Quantum Information Engineering: Concepts to Quantum Technologies

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Quantum Information Engineering: Concepts to Quantum Technologies.

Simon John Devitt, B.Sc (Honours) Melbourne.

Submitted in partial fulﬁlment of the requirements of the degree of Doctor of Philosophy.

November, 2007.

Center For Quantum Computing Technology. School of Physics, University of Melbourne. Victoria, Australia.

Acid Free Paper. ii Quantum Mechanics: It’s all fun and games until somebody loses an i.

iii The University of Melbourne. ABSTRACT Quantum Information Engineering: Concepts to Quantum Technologies.

by Simon John Devitt.

Academic Supervisor: Lloyd C. L. Hollenberg.

This thesis investigates several broad areas related to the effective implementation of quantum information processing, from large scale quantum algorithms and error correction, through to system identification and characterization techniques, efficient designs for quantum computing architectures and the design of small devices which utilize quantum effects. The discussion begins with the introduction of a quantum circuit appropriate for implementing Shor’s factoring algorithm on Linear Nearest Neighbor qubit arrays such as the Kane phosphorus in silicon system. Detailed numerical simulations are then presented, demonstrating the sensitivity of the circuit under coherent quantum errors. The concepts of Quantum Error Correction and Fault-tolerant computation are reviewed with original work carried out to show the relative robustness and practicality of Fault-tolerant computation for logical state preparation. Methods of intrinsic system identification and characterization are proposed. Protocols for characterizing both the confinement of a multi-level system to the qubit subspace and the Hamiltonian dynamics governing two-qubit interactions are presented as well as a brief review of characterization techniques already developed for single qubit dynamics. A quantum bus protocol is introduced which can be applied to several systems, allowing for a highly distributed architecture which is invulnerable to information loss through the transport bus. Several preliminary architecture designs are presented, including solid state and atom/cavity quantum processors. A more detailed, distributed system in ion traps utilizing this protocol is introduced which demonstrates both large scale quantum computation and a smaller proof of principle experiment which will be possible in the near future. Finally, I present a design for a small scale quantum device which can be used to deterministically prepare a large class of highly entangled quantum states using photonic qubits. This device, which is dubbed the “photnic module”, would be extremely versatile with a variety of uses, from communication to computation.

iv Declaration

This is to certify that:

(i) the thesis comprises only my original work towards the PhD except where indicated,

(ii) due acknowledgement has been made in the text to all other material used,

(iii) the thesis is less than 100,000 words in length, exclusive of table, maps, bibliographies, appendices and footnotes.

Simon John Devitt, July 2007.

v vi Publications

During the course of this project, a number of refereed journal and conference articles have been published based on the work presented in this thesis. They are listed here for reference.

Journal Articles

A.G. Fowler, S.J. Devitt and L.C.L. Hollenberg, Implementation of • Shor’s Algorithm on a Linear Nearest Neighbour Qubit Array, Quant. Inf. Comp. 4, 237-251 (2004).

S.J. Devitt, A.G. Fowler and L.C.L. Hollenberg, Robustness of Shor’s • algorithm, Quant. Inf. Comp. 6, 616-629 (2006).

A.D. Greentree, S.J. Devitt and L.C.L. Hollenberg, Quantum infor- • mation transport to multiple receivers, Phys. Rev. A. 73, 032319 (2006).

S.J. Devitt, J.H. Cole and L.C.L. Hollenberg, Scheme for direct mea- • surement of a general two-qubit Hamiltonian, Phys. Rev. A 73, 052317 (2006).

D.K.L. Oi, S.J. Devitt and L.C.L. Hollenberg, Scalable Error Cor- • rection in Distributed Ion Trap Computers, Phys. Rev. A. 74, 052313 (2006).

J.H. Cole, S.J. Devitt and L.C.L. Hollenberg, Precision character- • isation of two-qubit Hamiltonians via entanglement mapping, J. Phys. A: Math.Gen, 39 (2006), 14649-14658.

S.J. Devitt, A.D.Greentree and L.C.L. Hollenberg, Information free • quantum bus for generating stabilized states, Quant. Inf. Proc. 6, 229 (2007).

