0   V = 0 eiωt 0 (7.53)    0 0 e2iωt  Let φ = V ψ be a state in the rotating frame of V and ψ is a state in the labora- | i | i | i tory frame. Then the equation of motion for this state can be derived from Schr¨odinger’s Equation:

i~∂t φ = H˜ φ (7.54) | i | i where H˜ is the rotating frame Hamiltonian given by H˜ = V HV i~V ∂tV . This results † − † in 0 g(t)eiφ 0  iφ iφ  H˜ = g(t) E1 ~ω √2g(t)e , (7.55) − −  iφ   0 √2g(t)e E2 2~ω  − − where we have set E0 = 0. In this work, we focus only on excitation that is on-resonance with the 0 1 transition. Using E1 = ~ω10, ω = ω10, and deﬁning E2 E1 = ~δω, we | i ↔ | i − obtain 0 g(t)eiφ 0   ˜ iφ iφ H = g(t)− 0 √2g(t)e , (7.56)  iφ   0 √2g(t)e− ~δω  254 CHAPTER 7. APPLICATIONS OF NMRQC

We can now use this rotating frame Hamiltonian in a straightforward manner to numerically calculate the eﬀect of hard (square wave) and shaped pulses on the three-level system. This can be done by discretizing the shape into many steps, during each of which g has a constant amplitude. The unitary evolution in each slice is given by U = exp( iH˜ ∆t/~) where ∆t is − the slice length. We then vary g from slice to slice according to the modulation, and multiply all unitary evolutions together to obtain the overall evolution of the system, mathematically described as: iφ 0 gje 0  i∆t  iφ iφ  U = exp g− 0 √2gje (7.57) Y − ~ j j   iφ    0 √2gje− ~δω  From U, we can calculate the leakage out of the qubit manifold. Also note that from Eq.

7.56, it is clear that the eﬀects of δω scale as the product tδω/2π, thus all results shown here are plotted in this way.

7.4.2 Results: Pulse shaping

Clearly from chapter 6, we have extensively used shaped pulses to selectively excite a spin- 1/2 particle. In contrast to NMR however, here the two transitions share an energy level leading to much more complex system dynamics. Although it is not immediately obvious in what way amplitude modulating could be beneﬁcial, it is not unreasonable to assume some improvements.

In our simulations, we chose a ﬂip angle of 180◦ because this is usually the hardest selective rotation to achieve. Even though we show results for several pulse shapes, we only provide explicit functions of the time dependence of g(t) for Gaussian and Hermite 180◦ shapes. The RF envelope g(t) of the Gaussian shape is given by:

2 2 ggauss(t) = (a/τ)exp( t /2τ ) (7.58) − for t < ατ and g(t) = 0 otherwise, where α is the cut-oﬀ of the pulse in time (usually | | 3-5), a is the amplitude (a 1.25 for 180 pulses and for typical values for α), and τ is the ≈ ◦ characteristic pulse width. The total pulse width is Tpw = 2ατ.

The Hermite 180◦ shape [War84] is simply a Gaussian multiplied by a second order polynomial. The RF envelope g(t) of the Hermite 180◦ shape is

2 2 2 ghrm180(t) = (1 β(t/ατ) )(a/τ)exp( t /2τ ), (7.59) − − 7.4. NMR TOOLS FOR JOSEPHSON PHASE QUBITS 255

with the definition of parameters as before. The parameter β determines how strongly the Gaussian pulse is modulated. Here, a 2.2 for α = 3 and a 1.67 for α = 4, both for ≈ ≈ β = 4 and a 180◦ flip angle. Using these parameters and shapes we have calculated the performance of the pulses. We define the error as = 1 U(3, 3) 2 where U(3, 3) denotes the (3, 3) element of the − | | resulting unitary transform. Ideally, = 0 if we are only operating on the desired qubit subspace. Otherwise, the undesired energy level is involved in the operation. The error is the same as the maximum probability of being in the state 2 after the pulse, when starting | i from an arbitrary superposition of a 0 + b 1 . The measure serves as a lower bound error | i | i because in addition to leakage outside the qubit manifold, the desired qubit rotation may also slightly deviate from the ideal rotation even when = 0. For example, one deviation is due to transient Bloch-Siegert shifts (section 5.2), but these effects can be corrected using a similar method to the one described in section 5.3.

gauss, α=3 0 10 model, untruncated model, truncated

−4 ε 10

−8 10

0 1 2 3 4 5 6 7 8 t pw

Figure 7.20: Estimate of the error as a function of normalized pulse width tpw = Tpwδω/2π = 2ατδω/2π via Fourier analysis using the untruncated and truncated gaussian shape, compared with the exact calculation.

We can numerically calculate the error via Eq. 7.57, but in order to gain an intuitive understanding into which amplitude modulation could be best suited, it is useful to estimate the error ﬁrst. This system’s response is approximately linear for small rotation angles, and hence Fourier analysis provides useful insight. In fact, the Fourier approach has been used extensively in NMR even for large rotation angles in order to get a ﬁrst idea of the selectivity 256 CHAPTER 7. APPLICATIONS OF NMRQC

of a shaped pulse. The relative power of the frequency component of the untruncated gaussian shaped pulse at a frequency δω away from ω10 is given by

2 πtpw (tpw) exp " # . (7.60) ≈ − α

This is plotted in Fig. 7.20 and is compared with both the Fourier analysis of a truncated gaussian shaped pulse and the exact calculation. For small pulse widths, the exact calculation and the one based on the Fourier analysis of untruncated shapes are similar, but the exact calculation ﬂattens out for tpw > 4. Though approximate, it is evident that Fourier analysis still provides a rough estimate of the error, especially if truncation eﬀects are included. From Fourier analysis we expect the hard pulse to perform poorly compared with gaussian or Hermite 180 shapes. In order to quantify their performance accurately however, we must calculate the error exactly using Eq. 7.57.

hard 0 10 gauss, α=2 gauss, α=3 hrm180, α=3, β=4 hrm180, α=4, β=4

−4 ε 10

−8 10

0 2 4 6 8 10 12 14 16 t pw

Figure 7.21: Plot of the error as a function of the normalized pulse width tpw for three diﬀerent pulse shapes and several diﬀerent levels of truncation.

Fig. 7.21 plots as a function of the normalized pulse width for hard, gaussian and Hermite 180 shaped pulses. Clearly, for low error rates, a long pulse must be used. Since our goal is to apply the desired rotation as quickly as possible, it appears that the gaussian shape is best suited for this problem. Also, the gaussian shape could probably be further optimized using our calculational methods outlined here. 7.4. NMR TOOLS FOR JOSEPHSON PHASE QUBITS 257

There exist other pulse shapes which have been designed for NMR experiments to invert or select spins over a very sharp and speciﬁed bandwidth [GF91]. These belong to the class of BURP pulses, and their performance is shown in Fig. 7.22 compared with hard, gaussian and Hermite 180 shapes. Rather surprisingly, these specially designed shapes perform rather poorly in our three-level system.

hard 0 10 gauss hrm180 iburp reburp

−4 ε 10

−8 10

0 2 4 6 8 10 12 14 16 t pw

Figure 7.22: Plot of the error as a function of normalized pulse width tpw for several traditional NMR pulse shapes. The Gaussian shape here corresponds to α = 3.

It is also interesting to note that the error (or the maximum occupation probability of state 2 ) during the application of the pulse may be much higher than at the end of the | i pulse, as indicated in Fig. 7.23. Depending on the experiment, this may still be undesirable, for example when the lifetime of the third energy level is short. We discuss the impact of such short lifetimes later. While pulse shaping clearly oﬀers advantages in this three-level system, it may not necessarily be straightforward to do so experimentally, and hence it is useful to look at alternatives to pulse shaping. In the next section we discuss how composite pulses could provide an alternative approach towards improving single-qubit operations in a Josephson phase qubit.

7.4.3 Results: Composite pulses

Composite pulses are also used extensively in NMR [LF79, Lev86, Fre97] to reduce certain types of errors (eg. rf inhomogeneity, oﬀ-resonance eﬀects), but often at the cost of longer 258 CHAPTER 7. APPLICATIONS OF NMRQC

total durations. We have not used such pulses explicitly in any of our experiments, but we next describe the design of three diﬀerent composite pulses, which perform better than hard pulses in our three-level system.

Figure 7.23: Plot of the error (maximum occupation probability) as a function of time normalized with respect to the total pulse length tpw during a hard and gaussian shaped pulse and the second composite pulse. A Gaussian-shaped pulse with a pulse width of 8 tpw = 4 has a very small error (about 10− ), but during the pulse the error can be as high 2 as 5 x 10− . The error during the pulse can be reduced by applying the pulse for a longer duration since the maximum error scales as approximately 1/tpw, based on Fourier analysis of the truncated pulses.

Results: Composite pulses - solution 1

This first solution assumes the two transitions as being independent, and visualizing the process on the Bloch sphere - this is a correct first order assumption and only holds for small rotation angles. Suppose we apply a short pulse on-resonance with the undesired transition. The undesired transition is, of course, excited on-resonance but the desired transition is excited off-resonance. Suppose we now apply the same pulse but 180◦ out of phase with respect to the first. Then, whatever evolution was excited during the first pulse is approximately undone by the second for the undesired transition (this is only approximate because the two transitions share an energy level). However, since the desired transition was excited off-resonance, the second pulse does not unwind the evolution of the first. Hence, the net effect of the double-pulse is that only the desired transition is excited. 7.4. NMR TOOLS FOR JOSEPHSON PHASE QUBITS 259

Now repeat the same two-pulse block several times but each time with a different overall phase. If we choose these phases carefully, we can generate any arbitrary single qubit operation. We can find the optimal phases and power levels of each of the double-pulses via optimization programs that are readily available in MATLAB, however we will not show detailed results here. While this method is capable of producing a sequence of pulses with very small errors 7 (< 10− ) and a very short total duration (< 20π/δω), it requires the use of optimization programs which can be very sensitive to the initial guess values. Different rotation angles require different optimized parameters, which is not very practical. Finally, this method purposely excites the undesired transition before undoing it, temporarily populating state 2 . If this state only has a very short lifetime, this method may not be of much use. Despite | i the drawbacks of this method, its intuitive insight, stemming from the visualization of the dynamics of spin-1/2 particles, sets the stage for two useful designs for composite pulses, which are described next.

Results: Composite pulses - solution 2

We begin by noting that g = 0 is a valid choice for a pulse18. This corresponds to applying no electromagnetic radiation for some time. Working with the rotating frame Hamiltonian from Eq. 7.56, and letting it evolve for a time t = π/δω, we obtain the transformation:

1 0 0   Z2 = 0 1 0 , (7.61)    0 0 1  − where the subscript denotes that state 2 acquires a 180 phase shift with respect to all | i ◦ other states. Now, if we sandwich a driven evolution U(t) between two free evolution periods Z, we obtain 0 g 0  it   V (t) = Z2U(t)Z2 = exp g 0 √2g , (7.62) − ~  −    0 √2g ~δω  − iHt/˜ ~ where we assumed g to be real and U(t) = e− . Note that we can interpret the exponent of V (t) as applying anx ˆ-rotation on the 0 1 transition and a negative xˆ-rotation on | i ↔ | i the 1 2 transition, in the limit of fast pulses (δω/g 0). If we now apply a second | i ↔ | i → 18This method was developed by Isaac Chuang, following my proposed double-pulse idea. 260 CHAPTER 7. APPLICATIONS OF NMRQC

pulse, U(t), which can be regarded as ax ˆ-rotation on both transitions, the excitation of the undesired transition is undone. The overall transformation is then given by W (t) =

U(t/2)V (t/2) = U(t/2)Z2U(t/2)Z2, and is an interesting pulse sequence because it becomes a nontrivial unitary transform on just the desired qubit manifold in the limit of fast pulses with g δω. This procedure however only works for small rotation angles. Even though the interpretation of the exponents of V (t) and U(t) above is correct, the matrices V (t), U(t), and W (t) themselves have elements connecting the 0 and 2 states whose magnitudes are second | i | i order in time t. We can only ignore these for small rotation angles. For larger rotation angles, one may have to apply a sequence of W (t/n),

R(t) = [W (t/n)]n (7.63) to suppress the 0 2 excitation. Furthermore, if we chose n suﬃciently large, then | i ↔ | i state 2 also only has a very small transient population. The total operation time is equal | i to n(2π/δω) + τ (where τ is the total duration of the electromagnetic radiation), which becomes considerably long for large n. Surprisingly, this composite method still performs rather well even when the system is far away from the short pulse limit g δω (see Fig. 7.24).