S.J. Devitt, S.G. Schirmer, D.K.L. Oi, J.H. Cole and L.C.L. Hol- • lenberg, Subspace Conﬁnement: How good is your qubit? New J. Phys. 9 (2007) 384.

vii S.J. Devitt, A.D. Greentree, R. Ionicioiu, J.L. O’Brien, W.J. • Munro and L.C.L. Hollenberg, The Photonic Module: and on-demand source for photonic entanglement. Phys. Rev. A. 76, 052312 (2007).

Conference Articles

S.J. Devitt, A.G. Fowler and L.C.L. Hollenberg, Investigating the • practical implementation of Shor’s algorithm, Proc. SPIE 5650, 483 (2005).

A.G. Fowler, S.J. Devitt and L.C.L. Hollenberg, Constructing Steane • code fault-tolerant gates, Proc. SPIE 5650, 557 (2005).

S.J. Devitt, A.G. Fowler and L.C.L. Hollenberg, Practicality of Fault- • Tolerant Quantum Computation, Proc. Australian Institute of Physics 16th National Congress (2005).

L.C.L. Hollenberg, A.D. Greentree, C.J. Wellard, A.G. Fowler, • S.J. Devitt and J.H. Cole, Qubit Transport and Fault-tolerant Architec- tures in Silicon, Proc. 2006 International Conference On Nanoscience and Nanotechnology. (2006).

S.J. Devitt, S.G. Schirmer, D.K.L. Oi, J.H. Cole and L.C.L. Hol- • lenberg, Subspace conﬁnement of qubit systems, Proc. Australian Institute of Physics 17th National Congress (2006).

A.D. Greentree, S.J. Devitt and L.C.L. Hollenberg, Adiabatic proto- • cols for operator measurement based entanglement and quantum computing, Proc. Australian Institute of Physics 17th National Congress (2006).

L. C. L. Hollenberg, A. D. Greentree, C. J. Wellard, A. G. Fowler, • S. J. Devitt, J. H. Cole, and A. Stephens, Qubit Transport and fault- tolerant architectures for Si:P quantum computing, Proc. Australian Insti- tute of Physics 17th National Congress (2006).

Un-refereed Articles

A.M. Stephens, S.J.Devitt, A.G. Fowler, J.C. Ang, L.C.L. Hollen- • berg, Gate-Level Simulation of logical state preparation, quant-ph/0608112, (2006).

viii Patents

S.J. Devitt, Implementation of Shor’s Algorithm on a Linear Nearest • Neighbour Qubit Array, Australian Provisional Patent, QUCOR Pty. Ltd. Sydney. Patent number:2004900951.

ix x Acknowledgements

This thesis would never have left the ground if it were not for a number of people. First I would like to thank my supervisor, Lloyd Hollenberg for exceptional support and guidance and giving me the freedom to pursue aspects of quantum computing which I found interesting and compelling. Secondly to Jared Cole, Andrew Greentree and Austin Fowler, three people whom I have collaborated closely with over the last three years and who have not only brought out the best in my own research, but who have always been up for a few drinks wherever we have been in the world. During my Ph.D I have visited several institutions, including an extended 12 month visiting student-ship in 2006 to Cambridge which was generously funded by the Rae & Edith Bennett Foundation. I wish to acknowledge the tremendous help and entertainment provided by Sonia Schirmer and Daniel Oi of Cambridge University who have been close collaborators and graciously hosted me during my brief visit in 2005 and my extended student-ship in 2006, along with the extraordinary administrative skill of Kaija Hampson who helped me settle into a new environment. Thanks also go to Frank Wilhelm and Jan von Delft at LMU in M¨unchen for hosting me for a month in 2005. Personally I wish to thank the great people at the School of Physics, both the 1st and 2nd generation of inhabitance of room 408 and 412, the guys and girls on the 6th ﬂoor, Sean Crosby for continued access to my favorite tool of procrastination, the DMP group at large and scores of others who made my time extremely enjoyable. Finally I wish to acknowledge the support from both my family and non- physics friends (In Australia, the U.K. and Hungary). Their continued tolerance with my moods and tendency to disappear for weeks at a time gave me a much needed break from the world of physics.

xi xii Contents

1 Introduction 1

2 Background 5 2.1 Emergence of the Quantum from the Classical...... 5 2.2 Quantum Information Processing: Computational Models..... 8 2.2.1 The Quantum Circuit Model...... 8 2.2.2 ClusterStateModel ...... 12 2.3 Conclusion...... 18