Results: Composite pulses - solution 3

Another composite pulse design is based on the previous solution, supplemented by Bloch- sphere intuition and knowledge about the excitation proﬁle of a hard pulse.

Suppose we excite a two-level system δω/2π Hertz away from resonance via a hard pulse of duration τ. Whenever δω/2π equals multiples of 1/τ, the system does not get excited. However, in our system the two transitions share an energy-level. Hence, even if we excite the undesired transition oﬀ-resonance, there is still a substantial error as evident from Fig. 7.21. Nonetheless, such a carefully timed pulse is still useful for the design of our composite pulse. For the ﬁrst step of the composite pulse, let the hard pulse be applied on resonance with the 0 1 transition for a time of 2π/δω. Let this pulse be denoted by U(2π/δω) and let | i ↔ | i the power be such that we would obtain only a small rotation angle (less than 45◦). The resulting matrix element of U(2π/δω) connecting the 1 and 2 states is very small because | i | i 7.4. NMR TOOLS FOR JOSEPHSON PHASE QUBITS 261

this transition is 2π/δω Hertz oﬀ-resonance, in accordance with our Fourier intuition from

the previous paragraph. However, U(2π/δω) still has a matrix element connecting the 0 | i and 2 states because the two transitions share an energy level, leading to errors (similar to | i U(t) and V (t) from above). The last step of the composite pulse simply consists of undoing the 0 2 excitation. | i ↔ | i This can be done by using Z2 from Eq. 7.61 and by using the fact that a 180◦ zˆ- rotation sandwiched between twox ˆ-rotations leads to no net excitation. Suppose we apply the sequence C = U(2π/δω)Z2U(2π/δω). Let us investigate the effects of C on the three possible excitations: (1) The first and last pulse, U(2π/δω), excite the 0 1 transition, | i ↔ | i whereas Z2 has no effect on the states 0 and 1 . (2) Neither U(2π/δω) nor Z2 excite the | i | i 1 2 transition as described in the previous paragraph. (3) The two pulses U(2π/δω) | i ↔ | i lead to excitations between the states 0 and 2 , which can be regarded asx ˆ-rotations, | i | i whereas Z2 acts like a 180◦ zˆ-rotation on the same states. Hence, the sequence C leads to no net excitation between the 0 and 2 states. | i | i As a result, the sequence C acts non-trivially only on the desired qubit manifold. By carefully adjusting the power of the pulses we can create any arbitraryx ˆ-rotation on the desired transition. The sequence thus consists of repeated applications of two pulses of length 2π/δω interlaced with a free evolution of length π/δω. Similar to the previous composite pulse, this method also results in low errors only when U(2π/δω) implements small rotations. Hence in general, we may also have to apply n instances of U(2π/δω)Z2U(2π/δω) if our goal is to achieve large rotation angles. The resulting duration of this composite pulse is 2.5n(2π/δω). Fig. 7.24 shows the results of the composite pulse methods compared to a hard pulse and the Gaussian shaped pulse. The time axis denotes the total duration of the pulses. For the composite pulses, this includes the duration of the electromagnetic radiation plus the delay period implementing Z2. As can be seen, the two composite pulse methods outperform hard pulses but not a gaussian pulse with α = 3. It is clear that the two simple composite pulses may provide useful alternatives for implementing accurate single qubit rotations in a Josephson phase qubit. Furthermore, it should be possible to combine shaped and composite pulses to even further improve qubit operations. Nonetheless, as is the case with shaped pulses, the occupation probability of state 2 | i during the application of the composite pulse can be rather large. For the second composite 2 pulse, this number can be as large as 3 x 10− (see Fig. 7.23) which is much larger than 262 CHAPTER 7. APPLICATIONS OF NMRQC

Figure 7.24: Plot of the error as a function of normalized time tpw for the two composite pulses (n = 2), compared with a hard and gaussian shaped pulse. The ﬁrst composite method outperforms hard pulses whenever the pulses are multiples of 2π/δω even though the limit of g δ does not apply here. For the second composite pulse, when the applied pulses are integer multiples of 2π/δω, the error is minimized as expected.

the occupation probability at the end of the pulse. In the next subsection we show how transient populations in the third energy level can be highly undesirable in the presence of high tunneling rates.

7.4.4 Eﬀects of tunneling

Thus far we have only considered the ideal case where the two transitions of the three-level system are close in frequency but otherwise there exist no sources of decoherence. However, in real a Josephson phase qubit, the quantum state can tunnel through the barrier of the cubic well, and this process acts as a source of decoherence. What consequences do we expect from this? From ﬁrst principles we know that the tunneling rate depends exponentially on the barrier height and width. Hence in our cubic potential, we expect the upper level to be most susceptible to tunneling. When the tunneling rate out of this energy level is high, then there exists a signiﬁcant source of decoherence if the state becomes even transiently populated. It now becomes clear to keep the leakage as small as possible during our single qubit rotations. But as indicated in Fig. 7.23, the error (or occupation probability) during 7.4. NMR TOOLS FOR JOSEPHSON PHASE QUBITS 263

the shaped and composite pulses can be rather large and hence we wish to estimate the impact of this tunneling eﬀect.

A first model that gives insight into the importance of transient populations during the pulses can be written as follows. The probability of being in state 2 , and the tunneling | i rate out of 2 are defined as p2 and Γ2 respectively. The probability of tunneling out of | i state 2 can then be calculated by Pt = p2Γ2dt. We typically bias the system such that | i R Γ2 is on the order of δω/2π, which is about 10 times larger than the inverse of typical pulse widths [MNAU02], and hence we expect any transient populations in state 2 to tunnel out | i of the potential well during the pulse. Since the tunneling rates out of 1 (Γ1) and 0 (Γ0) | i | i 3 6 are about 10 and 10 times less than Γ2, we ignore their effects in this simple calculation [MNAU02].

In Fig. 7.25 we plot Pt as a function of normalized pulse width for a hard, gaussian, and the second composite pulse with a tunneling rate Γ2 (ω10/2π)/100 δω/2π. Clearly, ≈ ≈ the error is several orders of magnitude higher than , and tunneling thus appears to be

an important decoherence source for this system, as expected. Note that Pt only reﬂects the probability of tunneling, and does not include the occupation probability of being state 2 at the end of the pulse. For example, for inﬁnitely short pulses state 2 has a large | i | i occupation probability, but the state cannot tunnel because the pulse width is very short,

and hence Pt 0. However, being in state 2 at the end of the pulse is still an error ≈ | i which we must include in our calculations. We next describe a more rigorous approach that analyzes the eﬀects of tunneling and includes the leftover occupation probability of state 2 . | i We shall model the tunneling behavior similar to amplitude damping by using the operator sum representation [Kra83]. This type of model has been successfully used to predict the impact of decoherence in several NMR quantum computing experiments [VSB+01, SvDH+03] (see sections 6.6 and 6.8). As explained in section 2.4, in the operator sum

representation, an initial density matrix ρi is mapped to a ﬁnal density matrix ρf via

ρf = EkρiE†, where E†Ek = I and Ek are the Kraus operators. For amplitude Pk k Pk k damping acting on a single qubit, we only have two Kraus operators, which were described in Eq. 2.77. We show them here again for completeness sake.

1 0 E0 = , and (7.64) 0 √1 PΓ − 264 CHAPTER 7. APPLICATIONS OF NMRQC

Figure 7.25: Plot of Pt as a function of normalized pulse width for a hard, gaussian and composite pulse taking Γ2 δω/2π (ω10/2π)/100. Clearly, the estimate of the tunneling probability is several orders≈ of magnitude≈ higher than , indicating that tunneling eﬀects are important.

0 √PΓ E1 = (7.65) 0 0

tΓ where PΓ = 1 e with Γ denoting the inverse lifetime of the 1 state. From this, it − − | i becomes clear that any quantum state eventually collapses to the ground state 0 . | i

The tunneling mechanism out of state 2 is now modeled by using a fourth, fictional | i auxiliary energy level T which acts as a reservoir for the tunnel states. We now define | i a fictitious qubit with basis states 2 and T . The tunneling mechanism from 2 to the | i | i | i auxiliary level is then captured by modeling amplitude damping on the fictitious qubit. In this case the two Kraus operators take the form

1 0 0 0   0 1 0 0 E0 =   , and (7.66)  0 0 √1 PΓ 0   −   0 0 0 1  7.4. NMR TOOLS FOR JOSEPHSON PHASE QUBITS 265

0 0 0 0   0 0 0 0 E1 =   (7.67)  0 0 0 0     0 0 √PΓ 0 

with the basis states 0 , 1 , 2 , and T going from left to right. These operators also | i | i | i | i ensure that if the state tunneled, then all coherences to that state vanish. For example, if the initial state was an equal superposition of 0 and 2 , then after tunneling we have a | i | i mixed state without coherences, and find that the system is in state 0 with probability | i 0.5, or in state T also with probability 0.5. | i 3 6 The tunneling rates Γ1 and Γ0 are 10 and 10 times lower than Γ2, as described ∼ ∼ 5 earlier. When Γ2 δω/2π, we estimate their contributions to be only on the order of 10 ≈ − for typical pulse widths, using the simple tunneling model from above. Hence we do not include their effects in our operator-sum approach. Later, we show how to generalize our method to include tunneling from an arbitrary number of levels, and use this generalized method to verify that Γ1 and Γ0 are indeed negligible though we do not explicitly show our results here. The Kraus operators above only describe the tunneling mechanism, but they do not include excitation effects. The next step to model the single-qubit rotations in the presence of tunneling is to include the excitation of the transitions due to the applied radiation. The excitation of the two transitions in our three-level system do not commute with the tunneling mechanism. Thus, strictly speaking, one may have to derive a more complex formalism that captures both the tunneling effect and the qubit excitation simultaneously. However, we approximate the simultaneous behavior by slicing the microwave pulses into many discrete steps (as has been done in sections III and IV), and then simulate the tunneling and excitation one after the other in each step. In the limit as the number of steps goes to infinity this approximation becomes exact; we have used 256 steps for our simulation. From the operator-sum representation, it becomes clear that this approach takes density matrices as the input, which we have not yet specified. Our goal is to find the maximum possible error over all possible input states, which are arbitrary superpositions of the qubit, defined as a 0 + b 1 . We vary the values a and b to maximize the error, which is defined as | i | i the sum of the 2 2 and T T elements of the resulting density matrix. This corresponds | ih | | ih | to the maximum probability of being in the states 2 and T . It is a useful error measure, | i | i and in the absence of tunneling it yields the same error that we have defined before. 266 CHAPTER 7. APPLICATIONS OF NMRQC

Figure 7.26: Plot of the error as a function of time when including tunneling eﬀects with Γ2 δω (ω10/2π)/100. The error is given by the maximum of 2 2 + A A over all possible≈ input≈ superpositions a 0 + b 1 . Now, the hard pulse shows| theih | best| performance,ih | but still with a large error. | i | i

In Fig. 7.26 we plot the error as a function of normalized pulse width for a tunneling rate Γ2 ω10/100 δω/2π. From this figure, we notice that the error is on the order of ≈ ≈ 2 10− for typical pulse widths. Note that especially for the gaussian shaped pulse, the exact calculation matches the result of the simple tunneling model from Fig. 7.25 reasonably well. It is evident from both plots that for typical pulse widths the error is much larger than the 4 10− threshold required for fault-tolerant computations [Ste02]. How can we reduce the error resulting from tunneling without significantly increasing the pulse widths? From the simple theoretical model described earlier, the most straightforward method is to decrease the tunneling rates, which can be done by adjusting the bias current I to include more than three energy levels in the well. In this case, the frequency difference

δω/2π between the transitions becomes slightly smaller. However, the tunneling rate is exponentially dependent on the barrier thickness, and thus we can reduce the tunneling rate by many orders of magnitude while only slightly decreasing the frequency diﬀerence between the transitions. With four levels in the well, we can reduce Γ2 by three orders of magnitude to about (δω/2π)/1000 while decreasing δω/2π by only 30%. Though Γ3 is still high (about δω/2π), it does not get populated much because the 2 3 transition is | i ↔ | i about 2δω/2π Hertz oﬀ-resonance and hence should not be excited by much. 7.4. NMR TOOLS FOR JOSEPHSON PHASE QUBITS 267