3 Shor’s Factoring Algorithm 19 3.1 Introduction...... 20 3.1.1 FromFactoringtoPeriodFinding...... 21 3.1.2 The Quantum Period Finding subroutine ...... 22 3.1.3 Classicalpost-processing ...... 24 3.2 Linear Nearest Neighbor quantum circuit...... 27 3.2.1 Decomposing the QPF subroutine ...... 28 3.2.2 Conclusion...... 40 3.3 Stability of Shor’s Algorithm under errors ...... 42 3.3.1 Errormodelsandanalysis ...... 44 3.3.2 Stability under a ﬁxed number of errors ...... 47 3.3.3 Conclusion...... 55

4 Quantum Error Correction and Fault-tolerance. 57 4.1 Introduction...... 59 4.2 QuantumErrorCorrection ...... 61 4.2.1 The3-qubitcode ...... 62

xiii CONTENTS

4.2.2 StabilizerFormalism ...... 64 4.2.3 QECwithstabilizercodes ...... 66 4.3 DigitizationofQuantumErrors ...... 72 4.3.1 Systematicgateerrors ...... 72 4.3.2 Environmental decoherence ...... 73 4.4 Fault-tolerant Quantum Error Correction and the threshold theorem. 76 4.4.1 Fault-tolerance ...... 76 4.4.2 ThresholdTheorem...... 77 4.5 Fault-tolerant operations on encoded data ...... 80 4.5.1 SingleQubitOperations ...... 80 4.5.2 Two-qubitgate...... 82 4.6 Fault-tolerant circuit design for logical state preparation ..... 86 4.7 Simulations ofLogical StatePreparation ...... 90 4.7.1 LNN circuit for logical encoding ...... 90 4.7.2 Results...... 93 4.7.3 Conclusions ...... 94

5 Intrinsic Characterization of qubit systems 97 5.1 Introduction...... 99 5.1.1 Spectroscopy ...... 100 5.1.2 StateandProcessTomography ...... 101 5.2 Subspace Confinement: How good is your qubit? ...... 106 5.2.1 MotivationandPreliminaries ...... 107 5.2.2 Estimationofsubspaceleakage ...... 109 5.2.3 Finite Sampling Fourier analysis ...... 114 5.2.4 Efficiencyoftheprotocol...... 121 5.2.5 TheeffectofDecoherence ...... 125 5.2.6 Conclusions ...... 130 5.3 Singlequbitdynamics ...... 132 5.3.1 Characterization without decoherence ...... 132 5.3.2 Characterization in the Presence of decoherence ...... 135 5.3.3 Conclusions ...... 139 5.4 Characterization of two qubit dynamics ...... 141

xiv CONTENTS

5.4.1 Structure of two-qubit dynamics ...... 142 5.4.2 Periodic Entanglement and Two-qubit Hamiltonian Char- acterization...... 152 5.4.3 Characterization of a fully non-local interaction...... 156 5.4.4 Example: Spin-Orbit corrections in GaAs quantum dots. . 162 5.4.5 Conclusions ...... 166

6 Distributed Quantum Information Processing via Parity Mea- surements. 169 6.1 Introduction...... 171 6.2 Informationfreequantumbus ...... 176 6.2.1 Preparationofstabilizerstates...... 182 6.2.2 Universality using the Quantum bus ...... 188 6.2.3 Atom/cavity interaction with photonic qudit ...... 189 6.2.4 Implementation of the bus protocol in Solid State Systems 191 6.2.5 Conclusion...... 196 6.3 Scalable quantum error correction in distributed ion trap computers.199 6.3.1 Architecture...... 201 6.3.2 Operation ...... 201 6.3.3 NodeDesign...... 213 6.3.4 Conclusion...... 216 6.4 PhotonicModules...... 218 6.4.1 BellStateFactories ...... 222 6.4.2 Arbitrary entangled state preparation...... 224 6.4.3 OperationalTime...... 227 6.5 Conclusions ...... 229

7 Concluding Remarks. 231

xv CONTENTS

xvi List of Tables

3.1 Number of qubits required and circuit depth of diﬀerent implementations of the QPF subroutine. Where possible, ﬁgures are accurate to leading order in L. The circuit of Cleve and Wa- trous [CW00] introduced a fast, parallel implementation of the Quantum Fourier Transform but did not examine, in detail, the full circuit required for modular exponentiation...... 28