Figure 7.27: Plot of the error as a function of time when including tunneling eﬀects in a four level system. The tunneling rate Γ2 is (δω/2π)/1000, and Γ3 is 1000 times higher. The error is given by 2 2 + 3 3 + A A after maximizing over all possible input superpositions a 0 + b 1 .| Inih this| case,| ih the| | compositeih | pulse can perform about as well as a gaussian pulse. | i | i

The tunneling effects from 2 and 3 during single-qubit rotations can be calculated | i | i via the operator-sum representation as follows. The effects of the radiation and tunneling acting simultaneously is approximated as before by slicing the pulse into many steps and simulating tunneling and excitation one after the other in each step. The tunneling is modeled by sequentially modeling tunneling out of state 2 and 3 . In other words, we | i | i first apply the two Kraus operators that describe the tunneling from 2 to T . Then, we | i | i apply the two Kraus operators that the describe the tunneling from 3 to T . Strictly | i | i speaking this is not correct because the Kraus operators that describe tunneling from 2 | i do not commute with those describing the tunneling from 3 . However, since the pulse is | i already discretized into many steps (we used 256) during each of which we model tunneling separately, our approach becomes a good approximation. The resulting four Kraus operators can be easily derived from Eqs. 7.66 and 7.67. Furthermore, we can model the tunneling from an arbitrary number of states in this manner, and verified that the error for a three- level system is indeed dominated by the tunneling from state 2 and that tunneling from | i 1 and 0 is negligible. | i | i Fig. 7.27 shows the error for a four level system with tunneling rates Γ2 (δω/2π)/1000, ≈ and Γ3 δω/2π). As can be seen, the results are much improved compared with Fig. 7.26. ≈ 268 CHAPTER 7. APPLICATIONS OF NMRQC

In fact, now the composite pulse can be as good as the gaussian shaped pulse. It is possible to continue increasing the number of levels in the well to suppress tunneling even further. However, this may not be practical much beyond four or ﬁve levels because the state measurement needs a transition between state 1 and a higher energy state with | i a very large tunneling rate [MNAU02]. This becomes increasingly diﬃcult for more energy levels in the well. In the four level case, it is possible to directly excite the 1 to 3 | i | i transition [SLH+03]. Finally, we point out that other proposals for performing single qubit rotations in a three or multilevel system [WL03, TL00, PK02] are also problematic whenever energy levels with high tunneling rates are populated. We believe that our method for simulating the tunneling mechanism could be useful to estimate the feasibility of these and other methods.

7.4.5 Summary

In summary, we have shown how shaped and composite pulses can improve the accuracy of single-qubit operations in a Josephson phase qubit. We estimate the feasibility of our methods by including tunneling effects to show that tunneling can be a dominating source of decoherence, and we conclude that operating a Josephson phase qubit with three levels is not recommended. Instead, one may wish to operate the system with four energy levels to reduce the effects of tunneling. This work also takes first steps towards applying NMR quantum computing and spectroscopy techniques to related quantum systems. We believe that continued effort into this direction could prove fruitful for other implementations of quantum computers including solid state and trapped ion implementations. Furthermore, we applied fundamental ideas from quantum computing to simulate tunneling effects in Josephson junction qubits, illustrating how quantum computing is useful to study and simulate the physics of real quantum systems in a straightforward manner.

7.5 Quantum computing and quantum optical phenomena

This work emerged in discussions with HyungBin Son and Murali Kota. Together we worked out the theory and experiments. This work is also at the center of Murali’s [Kot03] and Bin’s [Son03] thesis. The resulting paper is in preparation for submission. 7.5. QUANTUM COMPUTING AND QUANTUM OPTICAL PHENOMENA 269

In this section we begin to explore some connection between the ﬁelds of quantum computing and Atomic, Molecular, and Atomic (AMO) Physics by studying Electromagnetically Induced Transparency (EIT) using the language of quantum computing.

7.5.1 The EIT eﬀect

In recent years, there has been a tremendous growth in developing quantum systems to harness the power of quantum computation and cryptography. While NMR-based systems have taken the lead in realizing quantum computers (see chapter 6), AMO physics is emerging as a key player in quantum communication. Recent experiments on storing and releasing pulses of light [HHDB99, LDBH01, PFWL01] in atomic gases using the Electro- magnetically Induced Transparency (EIT) effect [HFK92, Har97, HH99, Pan01] are being envisioned for applications in quantum memory for long distance quantum communications [LYF00, DLCZ01]. The two fields of NMR quantum computation and AMO physics, though tackling problems in quantum information in parallel, seem to be rather distant in their practical realizations. Given the similarities between AMO physics and NMR quantum computing, a question that naturally arises is if AMO phenomena can be efficiently realized by NMR techniques. One possible way to test this is by implementing an AMO phenomenon in NMR based quantum systems that are similar to AMO systems. Atom-like higher-order spin systems have been used to realize classical logic, and more recently, quantum algorithms via NMR techniques (see section 6.7 and references within). These systems are well suited to demonstrate AMO phenomena in NMR quantum systems. The physics of higher-order spins is quite similar to that of atom-clouds aside from the differences in techniques for their manipulation. Higher-order spins can be, in principle, used to demonstrate quantum optical phenomena, but these experiments require precision quantum control in order to be achieved. Over the past few years NMR has demonstrated techniques for efficient quantum control and manipulation of nuclear spins in the context of quantum computing. We exploit these techniques to experimentally realize the EIT phenomenon by NMR. We show how these techniques provide a framework for state preparation and their manipulation using well known quantum gates. The techniques for this are based on the universal set of quantum gates using multi- level systems (see section 4.7). Our results indicate that quantum computing techniques can be utilized for coherent control of quantum optical systems. In this section, we demonstrate EIT using an atom-like higher-order spin system. This is 270 CHAPTER 7. APPLICATIONS OF NMRQC

made possible by a careful selection of a molecule that induces an electrostatic quadrupole interaction in a liquid crystal environment. This is the same molecule as used in the experiments described in section 6.7. Three levels of the 133Cs spin states that forms an atom-cloud (Λ-like) configuration, with two transitions, were used to realize EIT by NMR. In optical EIT the transparency behavior in atom clouds is observed by probing the amount of electromagnetic radiation that passes through the cloud, observing liquid state NMR quantum systems for transparency is not straightforward as it is not possible to create a ground state of the system at room temperature. We circumvent this limitation by utilizing the mixed ensemble of spin states to create an effective pure state. This effective pure state is then used to realize the required “dark states” for various instances of EIT. We measure the transparency behavior by observing the spin states through density matrix reconstruction techniques of a higher-order spin system (see section 6.7). EIT theory also predicts that the phase of the quantum system does not change during the EIT evolution period. This property has not been demonstrated as it is very difficult to measure the phase evolution of atomic systems. Due to advanced NMR techniques and relatively long coherence times of spin systems, we show that the phase coherence of the dark states in spin systems is maintained during the EIT Hamiltonian evolution through Ramsey interferometry.

Theory of the EIT eﬀect

Let us start by analyzing the EIT effect in a three level quantum system. Consider a three level system [HHDB99, LDBH01] and denote its energy eigenstates as 1 , 2 and 3 . The | i | i | i energy differences between the 1 and 2 corresponds to ωprobe and between 2 and 3 | i | i | i | i states corresponds to ωcontrol. Two electromagnetic fields are applied simultaneously to the system: the probe field at frequency ωprobe with strength a and the control field at frequency

ωcontrol with strength b. The Hamiltonian describing the incident radiation is described by the following in the 1 , 2 and 3 basis: | i | i | i 0 a∗ 0   Heit = a 0 b∗ (7.68)    0 b 0  The numbers a and b can in general be complex, but we will choose them to be real here. The most interesting cases for EIT occur when (i.) the intensities of the two frequencies are such that b a. and when (ii.) we have a coherent superposition of 1 and 3 and | i | i 7.5. QUANTUM COMPUTING AND QUANTUM OPTICAL PHENOMENA 271

|2> I z (Spin Angular Momentum)

ω control −1/2 |1> ω probe ω probe 1/2 |2> |3> ω control |1> 3/2 |3> (a) (b)

Figure 7.28: (a) Energy level diagram of Λ-like spin states of a three-level atomic system. (b) Schematic energy level diagram for the Iz = 1/2, 1/2, 3/2 levels a multi-level system with quadrupole splitting. − the intensities a and b are tuned in the appropriate ratios. (i) The time evolution of the system under the inﬂuence of H can be written as R = iHt e− . An interesting case for EIT occurs when a = b = 1/√2, leading to the evolution

1 i a sin(bt) a (cos(bt) 1) b b −  a  R = i b sin(bt) cos(bt) i sin(bt) , (7.69)    a (cos(bt) 1) i sin(bt) cos(bt)  b − a to ﬁrst order in b . From the expression in Eq. 7.69, we ﬁnd that the matrix element connecting states 1 and 2 is zero in the b a limit. This means that the system does | i | i not absorb incident radiation at the frequency ωprobe! (ii). There exists yet another interesting instance of EIT, wherein certain quantum states of the EIT system, known as coherent dark states, do not evolve under the EIT

Hamiltonian. Consider the eigenstates of the Hamiltonian, Heit,

a a √2 b2  b b −  EV = s √b2+a2 √b2+a2 0 (7.70) eit 2(a2 + b2)  b − b   √2a   1 1 b  where the eigenvectors of Heit are the column vectors of EVeit. The corresponding eigen- 2 2 2 2 values of the ﬁrst, second, and third columns of EVeit are +√a + b , √a + b and 0 − respectively. Suppose the quantum system is in the eigenstate corresponding to the 0 eigenvalue, and Heit is applied by subjecting the system to radiation simultaneously at the probe 272 CHAPTER 7. APPLICATIONS OF NMRQC

and control frequencies. Since the eigenvalue of the quantum system is 0, it does not evolve under the Heit evolution. The quantum system is left unperturbed by the EIT Hamilto- nian, which means that the two applied frequencies ωprobe and ωcontrol do not interact with the quantum system. The quantum system is thus transparent to both frequencies simultaneously. By creating a unique superposition of 1 and 3 , also known as a “coherent | i | i dark state”, we have made the quantum system transparent to electromagnetic radiation at ωprobe and ωcontrol. As an example, consider the situation where we have a = b. In this a case, the unitary evolution due to the EIT Hamiltonian can be written as,

1 + 1 cos(lt) i sin(lt) 1 (cos(lt) 1)  2 2 √2 2 −  R = i sin(lt) cos(lt) i sin(lt) (7.71) √ √  2 2   1 (cos(lt) 1) i sin(lt) 1 + 1 cos(lt)  2 − √2 2 2 1 3 where, l = (a2 + b2). From Eq. 7.70, we find that the eigenstate | i−| i is a dark state p √2 of the EIT Hamiltonian, which is transparent to the incident electromagnetic radiation at frequencies ωprobe and ωcontrol. Hence, coherent trapping of populations in the two states 1 and 3 forms the key to | i | i demonstrating the EIT effect [LI01, Pan01]. Such a unique quantum state has the effect of completely eliminating the absorption or emission of light. Coherent trapping of populations has been used to achieve extremely low group velocities of wave packets of light in atom- clouds [HHDB99, LDBH01]. In the following sections, we show efficient schemes for coherent population trapping using quantum gates, and use these techniques to demonstrate EIT in NMR quantum systems. Before moving on to discuss techniques for EIT by NMR, we next show an interesting observation that EIT is an instance of quantum bang-bang control [VL98].

Quantum Control

Here we provide an explanation of EIT as being an instance of quantum bang-bang control19 in the limit b a. When we apply two Hamiltonians to a system, with one being much higher in intensity than the other, we can approximate the unitary evolution of the system

19Aram Harrow ﬁrst made this claim, and together Murali and Bin they worked out the details. 7.5. QUANTUM COMPUTING AND QUANTUM OPTICAL PHENOMENA 273

by quantum bang-bang control as,

U khU k = 0, (7.72) X † k where h is the weak Hamiltonian and U k are a set of unitary operators corresponding to the strong Hamiltonian.