3.2 Functions used for various values of L. Note that for 2L +4 = 14, 16, the values of N used are not products of two primes. How- ever, since we are only investigating the reliability of the QPF subroutine, this is not relevant to our analysis...... 47

3.3 Total circuit depths (K) for the LNN and non-LNN circuits, for L =5 to L =8 ...... 53

3.4 Minimum component precision required to apply the QPF subroutine to at least L = 10 for various scalings in component precision. 55

4.1 Final state of the ﬁve qubit system prior to the syndrome measurement for no error or a single X error on one of the qubits. The last two qubits represent the state of the ancilla. Note that each possible error will result in a unique measurement result (syndrome) of the ancilla qubits. This allows for a X correction gate to be applied to the data block which is classically controlled from the syndrome result. At no point during correction do we learn anything about α or β...... 64

4.2 The Four Stabilizers for the [[5,1,3]] quantum code, encoding ﬁve physical qubits into one logical to correct for a single X and/or Z error. Unlike the 7- and 9-qubit codes, the [[5,1,3]] code is a non- CSS code, since the stabilizer set does not separate into X and Z sectors...... 68

xvii LIST OF TABLES

4.3 The eight Stabilizers for the [[9,1,3]] quantum code, encoding nine physical qubits into one logical to correct for a single X and/or Z error. This code was the ﬁrst quantum code to be introduced and has since been shown to be a member of a much broader class of subsystem codes known as Bacon-Shor codes [Bac06]...... 68

4.4 Transformations of the [[7, 1, 3]] stabilizer set under the gate oper- 7 ation U =CNOT⊗ , where U † U. Note that the transformation does not take any stabilizerG → outsideG the group generated by i j 7 K K (i, j) [1, .., 6], therefore U =CNOT⊗ represents a valid ⊗ ∈ operationonthecodespace...... 84

4.5 Resources required for various types of circuits to prepare a logical 0 state. * indicates the minimum possible depth and gate counts |fori Fault-tolerant circuits, as these circuits can change depending on speciﬁc measurement results obtained during the calculation. . 92

2 5.1 Triplet values, [c1,c2,c3], and concurrence values, C , for some well known quantum gates. The SWAP and Identity gates represent non-entangling operations, while the √SWAP and CNOT gates are able to prepare Bell states from un-entangled product states. . 152

5.2 Functional form of C2(t) for the four separable input states ψ , i | ii ∈ [1, .., 4]. In each case the concurrence varies as a sinusoidal signal with frequency given by the Hamiltonian parameters, [J1,J2,J3]. . 155

5.3 Relevant data from the three power density spectra, one of which is shown in Fig. 5.18. Values for all the respective peak heights and µ (i),µ (i),µ (i),µ (i) can be used to ﬁnd K ...... 164 { 1 2 3 4 } Q

5.4 Calculations of C2 for all three input states using the 4 unitary operators experimentally obtained for t = 1. The last column represents the data measured for the three input states and unknown unitary operator at t = 1, U(1)...... 166

xviii LIST OF TABLES

6.1 Estimates on Cavity requirements for various systems. The Cav- ity from Boozer et. al. [BBM+06] has been experimentally demonstrated, while the atom chip cavity of Trupke et. al. [THE+05] and the photonic bandgap cavity of Song et. al. [SNAA05] in silicon have yet to couple the atomic qubit. Hence we use theoretical estimates for the coupling, β, and atomic decay rates, Γ, for Rubidium + and NV− [GSPH06, GOD 06] qubits. The ﬁrst column quotes the atom/cavity coupling while the second column estimates the required photon storage time in the cavity to invoke a π phase shift in the atom with a single photon absorption probability of 10%. The ﬁnal column quotes the current photon storage time which has been experimentally demonstrated (estimated) for each cavity system. For both the Boozer and Trupke cavities, approximately an order of magnitude improvement in either the coupling constant or cavity lifetime is required. Current estimates suggest that the photonic bandgap cavity will be able to exhibit the interaction with the fastest operational time of all the systems and is also more amenable to current cavity Q-switching protocols [GSPH06]. The last column details estimates on atomic decay rates for the systems considered for each cavity. The ratio of the required photon storage time to the coherence time of the atomic system dictates the maximum Parity-weight a single module can measure in any onestep...... 228