In our case, h and U k are given by,

h = ha, (7.73) h τ k i b k U = e N (7.74)

where ha and hb describes the interaction of the probe and control ﬁeld respectively with the system:

0 a 0 0 0 0     ha = a 0 0 , hb = 0 0 b . (7.75)      0 0 0   0 b 0 

The time evolution of the system, in the limit b a, is given by N 1 − ha hb i(ha+hb) i i lim e− lim e− N e− N (7.76) b a ≈ N Y →∞ k=0 N 1 − k ha k i hb i i hb ihb = lim " e− N e− N e N # e− N Y →∞ k=0 N 1 − ha i ihb = lim " Uke− N U †# e− (7.77) N Y k →∞ k=0

k i hb where Uk = e− N ,

N 1 − ha iUk U † ihb = lim " e− N k # e− (7.78) N Y →∞ k=0 N 1 i ( − UkhaU †) ( ihb) = lim e Pk=0 − N k e − (7.79) N →∞ iS ihb = e− e− (7.80) 274 CHAPTER 7. APPLICATIONS OF NMRQC

where,

N 1 − ha S = lim Uk U † (7.81) N X N k →∞ k=0 which can be rewritten as

bk bk N 1 0 a cos N ia sin N − 1  bk  S = lim a cos N 0 0 (7.82) N X N   →∞ k=0  ia sin bk 0 0  − N 0 a cos x ia sin x 1 1   = Z a cos x 0 0 dx (7.83) b 0    ia sin x 0 0  − where x = k/N,

0 a sin b i a (cos b 1) b − b −  a  = b sin b 0 0 (7.84)    i a (cos b 1) 0 0  b − From the above it is clear that, in the limit b a, the Hamiltonian S 0. We then have → from Eq. 7.82,

iS ihb ihb lim e− e− = e− (7.85) b a 1 0 0   = 0 cos b i sin b (7.86)    0 i sin b cos b 

This is very surprising as we have the same form from the analytical solution for R in Eq. 7.69. While it is not likely to be of practical use, the connection between the EIT eﬀect and quantum bang-bang control is interesting to note.

Coherent population trapping using quantum gates

In both the instances of EIT, state preparation is the key element for demonstrating the EIT eﬀect. Traditionally, the “dark state” in atomic systems is prepared by slowly ramping the probe in the presence of a control ﬁeld to adiabatically evolve the state from the 7.5. QUANTUM COMPUTING AND QUANTUM OPTICAL PHENOMENA 275

initial ground state 1 to the stationary state of the EIT Hamiltonian [HHDB99, LDBH01, | i PFWL01]. In this subsection, we show eﬃcient schemes for coherent population trapping using non-adiabatic discrete quantum gates. Our procedure is a concrete and systematic method of creating the required “dark state” for any given set of probe and control ﬁeld intensities.

Let us discuss the preparation of the dark state when both the probe and control ﬁeld intensities are equal. We start with the system ground state 1 , followed by the application | i of the Hadamard gate that creates an equal superposition of 1 and 2 . This gate can be | i | i described mathematically as,

1 1 0 1   1 H(1,2) 1 = 1 1 0 1 = ( 1 + 2 ) (7.87) | i √2  −  | i √2 | i | i  0 0 √2 

Here the subscript (p,q) indicates the subspace in which the operation is performed. We then apply the not gate, with an overall phase of π, in the ( 2 , 3 ) subspace. The not | i | i operation ﬂips the states 2 and 3 with the required phase to create the dark state. This | i | i gate can be written in matrix notation as,

1 0 0 iπ   (e NOT )(2,3) = 0 0 1 . (7.88)  −   0 1 0  − The quantum system is now in the state ( 1 3 )/√2. This state is transparent to | i − | i electromagnetic radiation applied simultaneously at the probe and control frequencies. For diﬀerent ratios of a and b, this procedure can be generalized to ﬁrst create an arbitrary superposition of 1 and 3 using single qubit rotations in the ( 1 , 2 ) subspace, followed | i | i | i | i iπ by the e NOT operator in the ( 2 , 3 ) subspace. For example, applying the Pauli σx for | i | i an amount of time cos 1 (a) creates an arbitrary superposition of the states 1 and 2 such − | i | i that

(1,2) 1 ( iσx cos (a)) Ψ = e − − 1 = a 1 ib 2 (7.89) | i | i | i − | i

2 where b = 1 a . This operation, followed by a σy pulse for a duration π in the ( 2 , p − | | | i 276 CHAPTER 7. APPLICATIONS OF NMRQC

3 ) subspace leads to an arbitrary superposition of the states 1 and 3 . | i | i | i

(2,3) ( iσy π) e − Ψ = b 1 a 3 (7.90) | i | i − | i

This is the dark state corresponding to an arbitrary probe field strength of a and control (p,q) field strength of b. Here, σj indicates the Pauli operation on the (p, q) subspace along the j axis. Hence for any given strengths of the probe and control fields, we can prepare the required superposition dark states. Our described method is systematic and concrete, thus making quantum gates very useful in realizing coherent states.

7.5.2 Simulation of the EIT eﬀect (3 energy levels)

We now show the experimental realization of the EIT effect using higher-order spin systems. These atom-like systems, are multi-leveled, as described in section 4.7.2 and are thus suitable for atom quantum optical experiments by NMR. Similar to the description of the experiment in section 6.7, we used 50 percent of cesium pentadecafluorooctanoate by weight in D2O. The levels 1 , 2 , and 3 correspond to the | i | i | i 1/2, 1/2 and 3/2 spin states. All other experimental equipment was identical as well, − − including the procedure to synthesize an effective pure state.

EIT in the presence of a strong control ﬁeld

The EIT experiments in the strong control field limit were performed at fixed probe field strengths, while sweeping the control field strength from 0 to a strength corresponding to 10π pulses. The transparency behavior is verified by observing the spin states. This is made possible by observing the NMR spectrum of the spins, which is the Fourier transform of the free induction decay signal. The time-varying complex-valued voltage V (t), measured by a pick-up coil around the sample is

iHt iHt ˆ t/T2 V (t) = T r he− RρR†e Oi e− , (7.91)

k k where, Oˆ is the measurement observable, Oˆ = (σ iσ )/2 = 0 1 + 1 2 and T2 is Pk=1,2 x − y | ih | | ih | the transverse (phase) relaxation time constant. The Fourier transform of the time varying voltage V (t) is expected to show the characteristic 1/b behavior with a ﬁxed intensity of the probe frequency ωa. This behavior is experimentally veriﬁed by observing the single 7.5. QUANTUM COMPUTING AND QUANTUM OPTICAL PHENOMENA 277

quantum coherence corresponding to the matrix element 1 2 . This can be done without | ih | applying any read-out pulse. The pulse sequence implementing this experiment is given by,

Ub a = (Ueit), (7.92) where Ueit = exp( iHeitt) and Heit is from Eq. 7.68. This is followed by the ﬁnal read-out − of the 1 2 matrix element, which does not require additional pulses. | ih |

Figure 7.29: Experimental data verifying the transparency behavior of an NMR EIT system in the strong control field limit. Here, we observe the signal corresponding to the 1 2 element of the density matrix. The intensity of the probe field was set to a strength| ih | equivalent to a π pulse. The dash-dot line represents the 1/b trend of Rabi oscillations in the 1 2 element. The dotted line is the simulation result, which includes the RF inhomogeneity| ih | of the control field and the Bloch-Siegert shifts. The points are experimental results obtained by averaging over five experiments. The height of the error bars were obtained from standard deviations normalized by Nexperiments 1, where Nexperiments is the number of repeated experiments. −

The experimental results are shown in Figures 7.29 and 7.30 for a probe ﬁeld strength corresponding to π and 3π/2 pulses. These results are compared with the expected results. Clearly, we observe the expected 1/b trend of Rabi the oscillations of the 1 2 transition. | i → | i The simulation takes into account Bloch-Siegert shifts, and exponential decay due to RF 278 CHAPTER 7. APPLICATIONS OF NMRQC

Figure 7.30: Experimental data verifying the transparency behavior of an NMR EIT system in the strong control field limit. Here, the intensity of the probe field was set to a strength equivalent to 3π/2 units. inhomogeneity of the NMR probe. Our results show that the EIT system approaches the dark state 1 as the strong control field increases in strength. The signal strength, as seen | i in Figs. 7.29 and 7.30, progressively gets smaller indicating that the system does not absorb at radiation at frequency ωprobe in the asymptotic limit of b , showing the first evidence → ∞ of the EIT effect by NMR.

EIT with coherent dark states: theory and experiments

We now discuss EIT with coherent dark states, not just in the limit b a, and present experimental results. As we analyzed in the beginning of this section, a coherent superposition of levels 1 and 3 creates a situation wherein the quantum system becomes transparent to | i | i both the probe and control frequencies simultaneously. The required dark state is prepared using an effective pure state 1 through temporal labeling followed by the application of | i the required quantum gates. Then the EIT Hamiltonian is applied by turning on the probe and control fields simultaneously, and the transparency behavior is verified by observing the dynamics of the density matrix of the spin system under the EIT Hamiltonian evolution. We first experimentally reconstruct the deviation density matrix of the EIT dark 7.5. QUANTUM COMPUTING AND QUANTUM OPTICAL PHENOMENA 279

state using techniques that were developed in section 6.7. The dark state is then allowed to evolve under the EIT Hamiltonian Heit. We verify transparency by reconstructing the time evolution of the dark state deviation density matrix. Since the system is in the stationary state of the EIT Hamiltonian, we expect the density matrix to remain unchanged when applying Heit. The pulse sequence implementing this experiment is given by,

Ucoherent = (Ueit), (7.93) where Ueit = exp( iHeitt) and Heit is from Eq. 7.68. This is followed by the procedure − to reconstruct the deviation density matrix, which is realized by performing a series of experiments to convert all the elements of the density matrix into single quantum coherences that are observable by NMR (see section 6.7).

(a) (b)

Figure 7.31: Experimentally reconstructed deviation density matrix of the “dark state” 1 3 − (a) before applying the EIT Hamiltonian and (b) after applying the EIT Hamil- | i√2| i tonian for a duration equivalent to 3π/2 pulses. Note that the density matrices presented here show only the absolute values of the elements for the sake of visual clarity.

The results are shown in Fig. 7.31 and conﬁrm this behavior. Furthermore, the results are within 15 percent of the expected density matrix; here 15 percent indicates the highest error in an element of the experimental density matrix compared to the expected density matrix. From additional control experiments, it was found that the errors are mainly due to decoherence eﬀects and RF inhomogeneity of the probe. Another important aspect to note from our theoretical analysis is that the phase of the 280 CHAPTER 7. APPLICATIONS OF NMRQC

dark state does not change during the EIT evolution period. But how can we test this experimentally? Fortunately, we have additional energy levels in the 133Cs spin system that are not used in our experiments and that could be useful. It is possible to couple the dark state to an additional level, the Iz = 3/2 spin state. This permits us to measure the phase − of the dark state during the EIT evolution period by performing a Ramsey interferometry experiment, which is an indirect measure of the phase coherence of a quantum system.

Ramsey interferometry for measuring phase coherence: theory and experiments

In this section, we discuss the phase evolution of the EIT system through Ramsey interferometry.

(a) (b)

Figure 7.32: (a) Energy level diagram of Λ-like spin states of a three-level atomic system coupled to an additional reference level R . (b) Schematic energy level diagram for the | i Iz = ( 3/2, 1/2, 1/2, 3/2) levels a multi-level system with quadrupole splitting. Here, 11 = −R , 10− = 1 , 01 = 2 , and 00 = 3 . | i | i | i | i | i | i | i | i

The basic idea of Ramsey interferometry is to measure the relative phase between any two energy levels by creating interference between the two levels. To test for the phase evolution of the dark state, we apply the following quantum operations to create an interference between the fourth level, labeled as the reference state R (see Fig. 7.32) and the | i dark state D . | i

iHeit iθσz URamsey = HRDe− e HRD (7.94) 7.5. QUANTUM COMPUTING AND QUANTUM OPTICAL PHENOMENA 281

where time goes from right to left, and HRD is a hadamard gate, creating the superposition

( R + D )/√2, and σz is the Pauli z-operator in the reference state R and dark state D | i | i | i | i basis. The operator σz changes the relative phase between the reference and dark state. Now we calculate the expectation value of the reference state with the application of U.