xix LIST OF TABLES

xx List of Figures

2.1 Circuit representation for a single qubit gate. Time runs left to right and the solid line represents the time evolution of the qubit, φ = 0 . The gate H is represented by a 2 2 unitary matrix |andi after| i the operation the output state is given× by H 0 ..... 9 | i 2.2 Circuit representation and truth table for the two-qubit CNOT gate. If the control qubit is in the state 1 , the target is ﬂipped, while if the control is in the 0 state the| gatei does nothing. . . . 10 | i 2.3 Quantum circuit to prepare a four qubit GHZ state. After the state has been prepared, we measure the bottom most qubit, collapsing the state to 0000 or 1111 with an equal probability of 50%. . 12 | i | i 2.4 Circuit representation of a controlled-PHASE (CZ) gate...... 13

2.5 Topological conﬁguration of a cluster state appropriate for universal computation. Each node represents a physical qubit which can by physically located anywhere within the qubit array. Initially, each qubit is prepared in the + state and CZ gates are applied | i between any two connected qubits...... 13

2.6 Topological geometries and state vectors for two, three and four qubit cluster states. Two and three qubit cluster states are actually equivalent to Bell states and 3-qubit GHZ states (up to local rotations), however as the cluster gets larger the total number of basisstatesgetsexponentiallylarge...... 14

2.7 Quantum circuit representation for an Rz(θ) gate applied using the cluster state model. Initially an arbitrary qubit is linked to a cluster qubit via a CZ gate. Next the original qubit is measured in the R (θ)H basis, corresponding to single qubit R (θ) and H { z } z gates and measurement in the 0 , 1 basis. The classical measurement result is then fed-forward{| i | toi} the second qubit and used to classically control a Z ﬂip. After this classical correction, the second qubit is in the state Rz(θ) φ corresponding to the desired rotation...... | i 15

xxi LIST OF FIGURES

2.8 Topological geometry (a) and equivalent quantum circuit (b) to realize a CNOT operation over the two qubit state CT in the | i cluster model. Unlike single qubit operations, the CNOT gate requires a two-dimensional topological geometry where the control qubit, C , remains at a ﬁxed physical location and the target | i qubit, T , is teleported from site two to four. After all four qubits have been| i linked together, X basis measurements are performed on qubits two and three (corresponding to Hadamard gates and measurement in the 0 , 1 basis). The results are then fed- forward and used to classically{| i | i} correct the ﬁnal state...... 17

3.1 (From Ref. [Fow05]) Plot of Eq. 3.7 for the case, 22L = 256 with a) r =8 and b) r =10...... 24

3.2 Quantum circuit required to implement the QFT on an array of four qubits which allow for arbitrary couplings. Extending this circuit to a larger number of qubits is trivial, since for each additional qubit the controlled phase rotations simply decrease by powers of two. Also note that for the standard circuit the order of the qubits swap...... 29

3.3 Quantum circuit required to implement the QFT on a LNN array of four qubits. Unlike the standard circuit, the qubit order remains the same. Each dotted box represents compound gates which can be obtained via the canonical decomposition and are counted as a singlegate[section5.4]...... 30

3.4 Quantum circuit required to implement a) addition and b) controlled addition on a four qubit binary register. The initial state b is ﬁrst placed into Fourier space b φ(b) through the use| i of a QFT. Secondly, single qubit| phasei → | rotationsi are performed to add a, with each angle classically calculated such that φ(b) φ(b + a) . If a controlled addition is desired, then |eachi single → | qubit phasei rotation becomes a controlled phase rotation + SWAP. Finally, an inverse QFT is performed, taking φ(b+a) b+a , after which the control qubit has been swapped fromoneendoftheregistertotheother.| i→| i ...... 32