2 iHeit iθσz 2 R URamsey R = R HRDe− e HRD R (7.95) |h | | i| |h | | i| iHeit 2 = R HRDe− (cos(θ/2) + i sin(θ/2)σz)HRD R (7.96) |h | | i| 2 = R HRD(cos(θ/2) + i sin(θ/2)σz)HRD R (7.97) |h | | i| = cos2(θ/2) (7.98)

From the above we expect to see a cos2(θ/2) behavior in the expectation value of the reference state. We conﬁrm this phase evolution by observing the visibility of the expectation of the

Imax Imin R deﬁned as V(a, b) = − as a function of the varying ratios of b to a, where I | i Imax+Imin is the NMR signal strength corresponding to the expectation of R . The intensity of the | i expectation of the state R is given by | i 1 + p2 + 2p cos θ I(θ) = R U R 2 = (7.99) |h | | i| 4 where p is the expectation of the state R over the EIT evolution: p D e iHeit D . Note | i ≡ h | − | i that R U R 2 reduces to cos2 (θ/2) when p = 1, which is the condition for the maximum |h | | i| visibility.

Visibility in the strong control ﬁeld limit: theory and experimental results

In this section, we study the Ramsey interferometry in the limit b a and present experimental results to measure the quantum mechanical phase coherence of the EIT system. The unitary operation corresponding to this experiment is given by,

iHeit iθσz URamsey = HR1e− e HR1 (7.100) where HR1 is the hadamard gate in the R and 1 basis and σz is the Pauli sigma-z | i | i operator in the same basis. Let us systematically analyze the eﬀect of each of the individual operations in URamsey. 282 CHAPTER 7. APPLICATIONS OF NMRQC

1 + R After the ﬁrst hadamard operation, the quantum state is in the state ψ0 = | i | i. | i √2 The state of the system due to EIT evolution is given by,

1  √2  2 2 eiθ( b + a cos l) iHeit  l2 l2  ψ1 = e− ψ0 =   (7.101) | i | i  eiθ( i a sin l)   − l   eiθ ab (cos l 1)   l2 − 

2 2 where l = √a + b . Here ψ1 is written in the R , 1 , 2 , and 3 basis. The application | i | i | i | i | i of the second hadamard operator to the state ψ1 gives, | i

iθ 1 + e c  2 √2  iθ 1 e− c  2 √2  ψ2 = HR1 ψ1 =  −  (7.102) | i | i  eiθ( i a sin l)   − l   eiθ ab (cos l 1)   l2 −  where c is given by, b2 a2 c = + cos l (7.103) l2 l2 When we write out the density matrix corresponding to the above state, we ﬁnd the expectation value of R to be, | i 2 2 1 + c + 2c cos(θ) R URamsey R = (7.104) |h | | i| 4

With the above analytic expression in the expectation of R , we estimate the observable in | i NMR corresponding to the single quantum coherence R 1 which is obtained by applying | ih | a readout pulse on the R 1 transition. Summarizing the above, we can write the pulse | i → | i sequence implementing URamsey as

(Y (3, 1)X(3, 1))A(Z(3, θ))B(Ueit)C (Y (3, 1)X(3, 2))D(X(3, 1))E (7.105)

Note that because this is a pulse sequence, time goes from left to right. The symbols X(r, s), Y (r, s), and P (r, s) denote rotations aboutx ˆ,ˆy, andz ˆ axes respectively, r indicates the transition number, and s indicates the strength of the pulse in units of π/2. The EIT evolution period is denoted by Ueit. The symbol A indicates the hadamard operation in 7.5. QUANTUM COMPUTING AND QUANTUM OPTICAL PHENOMENA 283

the R and D basis , where D = 1 in the strong control field limit; B denotes thez ˆ- | i | i | i | i rotation on the dark state D in the R and D basis; C denotes the EIT evolution period; | i | i | i D denotes the hadamard operation in the R and D basis; E is the final read-out pulse | i | i on R 1 transition. | i → | i The individual x or y pulses were realized using Gaussian shaped pulses of duration 620 µs set at a power corresponding to a π pulse. The EIT pulse was realized using Gaussian shaped pulses of duration 6 ms. This is much longer than 620 µs and is needed because the control field power is swept from 0 to a setting corresponding to a 21π/2 pulse. At such a high power setting, and with a pulse width of 620 µs, the coil would arc, or possibly be destroy sensitive electronics. For a much longer pulse, we are able to tune the power to a lower setting, allowing us to sweep the power over a much larger dynamic range. The probe field power was set to a power corresponding to 3π/2 pulses. The pulse lengths and power setting were carefully calibrated such that we were able to maintain the selectivity of the transition pulses. Also, the length of non-EIT pulses was selected so that the experimental time was much shorter than relaxation times of the transitions, but still selectively excite the individual transitions. We plot the visibility of R 1 quantum coherence as a function of b/a is shown in | ih | Fig. 7.33. The experimental results are shown as error bars. The height of the error bars indicate random errors occurring over three identical experiments, which were obtained from standard deviations normalized by Nexperiments 1, where Nexperiments is the number of − repeated experiments. The simulated curve, shown as a dashed line, is obtained by including decoherence effects20. Decoherence effects due to the total run time for implementing all the pulses in Eq. 7.105 are significant in estimating the experimental data.

Visibility for EIT with coherent dark states: theory and experimental results

In this section, we discuss the veriﬁcation of quantum mechanical phase coherence for coherent dark states. As seen in Eq. 7.98, we ﬁnd that the expectation of R for Ramsey | i interferometry to be a cos2(θ/2) function, and this is plotted in Fig. 7.34.

20Modeling decoherence eﬀects in multi-level systems is out of the scope of this thesis, but has been described in detail in Murali Kota’s Master Thesis [Kot03]. This has been done via Lindblad Equations and also Kraus operators. 284 CHAPTER 7. APPLICATIONS OF NMRQC

0.9

0.8

0.7

0.6

0.5

Visibility 0.4

0.3

0.2

0.1

0 0 1 2 3 4 5 6 7 b/a

Figure 7.33: Simulations and experimental results of visibility as a function of b/a in the strong control ﬁeld limit. The solid line indicates the ideal expectation of visibility, the dashed line indicates the simulated behavior including decoherence eﬀects and pulse imperfections, and the points indicate the experimental results. The experimental results are in close agreement with simulations.

For this experiment, we start with a superposition of R and the dark state such that, | i

1  √2  eiθ 1  2  φ0 =   (7.106) | i  0     eiθ 1   − 2  1 3 The dark state D = | i−| i corresponds to the case when a = b. The time evolution of | i √2 7.5. QUANTUM COMPUTING AND QUANTUM OPTICAL PHENOMENA 285

Figure 7.34: The expectation of the reference state R as a function of θ. The solid line indicates the ideal value in the expectation of R , the| dashed-linei the simulated expectation of R including decoherence eﬀects and the| squaresi indicate experimental results. They show| i the characteristic cos2 θ behavior derived.

φ0 under the EIT Hamiltonian evolution results in the following state, | i

1  √2  iθ b2+a2 cos(l) ab cos(l)+ab e −2 φ1 =   (7.107)  iθ i sin(l)(b a)  | i  e − −   2l   iθ ab cos(l)+ab+a2+b2 cos(l)   e − l2 

Applying the second hadamard transform HRD results in the state,

1  √2  eiθ q  2  φ2 =   (7.108) | i  eiθ 1 q2   q 2 −   eiθ q   √2 

2 2 (a+b) +(a b) cos(l) √ 2 2 where q = 2l−2 and l = a + b . When we write out the density matrix corresponding to the above ﬁnal state, we ﬁnd 286 CHAPTER 7. APPLICATIONS OF NMRQC

the expectation value of R to be, | i 2 2 1 + q + 2q cos(θ) R URamsey R = (7.109) |h | | i| 4 which is similar to the expression we obtained for EIT in the strong control ﬁeld limit. The pulse sequence for implementing URamsey is given by,

(Y (1, 1)X(2, 1)X(3, 2))A(Z(3, θ)Z(1, θ))B(Ueit)C ((X(3, 2)X(2, 1)Y (1, 1)Z(3, 2)Z(2, 4)Z(1, 2) − X(2, 1)X(3, 2)Z(1, 3)Z(2, 2)Z(3, 1))D (7.110)

Here, time goes from left to right. X(r, s), Y (r, s), and Z(r, s) denote rotations aboutx ˆ,ˆy, andz ˆ axes respectively. r indicates the transition and s indicates the strength of the pulse in units of π/2. Ueit denotes the EIT evolution period. A indicates the first pulse sequence R 1 3 implementing the superposition ψ = | i + | i−| i. B denotes the z-rotation on the | iRD √2 2 1 3 dark state D = | i−| i in the R and D basis. C denotes the EIT evolution period, D | i √2 | i | i denotes the hadamard operation in the R and D basis. It should be noted that the | i | i sequence for A should have been the same as D. But as we are free to prepare the state ψ is any way we want, we use the shortest possible pulse sequence to minimize the | iRD decoherence effects. Following this sequence, we reconstruct the deviation density matrix, which involves a series of experiments to read-out all the elements of the density matrix (see section 6.7). b The visibility of the reference state for varying ratios of a is shown in Fig. 7.35. The maximum visibility in the above results is about 75 percent; the deviation from the ideal expected curve is largely due to relaxation effects during the time of the experiments and also RF inhomogeneity in the NMR probe. We have included the effects of relaxation and experimentally measured pulse imperfections due to RF inhomogeneity of the probe. Our simulation and experimental results are in agreement with each other.

Summary

In this section we have discussed how quantum computing and quantum optical phenomena have deep interconnections, as illustrated via the EIT eﬀect. We have shown how EIT can be re-interpreted as an instance of quantum bang-bang control. We have applied the 7.6. SUMMARY 287

Figure 7.35: Visibility of of the expectation value of R state as a function of b/a for EIT with a coherent dark state. The solid line is the ideal case| i scenario, the dashed-line indicates simulated expectation with decoherence effects and the points indicate experimental results. 1 3 Here, the dark state is − , while the ratio b/a is varied with a fixed probe field and | i√2| i varying control field. ideas of quantum gates to provide a general method of creating dark states for atomic physics systems and showed, using NMR techniques, that the phase of dark states remains unchanged. We believe that these are just the first few steps towards discovering new and exciting connections between the two fields of quantum computing and AMO Physics. It is our hope that continued effort into this direction leads to efficient applications of quantum computing techniques.

7.6 Summary

This chapter illustrated how quantum computing and NMR quantum techniques can be transferred to other implementations of quantum computers. We have done this using three examples:

Application to an optical lattice quantum computer • Application to a Josephson phase qubit • 288 CHAPTER 7. APPLICATIONS OF NMRQC

Quantum computing and quantum optical phenomena (EIT by NMR) • We are excited to observe that NMR quantum computing is beginning to extend its applications, and that the language of quantum computation can shed new light on a wide variety of other theoretical and experimental problems, even in areas that at ﬁrst appear to be unrelated to quantum computing! Considering the parallels between two of the proposals for implementing a quantum computer (sections 7.1 and 7.2) and NMR quantum computing, we believe many more applications soon emerge. The results in each of the three examples are mostly my own contributions, except for section 7.5 which was worked out together with Murali Kota and HyungBin Son. Chapter 8

Conclusions

Quantum computation has moved from theoretical fancy to experimental reality, using nuclear spins in molecules dissolved in liquid solution. Over the years we continued to push the state-of-the-art in NMR Quantum Computing by implementing a series of meaningful quantum algorithms. My goal has been to not just demonstrate the proof of principle of quantum algorithm, but to choose the experiment such that I really put the quantum computer to the test, and also learn something which I could not have learned, had I not done the experiment. I furthermore aimed at inventing techniques that could beneﬁt other implementations of quantum computers. These goals have been met successfully:

Proof of principle • In the experiments described in chapter 6, I have unambiguously shown that quantum computers can in principle be built, and that the developed quantum algorithms work as expected.

These experiments were successful, not just due to the developed tools, but also due to the careful choice of molecules, which exhibited phenomenal NMR properties.

Stimulation of theoretical questions • Throughout this thesis, the importance and potential applications of quantum computers has become increasingly clear. In addition to showing the proof-of-principle of quantum computing, NMR quantum computing techniques have also raised an important theoretical question. For example, NMR quantum computing started the interesting debate of what resources are required to speed-up quantum algorithms, as

289 290 CHAPTER 8. CONCLUSIONS

brieﬂy described in section 4.9.

Another theoretical question that was raised by NMR techniques is how decoherence can be modeled in a predictive and workable manner on real systems. We have developed such a model and used its predictive value in several experiments (see sections 6.6, and 6.8). We found the model to be immensely useful for estimating the effects of decoherence on these two experiments. This has helped us determine the source of experimental errors, coming to the conclusion that we had excellent control over the coupled spins, except for decoherence effects. We also found that the same model can be adapted to model tunneling in a Josephson phase qubit (see section 7.4), coming to the conclusion that one may want to operate the device differently than previously proposed.

Transfer of developed techniques • In chapter 5 I have explicitly introduced several methods to improve the coherent control of nuclear spins: more efficient and effective state initialization schemes which were extremely beneficial for the implementation for the five and seven qubit experiments; methods to reduce the cross-talk between spins, and minimize several artifacts that arise when using shaped pulses; refocusing schemes to help reduce the effect of undesired terms in the Hamiltonian; methods to unwind J-couplings during RF pulses; methods to simplify arbitrary pulse sequences; ways to interpret the ensemble- averaged measurements; read-out schemes based on knowledge of the J-couplings. These techniques are beginning to find applications in traditional NMR experiments.

Chapter 7 shows several examples of how the language of quantum computing can benefit other implementations of quantum computers. I show how the language of quantum computing is useful to synthesize a composite pulse for an optical lattice quantum computer. I also show that NMR techniques are useful to improve single- qubit operations in a Josephson phase qubit. In addition, I estimate the feasibility of these methods in a real system by modeling tunneling effects via the operator-sum representation. Finally, I have discovered intriguing connections between AMO physics and quantum computing by implementing the EIT effect using NMR techniques.

The tools and techniques we developed are mostly practical in nature. The signiﬁ- cance of such practical problems will increase as other, perhaps more scalable implementations of quantum computers emerge. I am convinced that the signiﬁcance of 291

our techniques will grow, as they ﬁnd useful applications in other implementations of quantum computers.

Despite all the positive aspects that emerged from the experimental study of experimental quantum computing using NMR techniques, it is known that NMR quantum computing has some severe scaling limitations, the most common issue being the thermal equilibrium state at room temperature (the probability that all n spins start off in the ground state scales as n/2n). Even though efficient schemes exist to circumvent this problem, they are impractical unless the thermal spin polarization can be significantly increased. Another problem is that the decoherence times typically decrease for larger molecules, while pulse widths typically have to increase because the frequency separation between different spins becomes smaller. Despite this drawback, I firmly believe that NMR will continue to be very valuable for implementing small scale quantum algorithms and it will continue to raise several interesting theoretical questions. Through our experiments, we have learned a tremendous amount about what it would take to build a useful quantum computer: it is very challenging. A remaining important question is whether a “useful” quantum computer will ever exist? Regardless of what the answer may be, we will learn something fundamentally new about physics - either the Church-Turing thesis will have to be overhauled, or it will prevail as being correct, even in the quantum world. I also expect new quantum devices to emerge from the study of quantum computation. While the answers to these questions may still lie decades into the future, NMR quantum computing provides useful insight into the realm of quantum computing today, and it is my hope that this thesis gives us a glimpse of what may lie ahead. Appendix A

Numerical Models

We provide several numerical models that we have used extensively over the years, written for MATLAB:

1. def.m: set-up Hamiltonians and Pauli matrices.

2. X.m, Y.m, and Z.m: Ideal single-qubit rotations.

3. de.m” ideal free evolution under the coupling Hamiltonian.

4. rho.m: calculates the output density matrix based on an input density matrix and unitary evolution.

5. gop.m: generalized operator.

6. mykron.m: kronecker product function.

7. single-qubit rotations including J-coupling for arbitrary pulse shapes.

8. excitation proﬁle for arbitrary pulse shapes.

9. calculation of transient Bloch-Siegert shift.

10. gad7.m: generalized amplitude damping for 7 spins.

11. pd7.m: phase damping for 7 spins.

12. sample code simulation decoherence eﬀects.

292 A.1. SET-UP HAMILTONIANS AND PAULI MATRICES 293

The function def.m is called in the beginning to set up the appropriate Hamiltonians and Pauli matrices. This requires a prior definition of a variable called nq which defines the number of qubit that we are working with. For several of the shown codes, we read in an arbitrary pulse shape (eg. Gaussian or Hermite-shaped). The pulse shapes for our system are stored in files with the extension ’RF’ (eq. gauss.RF). The file contains three columns; the first lists the phase value for each slice (generally 0 or 180 but these can be changed when using phase-ramped pulses); the second column lists the amplitude of the pulse in some arbitrary units (between 0 and 1023); the third column lists the slice length in some arbitrary units (typically this is always set to 1). All of our simulations will read in all three of the columns. In the next few subsections, we shall give excerpts from these codes, as enumerated above.

A.1 Set-up Hamiltonians and Pauli matrices

% FILE: def.m % DATE: 1997 % Author: Lieven Vandersypen % Matthias Steffen % % Declares variables used in simulation programs. % Use: declare nq (number of qubits), then type def

global nqubits H Si Sx Sy Sz

nqubits=nq;

% Pauli matrices

Si=eye(2); Sx=[0 1 ; 1 0]; Sy=[0 -i; i 0]; Sz=[1 0 ;0 -1];

switch nqubits ... case 7 % for seven qubits

% define J-coupling values of our molecules % spin labeling is in order of the frequency. 294 APPENDIX A. NUMERICAL MODELS

Jab=-114; Jac=25; Jad=6.6; Jae=14.5; Jaf=-221; Jag=38; Jbc=80; Jbd=2; Jbe=13; Jbf=54; Jbg=-5.7; Jcd=2.5; Jce=3.9; Jcf=18.5; Jcg=-3.8; Jde=41.6; Jdf=19.4; Jdg=60; Jef=1; Jeg=-13.5; Jfg=69;

% define 7 spin Pauli matrices.

Sziiiiii=mykron(Sz,Si,Si,Si,Si,Si,Si); Siziiiii=mykron(Si,Sz,Si,Si,Si,Si,Si); Siiziiii=mykron(Si,Si,Sz,Si,Si,Si,Si); Siiiziii=mykron(Si,Si,Si,Sz,Si,Si,Si); Siiiizii=mykron(Si,Si,Si,Si,Sz,Si,Si); Siiiiizi=mykron(Si,Si,Si,Si,Si,Sz,Si); Siiiiiiz=mykron(Si,Si,Si,Si,Si,Si,Sz);

% define natural coupling Hamiltonian.

H=2*pi*Jab*Sziiiiii/2*Siziiiii/2 + 2*pi*Jac*Sziiiiii/2*Siiziiii/2 + ... 2*pi*Jad*Sziiiiii/2*Siiiziii/2 + 2*pi*Jae*Sziiiiii/2*Siiiizii/2 + ... 2*pi*Jaf*Sziiiiii/2*Siiiiizi/2 + 2*pi*Jag*Sziiiiii/2*Siiiiiiz/2 + ... 2*pi*Jbc*Siziiiii/2*Siiziiii/2 + 2*pi*Jbd*Siziiiii/2*Siiiziii/2 + ... 2*pi*Jbe*Siziiiii/2*Siiiizii/2 + 2*pi*Jbf*Siziiiii/2*Siiiiizi/2 + ... 2*pi*Jbg*Siziiiii/2*Siiiiiiz/2 + 2*pi*Jcd*Siiziiii/2*Siiiziii/2 + ... 2*pi*Jce*Siiziiii/2*Siiiizii/2 + 2*pi*Jcf*Siiziiii/2*Siiiiizi/2 + ... 2*pi*Jcg*Siiziiii/2*Siiiiiiz/2 + 2*pi*Jde*Siiiziii/2*Siiiizii/2 + ... 2*pi*Jdf*Siiiziii/2*Siiiiizi/2 + 2*pi*Jdg*Siiiziii/2*Siiiiiiz/2 + ... 2*pi*Jef*Siiiizii/2*Siiiiizi/2 + 2*pi*Jeg*Siiiizii/2*Siiiiiiz/2 + ... 2*pi*Jfg*Siiiiizi/2*Siiiiiiz/2;

% NOTE: This is only an excerpt and the code can similary be written for % any number of qubits. end

A.2 Ideal single-qubit rotations

% FILE: X.m % DATE: 1999 A.3. TIME EVOLUTION UNDER THE NATURAL HAMILTONIAN 295

% Author: Lieven Vandersypen % % Rotation of spin ’spinname’ about X over angle*pi/2 (right hand rule) % example: X(a,3) does rotation about X of 3pi/2 % X(1,3) does the same thing function X=X(spinname,angle) global nqubits if nargin == 1 if nqubits == 1 angle=spinname; spinname=1; else error(’X.m requires two arguments if nqubits > 1 !’) end end operator=expm(-i*angle*pi/2*[0 1;1 0]/2); % calculate 1-spin operator X=gop(spinname,operator); % turns 1-spin operator into n-spin operator % gop essentially creates a Kronecker product of ’nq’ identity matrices, % except for the ’spinname’-th location where we have ’operator’.

% NOTE: The code for Y.m and Z.m is very similar.

A.3 Time evolution under the natural Hamiltonian

% FILE: de.m % Author: Lieven Vandersypen % % de(time) simulates an evolution under the defined Hamiltonian % for the duration ’time’. function R=de(t) global H R=expm(-i*H*t); 296 APPENDIX A. NUMERICAL MODELS

A.4 Action of a unitary matrix on a density matrix

% FILES: rho.m % Lieven Vandersypen % % calculates the final density matrix rf when the unitary operation U % acts on an initial density matrix ri. function rf = rho(ri,U) global nqubits rf=U*ri*U’;

A.5 Generalized operator

% File: gop.m "generalized operator" % Date: 08-Oct-99 % Author: Lieven Vandersypen % % In: single-spin unitary operator U, spinname s % Out: n-spin unitary operator which acts on s with U and % trivially on the remaining spins function gop=gop(s,U) global nqubits goplocal=U; for position=1:(s-1) goplocal=kron(eye(2),goplocal); end for position=s+1:nqubits goplocal=kron(goplocal,eye(2)); end gop=goplocal; A.6. KRONECKER PRODUCT FUNCTION 297

A.6 Kronecker product function

% File: mykron.m % Date: 17-Aug-98 % Author: I. Chuang % % My kronecker product function for matlab: accepts multiple arguments. function out = mykron(ma,mb,varargin) if nargin < 2, error(’mykron requires at least two input arguments’);end lenVarIn = length(varargin); if (lenVarIn == 0) out = kron(ma,mb); return; else out = mykron(kron(ma,mb),varargin{:}); return; end

A.7 Simulated single-qubit rotations

% FILE: Qlv.m % DATE: 18-Jan-00 % Author: Matthias Steffen % Lieven Vandersypen % Simulation of spin dynamics during arbitrary RF pulses % including the natural coupling Hamiltonian. % % Usage: Uout=Qlv(spinname,maxamp,pulse_shape,pulse_width) % % pulse_shape -- pulse shape such as "gauss.RF" % pulse_width -- length of pulse in units of micro seconds % maxamp -- RF field strength in units of pi/2 (eg. 2 for a % pi pulse) % spinname -- an integer number indicating which spin the pulse is % applied to % Uout -- output unitary matrix % 298 APPENDIX A. NUMERICAL MODELS

% EXAMPLE: Qlv(2,2000,’gauss.RF’,410) % % Note: .RF file must be present in the defined directory function Qlv=Qlv(spinname,maxamp,pulse_shape,pulse_width) if nargin ~= 4 error(’Q.m requires four arguments !’) end global nqubits H

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % read in pulse shape from specified directory % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dir=sprintf(’/home/msteffen/matlab/pulseshapes/%s’,pulse_shape); input=fopen(dir,’r’); in=1; time(1)=0; line=fgets(input); while (line(1)==’#’), line=fgets(input); end while (line>0), line1=str2num(line); slicephase(in)=line1(1); sliceamp(in)=line1(2); slicelength(in)=line1(3); line=fgets(input); in=in+1; end slicephase=slicephase*pi/180; % convert to radians sliceamp=sliceamp/1023; % renormalize such that max value is 1 slicelength=slicelength/sum(slicelength)*10^-6*pulse_width; % convert to seconds

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % use predefined (from def.m) natural Hamiltonian % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% operator=eye(2^nqubits); H_nat=H; A.8. PLOTS EXCITATION PROFILE OF A SHAPED RF PULSE 299

index=length(sliceamp); phasevalue(index+2)=0;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % calculate RF field amplitude % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% maxamp=0.25*maxamp/abs((sum(sliceamp.*exp(i*slicephase).*slicelength)));

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % calculate the unitary matrix for each slice % % and multiply through the entire shape. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% while(index>0),

H_RF=sliceamp(index)*maxamp*2*pi*... gop(spinname,(cos(slicephase(index))*[0 1;1 0] - ... sin(slicephase(index))*[0 -i;i 0])/2); operator = operator * expm ( -i * (H_RF + H_nat) * slicelength(index) ); index=index-1; end

Uout=operator; % NOTE: A similar code was used to calculate the ideal negative % time delay surrounding the non-ideal pulse in order to cancel % out the J-coupling during the pulse.

A.8 Plots excitation proﬁle of a shaped RF pulse

% FILE: profile.m % DATE: 07-Nov-02 % Matthias Steffen % % calls Qlv and plots excitation profile % % USAGE: profile(maxamp,pulse_shape,pulse_width,spect_width) % % maxamp -- RF field strength in units of pi/2 (eg. 2 for a % pi pulse) 300 APPENDIX A. NUMERICAL MODELS

% pulse_shape -- pulse shape such as "gauss.RF" % pulse_width -- length of pulse in units of micro seconds % spect_width -- spectral width over wihich to plot excitation profile. % % e.g: P=profile(2,’gauss.RF’,300,10000) function P=profile(maxamp,pulse_shape,pulse_width,spect_width) nq=1; def numpoints=337; % number of points in the plot

%%%%%%%%%%%%%%%%%%%%%%%%%%% % create frequency window % %%%%%%%%%%%%%%%%%%%%%%%%%%% freq=-spect_width:2*spect_width/(numpoints-1):spect_width;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % - define off-resonant Hamiltonian % % - calculate resulting unitary evolution for each % % frequency step % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for index=1:numpoints freq(index) H=2*pi*freq(index)*Sz/2;

Q_oscframe=Qlv(1,maxamp,pulse_shape,pulse_width); rout=Q_oscframe*rit*Q_oscframe’; xcomp(index)=trace(rout*Sx/2); ycomp(index)=trace(rout*Sy/2); zcomp(index)=trace(rout*Sz/2); end figure(2) plot(freq,zcomp,’k-’,freq,sqrt(xcomp.^2+ycomp.^2),’k:’) axis([-freq(length(freq)) freq(length(freq)) -1 1]) set(gca,’FontSize’,16) xlabel(’\Delta \omega [Hz]’,’FontSize’,16); ylabel(’normalized amplitude’,’FontSize’,16); legend(’Z’,’XY’) A.9. CALCULATION OF THE TRANSIENT BLOCH-SIEGERT SHIFT 301

def; % restore default values

A.9 Calculation of the transient Bloch-Siegert shift

% FILE: bsshift.m % DATE: 13-Mar-00 % Author: L. Vandersypen % Modified: Matthias Steffen

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % read in original pulse_shape % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dir=sprintf(’/home/msteffen/matlab/pulseshapes/%s’,pulse_shape); input=fopen(dir,’r’); in=1; time(1)=0; line=fgets(input); while (line(1)==’#’), line=fgets(input); end while (line>0), line1=str2num(line); slicephase1(in)=line1(1); sliceamp1(in)=line1(2); 302 APPENDIX A. NUMERICAL MODELS

slicelength1(in)=line1(3); line=fgets(input); in=in+1; end fclose(’all’); numslices=length(sliceamp1); sliceamp2=sliceamp1; slicelength2=slicelength1;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % set phase ramp 2 to excite at freq=offset (no BS correction yet) % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% slicephase2=slicephase1; if offset~=0 final_phase2=pulse_width*1e-6*(-offset)*360; phasestep2=final_phase2/numslices; slicephase2= slicephase2 + ... [phasestep2/2:phasestep2:final_phase2-phasestep2/2]; slicephase2=mod(slicephase2,360); end

%%%%%%%%%%%%%%%%%%%%%%%%% % renormalize variables % %%%%%%%%%%%%%%%%%%%%%%%%% slicephase1=slicephase1*pi/180; % convert to radians slicephase2=slicephase2*pi/180; % convert to radians sliceamp1=sliceamp1/1023; % renormalize such that max value is 1 sliceamp2=sliceamp2/1023; % renormalize such that max value is 1 slicelength1=slicelength1/sum(slicelength1)*10^-6*pulse_width; slicelength2=slicelength2/sum(slicelength2)*10^-6*pulse_width; % convert to seconds maxamp=0.25*maxamp1/... abs((sum(sliceamp1.*exp(i*slicephase1).*slicelength1)));

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % calculate phase accumulation % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% operator=eye(2); for index = 1:numslices A.10. GENERALIZED AMPLITUDE DAMPING MODEL FOR 7 SPINS 303

H_RF=sliceamp2(index)*maxamp*2*pi*... gop(1,(cos(slicephase2(index))*[0 1;1 0] - ... sin(slicephase2(index))*[0 -i;i 0])/2); operator = expm ( -i * H_RF * slicelength1(index) ) * operator; cumul_phase(index)=2*phase(operator(1,1)); % is in radians end figure(1) plot((1:(numslices-1))/(numslices-1)*pulse_width,... (cumul_phase(2:numslices)-cumul_phase(1:numslices-1)) ./ ... (2*pi*slicelength1(1:numslices-1))) set(gca,’FontSize’,16) ylabel(’\omega_{BS} [Hz]’,’FontSize’,16); xlabel(’time [\mus]’,’FontSize’,16); figure(2) plot((1:(numslices))/(numslices)*pulse_width,cumul_phase/2/pi*360); set(gca,’FontSize’,16); ylabel(’phase [degrees]’,’FontSize’,16) xlabel(’time [\mus]’,’FontSize’,16);

PHASE=360*cumul_phase(length(cumul_phase))/(2*pi); % NOTE: A similar code was used to calculate the phase-ramp amount % that is necessary to apply simultaneous soft pulses.

A.10 Generalized amplitude damping model for 7 spins

% FILE: gad7.m % DATE: 14-Nov-00 % % Lieven Vandersypen % Matthias Steffen % % simulates generalized amplitude damping (GAD) for 7 spins % % USAGE: rout = gad7(rin,p,t,ratio) % % rout: output density matrix % rin: input density matrix % p: equilibrium polarization 304 APPENDIX A. NUMERICAL MODELS

% t: duration of GAD acting on ’rin’ % ratio: set this equal to ’1’ to simulate GAD using the % measured T1 values; set this equal to something very % high (1e9) to simulate without GAD. function rout=gad7(rin,p,t,ratio)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % define measured T1 values % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

T1_1=ratio(1)*5.0; T1_2=ratio(1)*10.0; T1_3=ratio(1)*13.7; T1_4=ratio(1)*2.8; T1_5=ratio(1)*3.0; T1_6=ratio(1)*31.6; T1_7=ratio(1)*45.4; p1=p; % fluorine polarization p2=0.5+(p-0.5)*1.25/4.7; %carbon polarization t=abs(t);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % T1 effects on spin 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g=1-exp(-t/T1_1); E{1}=sqrt(p1)*[1 0; 0 sqrt(1-g)]; E{2}=sqrt(p1)*[0 sqrt(g);0 0]; E{3}=sqrt(1-p1)*[sqrt(1-g) 0;0 1]; E{4}=sqrt(1-p1)*[0 0 ;sqrt(g) 0]; r1=0; for k=1:4 r1 = r1 + mykron(E{k},eye(2),eye(2),eye(2),eye(2),eye(2),eye(2))*... rin*mykron(E{k},eye(2),eye(2),eye(2),eye(2),eye(2),eye(2))’; end

...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % T1 effects on spin 7 A.11. PHASE DAMPING MODEL FOR 7 SPINS 305

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g=1-exp(-t/T1_7); E{1}=sqrt(p2)*[1 0; 0 sqrt(1-g)]; E{2}=sqrt(p2)*[0 sqrt(g);0 0]; E{3}=sqrt(1-p2)*[sqrt(1-g) 0;0 1]; E{4}=sqrt(1-p2)*[0 0 ;sqrt(g) 0]; r7=0; for k=1:4 r7 = r7 + mykron(eye(2),eye(2),eye(2),eye(2),eye(2),eye(2),E{k})*... r6*mykron(eye(2),eye(2),eye(2),eye(2),eye(2),eye(2),E{k})’; end rout=r7;

A.11 Phase damping model for 7 spins

% FILE: pd7.m % DATE: 30-Apr-01 % % Lieven Vandersypen % Matthias Steffen % % simulate phase damping process on 7 sipns % USAGE: rout = pd7(rin,t,ratio) % % rout: output density matrix % rin: input density matrix % t: duration of PD acting on ’rin’ % ratio: set this equal to ’1’ to simulate PD using the % measured T2 values; set this equal to something very % high (1e9) to simulate without PD. function rout=pd7(rin,p1,t,ratio)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % define measured T2 values % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

T2_1=ratio(2)*1.3; T2_2=ratio(2)*1.7; 306 APPENDIX A. NUMERICAL MODELS

T2_3=ratio(2)*1.8; T2_4=ratio(2)*1.6; T2_5=ratio(2)*1.5; T2_6=ratio(2)*2.0; T2_7=ratio(2)*2.0; t=abs(t);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % T2 effects on spin 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g=(1+exp(-t/T2_1))/2;

E{1}=sqrt(g)*[1 0; 0 1]; E{2}=sqrt(1-g)*[1 0;0 -1]; r1=0; for k=1:2 r1 = r1 + mykron(E{k},eye(2),eye(2),eye(2),eye(2),eye(2),eye(2))*... rin*mykron(E{k},eye(2),eye(2),eye(2),eye(2),eye(2),eye(2))’; end

...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % T2 effects on spin 7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g=(1+exp(-t/T2_7))/2;

E{1}=sqrt(g)*[1 0; 0 1]; E{2}=sqrt(1-g)*[1 0;0 -1]; r7=0; for k=1:2 r7 = r7 + mykron(eye(2),eye(2),eye(2),eye(2),eye(2),eye(2),E{k})*... r6*mykron(eye(2),eye(2),eye(2),eye(2),eye(2),eye(2),E{k})’; end rout=r7; A.12. SAMPLE CODE SIMULATING DECOHERENCE EFFECTS 307

A.12 Sample code simulating decoherence eﬀects nq=7; def; p1=(0.5000)+1.9356e-05; % define fluorine polarization p2=p1/4.7*1.25; % define carbon polarization rt1=(1-p1)*eye(2)+(2*p1-1)*[1 0; 0 0]; % fluorine spin density matrix rt2=(1-p2)*eye(2)+(2*p2-1)*[1 0; 0 0]; % carbon spin density matrix rit=mykron(rt1,rt1,rt1,rt1,rt1,rt2,rt2); % 7-spin molecule density matrix

% not shown here: code that produces the effective pure state ’R’ % from ’rit’ via temporal labeling.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % simulation of a cnot_52, used in the first % % part of the modular exponentiaion of the % % ‘‘easy’’ case of Shor’s algorithm. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% time=abs(1/8/J25);

R=rho(R,X(2,1)*Z(2,-1));

R=rho(R,expm(-i*H*time)); R=gad7(R,p1,time,ratio); R=pd7(R,p1,time,ratio); R=rho(R,X(1,2)*X(3,2));

R=rho(R,expm(-i*H*time)); R=gad7(R,p1,time,ratio); R=pd7(R,p1,time,ratio); R=rho(R,X(1,2)*X(2,2)*X(5,2));

R=rho(R,expm(-i*H*time)); R=gad7(R,p1,time,ratio); R=pd7(R,p1,time,ratio); R=rho(R,X(1,2)*X(3,2));

R=rho(R,expm(-i*H*time)); R=gad7(R,p1,time,ratio); R=pd7(R,p1,time,ratio); 308 APPENDIX A. NUMERICAL MODELS

R=rho(R,X(1,2)*X(2,2)*X(5,2));

R=rho(R,Y(2,-1)*X(7,1)*Z(7,1)*Z(5,1)); Appendix B

The three-spin Grover pulse sequence

Here we will give the pulse sequence for the three-spin Grover experiment (section 6.3). The sequence is taken from the C code which submits the sequence to the spectrometer.

%%%%%%%%%%%% % NOTATION % %%%%%%%%%%%% Xa(m): An m*pi/2 rotation about the X axis of spin a. mYc(m): An m*pi/2 rotation about the negative Y axis of spin c.

AB(m1,m2,phase1,phase2), AC(m1,m2,phase1,phase2), BC(m1,m2,phase1,phase2): An m1*pi2/ and m2*pi/2 rotation of two spins (a and b, a and c, or b and c) about the X, Y, -X or -Y axis. EXAMPLE: AC(1,0.5,PHX,PHmY)

ABC(): similar to the previous function but this time on all three spins. EXAMPLE: ABC(1,2,1,PHX,PHY,PHmX)

%%%%%%%%%%%%%%%%%% % Pulse Sequence % %%%%%%%%%%%%%%%%%% status(A); /* THERMALIZATION */ delay(d1); status(B); /* TEMPORAL LABELING */

309 310 APPENDIX B. THE THREE-SPIN GROVER PULSE SEQUENCE

tom3htemplab1(); status(C); /* COMPUTATION */

ABC(1,1,1,PHY,PHY,PHY); /* superposition via Hadamards */ if (ctype){ /* if ctype=0, skip search */

if (((ctype-1)/4)%2) pfa=0; if (((ctype-1)/2)%2) pfb=0; /* x0 depends on ctype */ if (((ctype-1)/1)%2) pfc=0; loop(v9,v10); /******** BEGIN GROVER LOOP ********/ /* v9 is number of iterations, v10 is index */

/* function evaluation */ ABC(pfa,pfb,pfc,PHX,PHX,PHX); /* depends on ctype */ tom3hphaseflip4(); /* flip sign 111 term */ ABC(pfa,pfb,pfc,PHmX,PHmX,PHmX); /* depends on ctype */

/* inversion about average */ ABC(1,1,1,PHY,PHY,PHY); tom3hphaseflip4(); /* flip sign 111 term */ ABC(1,1,1,PHmY,PHmY,PHmY); endloop(v10);

} /* close if (ctype) */ status(D); /* TOMOGRAPHY PULSES */ tomoPULSE;

%%%%%%%%%%%%%%%%%%% % tom3htemplab1() % %%%%%%%%%%%%%%%%%%% tom3htemplab1() { switch (ptype){ case 0: break; /* I */ case 1:{ /* cnotab */ Yb(1); delay(1/4/Jab); Xc(2); delay(1/4/Jab); BC(1,2,PHX,PHX); break; } 311

case 2:{ /* cnotac */ Yc(1); delay(1/4/Jac)); Xb(2); delay(1/4/Jac); BC(2,1,PHX,PHX); break; } case 3:{ /* cnotbc, Jbc<0 */ mYc(1); delay(1/4/Jbc); Xa(2); delay(1/4/Jbc); AC(2,1,PHX,PHX); break; } case 4:{ /* cnotba */ Ya(1); delay(1/4/Jab); Xc(2); delay(1/4/Jab); AC(1,2,PHX,PHX); break; } case 5:{ /* cnotca */ Ya(1); delay(1/4/Jac); Xb(2); delay((1e6/4/Jac)/1e6); AB(1,2,PHX,PHX); break; } case 6:{ /* cnotcb, Jbc<0 */ mYb(1); delay(1/4/Jbc); Xa(2); delay(1/4/Jbc); AB(2,1,PHX,PHX); break; } case 7:{ /* cnotab.cnotca (time -->) */ Yb(1); delay(1/4/Jab); Xc(2); delay(1/4/Jab); BC(1,2,PHX,PHX); Ya(1); delay(1/4/Jac); Xb(2); delay(1/4/Jac); AB(1,2,PHX,PHX); break; } } }

%%%%%%%%%%%%%%%%%%%%% % tom3hphaseflip4() % %%%%%%%%%%%%%%%%%%%%% tom3hphaseflip4() { Yb(2); delay(1e6/8/Jbc); Xa(1); Ya(0.5); Xa(1); delay(1/8/Jbc); Yb(1); Xb(0.5); delay(1e6/4/Jab); mYc(2); delay(1/4/Jab); Yb(1); mXb(1); delay(1/8/Jbc); Ya(2); delay(1/8/Jbc); mXb(1); Yb(0.5); delay(1/4/Jab); Yc(1); Xc(0.5); Yc(1); delay(1/4/Jab); Yb(1); delay(1/8/Jac); mYb(2); delay(1/8/Jac); } Appendix C

The 7-spin framework

Pseudo-code for the 7-spin framework is shown here. As described in chapter 5, homonuclear systems require several extra techniques just be able to address all spins and to improve unitary control. These techniques are best incorporated into a framework so we do not need to explicitly worry about them for every applied pulse. The techniques we have incorporated into the 7-spin framework include the compensation of transient Bloch-Siegert shifts, a frequency-shift method during the application of simultaneous pulses, and extra software-reference frames in case we have more spins than RF channels. Before showing the pseudo-code, let us explain what is necessary. The software-reference frame requires us to keep track of all timing events of the spectrometer. These include explicit time delays during two-qubit gates and implicit times delays such as phase and power settling times of the spectrometer, amplifier gating times, pre and post-pulse delays, and so forth. We use a “time counter” and the knowledge of the frequency separation between the RF channel frequency and the spin frequency to set the phase of the appropriate channel just prior to the pulse. In our case, we set spin 1 on-resonance with channel 1. Channel 2 addresses spins 2 and 3, channel 3 addresses spins 4 and 5, and channel 4 addresses spins 6 and 7. So for example, if spin 5 had acquired a phase of 43◦ just before the application of a pulse on the same spin, we adjust the phase of channel 3 prior to this pulse by 43◦. For the next pulse using the same channel, we may need to adjust by a different phase because some time has elapsed. The calculation of the transient Bloch-Siegert shift is done by calling a function which simply outputs the correct phase shift. This function is the result of curve-fitting the plot of the transient Bloch-Siegert shift as a function of pulse width for the given frequency

312 C.1. USED VARIABLES AND FUNCTIONS 313

separations of the 7 spins and the used pulse shape. We then keep track of this extra phase for all spins after every applied pulse, and adjust the phase of the appropriate channel prior to each pulse. This is in addition to the phase required for the software reference frames. The application of simultaneous soft pulses can take on two different scenarios. First, the two spins are addressed by two different channels. Second, the two spins are served by the same channel. The pulse shapes (i.e. the old pulse shape with the frequency-correction method added on) for all pulses that fit the first scenario are pre-computed via MATLAB. Depending on which spins we wish to pulse, we load the appropriate pulse shapes, adjust the phases of the two channels (Bloch-Siegert shifts, software-reference frames) and apply the pulse. If the pulses fit the second scenario, we call MATLAB in the framework itself. When we execute a pulse sequence on the spectrometer, the compiler will go through the code first, and do all calculations including the MATLAB function calls and only then applies the pulse sequence. Since the two spins served by the same channel may have different initial phases, we need to hardwire these phases into the pulse shape itself and then apply it (one channel can only be adjusted by one phase, not two). Since these phase values differ depending when this simultaneous pulse is applied, we cannot precompute the shape and must call MATLAB in the framework. Let us now show several variables and functions that are used and called in the pseudo- code:

C.1 Used variables and functions variables

spintype: integer number from 1 to 7, depending on the spin. spintype1: same as ’spintype’ but used in the simultaneous pulse code. spintype2: same as ’spintype’ but used in the simultaneous pulse code. phase[spintype]: array of 7 numbers, storing the accumulated phase values for the 7 spins. rof1: amplifier gating time before each pulse rof2: amplifier gating time after each pulse pw: pulse width shape: pulse shape 314 APPENDIX C. THE 7-SPIN FRAMEWORK

phase: phase (’X’ for x-rotation and ’Y’ for y-rotation). timecounter: value of elasped time throughout the pulse sequence.

functions

do_phases(spintype): adjusts ’phase[spintype]’ following the application of an RF pulse on spin ’spintype’ due to transient Bloch-Siegert shifts. calc_phases(): calculates the phase shifts for all spins that are necessary based on ’timecounter’ and the known frequency differences between the spin frequencies and the RF channel frequencies. This value gets added to ’phase[spintype]’. set_phases(spintype): adjusts the phase of the appropriate channel by the amount ’phase[spintype]’. set_power(spintype): sets the power of appropriate channel depending on whether we apply a 90 or 180 pulse, and which spin we are addressing. shaped_pulse(shape,pw,phase): apply a shaped pulse of shape ’shape’, duration ’pw’. The variable ’phase’ defines what kind of rotation we are doing. Two shapes will be used in the case of a simultaneous shaped pulse. calc_pulse_shapes(phase[spintype1],phase[spintype2],’shape’): calculate the pulse shapes including the frequency-correction method and the initial phases ’phase[spintype1]’ and ’phase[spintype2]’, add the two shapes, and stores the resulting pulse away under the name ’shape’). This functions uses MATLAB.

C.2 The 7-spin pseudo-code

We are now ready to show parts of the pseudo-code of the 7-spin framework. We will only show code for single and multiple spin rotations. Delay times simply involve applying a delay time, and incrementing ’timecounter’ (see below).

Single-spin rotation

This describes the framework for a single spinx ˆ-rotation on spin ’spintype’. Note that we include the pre-shaped-pulse delays into the timecounter before ’calc phase’ because when C.2. THE 7-SPIN PSEUDO-CODE 315

we actually apply ’shaped pulse’ we will ﬁrst execute an intrinsic pre-shaped pulse delay time plus the ampliﬁer gating time. The same argument applies to the phase settling times. We can performy ˆ-rotations in a very similar way. Thez ˆ-rotations are done by simply incrementing the variable ’phase[spintype]’.

set_power(spintype); % set power level of the channel timecounter=timecounter+0.5e-6; % power settling time timecounter=timecounter+1.75e-6+rof1; % pre-simultaneous-shaped-pulse % delay % and amplifier gating time timecounter=timecounter+0.5e-6; % phase settling time calc_phases(); % calculate the phase shift necessary % right when the pulse starts set_phases(spintype) ; % set phase for the used channel shaped_pulse(shape,pw,X); % apply pulse timecounter=timecounter+0.5e-6+rof2+pw; % post-shaped-pulse delay, % amplifier gating time and pulse % width do_phases(spintype); % Bloch-Siegert phase shifts

The framework will now call these statements every time we wish to apply a single-spin rotation. This allows us to write pulse sequences at a higher level because all artifacts are incorporated at the lower level of the framework.

Simultaneous spin rotations

We now describe the framework for a simultaneousx ˆ-rotation on two spins, ’spintype1’ and ’spintype2’. Note that we added small variable attenuators into the path of the four RF channels to make the pulse width the same on all spins.

case (different channels) set_power(spintype1); % set power level for first used channel set_power(spintype2); % set power level for second used channel timecounter=timecounter+0.5e-6; % power settling time timecounter=timecounter+0.5e-6; % power settling time timecounter=timecounter+3.0e-6+rof1; % pre-shaped-pulse delay % and amplifier gating time 316 APPENDIX C. THE 7-SPIN FRAMEWORK

timecounter=timecounter+0.5e-6; % phase settling time timecounter=timecounter+0.5e-6; % phase settling time calc_phase(); % calculate the phase shifts necessary % right when the pulses start set_phases(spintype1); % set phase of first used channel set_phases(spintype2); % set phase of second used channel load shape1; % load precomputed pulse shape 1 load shape2; % load precomputed pulse shape 2 shaped_pulse(shape1,shape2,pw,X); % apply pulse timecounter=timecounter+0.5e-6+rof2+pw; % post-shaped-pulse delay, % amplifier gating time and pulse % width do_phases(spintype1); % Bloch-Siegert phase shift. do_phases(spintype2); % Bloch-Siegert phase shift.

case (same channel) set_power(spintype1); % set power of the used channel timecounter=timecounter+0.5e-6; % power settling time timecounter=timecounter+1.75e-6+rof1; % pre-shaped-pulse delay % and amplifier gating time calc_phase(); % calculate the phase shifts necessary % right when the pulses start calc_pulse_shapes(phase[spintype1],phase[spintype2],shape) % calculate % shapes load shape; % load the pulse shape shaped_pulse(shape,pw,X); % apply pulse timecounter=timecounter+0.5e-6+rof2+pw; % post-shaped-pulse delay, % amplifier gating time and pulse % width do_phases(spintype1); % Bloch-Siegert phase shift. do_phases(spintype2); % Bloch-Siegert phase shift. Bibliography

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