xxii LIST OF FIGURES

3.5 Quantum circuit required to implement controlled modular addition, for a L-bit quantum register. The control qubits ki and xj are required later, once this circuit element has been integrated into the full modular exponentiation. The diagonal circuit elements labeled swap simply represent a chain of two qubit SWAP gates, shuttling a qubit from one end of a block to the other. The circuit introduces three ancilla qubits, k stores the control information | xi to avoid doubly controlled phase rotations in the addition elements, MS is used as a control qubit if addition overflow occurs and the |extrai qubit contained in the φ(b) register is used to indicate | iL+1 if overflow has occurred during any addition...... 34 3.6 Quantum circuit required to mesh together two quantum registers. Once meshed together, performing two qubit SWAP gates between each pair and then applying the reverse of the mesh circuit will SWAP both registers. This circuit element is not needed in the non-LNN version since it is assumed that any pairs of qubits can beinteracteddirectly...... 36 3.7 a) Circuit element required to perform a controlled-SWAP between two qubits ( a , b ), conditional on a third ( c ). The control-(+) and control-(| i)| gatesi are identical to the gates| i shown in Fig. 3.5. b) The full circuit− to perform a controlled SWAP of two LNN quantum registers. First the qubits are meshed, three qubit control- SWAP gates are applied in a ladder structure and finally the two registers are un-meshed. After the operation the control qubit has swappedposition...... 37 3.8 Circuit to implement a controlled modular multiplication, requiring L modular additions, a controlled register swap and finally L modularsubtractions...... 38 3.9 Final circuit to implement the Quantum Period Finding subroutine. The entire 2L, k-register is replaced by a single master control qubit, which is used to control each multiplication gate in the modular exponentiation, interleaved with classically controlled phase gates and measurement. This decomposition removes the need for a large k-register and for controlled phase rotations to implement the final QFT. Without these classically controlled phase rotations and measurement, this circuit would represent simple modular exponentiation...... 41 3.10 Map showing how the location of a single bit flip error plays a major role in the final output success of the LNN circuit. This image is for L = 5 (14 qubits), and shows the first modular multiplication section of the circuit. Each horizontal block represents one of the 14 qubits while each vertical slice represents a single time step. Darker areas represent successively lower values for s...... 46

xxiii LIST OF FIGURES

3.11 Plot showing the relative probability of measuring j = 22L/6 as a function of the speciﬁc number of errors for the non-LNN⌊ circuit.⌋ The curves represent L = 5 to L = 8. The horizontal lines show the point of random output for each successive value of L. .... 48

3.12 Plot showing the relative probability of measuring j = 22L/6 as ⌊ ⌋ a function of the speciﬁc number of errors for the LNN circuit. The curves represent L = 5 to L = 8. The horizontal lines show the point of random output for each successive value of L...... 49

3.13 Plot showing the stability of the non-LNN (right plot) and LNN (left plot) QPF circuit for L = 5, included on this plot are errors associated with the stochastic nature of simulations, representing the variance on the 50 run data set used in simulations...... 50

3.14 Plot showing the stability of the non-LNN (right plot) and LNN (left plot) QPF circuit for L = 6, included on this plot are errors associated with the stochastic nature of simulations, representing the variance on the 50 run data set used in simulations...... 50

3.15 Plot showing the stability of the non-LNN (right plot) and LNN (left plot) QPF circuit for L = 7, included on this plot are errors associated with the stochastic nature of simulations, representing the variance on the 50 run data set used in simulations...... 51

3.16 Plot showing the stability of the non-LNN (right plot) and LNN (left plot) QPF circuit for L = 8, included on this plot are errors associated with the stochastic nature of simulations, representing the variance on the 50 run data set used in simulations...... 51

3.17 Estimate on the maximum number of errors possible for each value of L before the LNN circuit becomes equivalent to a classical random search. non-LNN circuit, LNNcircuit...... 52 × ≡ ≡

3.18 Required component precision in order to observe quantum processing for small values of L in the QPF subroutine. Each curve represents a separate type of additional scaling indicated by our simulations, 1/n , L/n , L2/n and L3/n . .. 54 × ≡ p ⋄ ≡ p △ ≡ p ≡ p

4.1 Quantum Circuit to prepare the 0 L state for the 3-qubit code where an arbitrary single qubit state,| i ψ is coupled to two freshly initialized ancilla qubits via CNOT gates| i to prepare ψ ...... 62 | iL

xxiv LIST OF FIGURES

4.2 Circuit required to encode and correct for a single X-error. I assume that after encoding a single bit-ﬂip occurs on one of the three qubits (or no error occurs). Two initialized ancilla are then coupled to the data block which only checks the parity between qubits. These ancilla are then measured, with the measurement result indicating where (or if) an error has occurred, without directly measuring any of the data qubits. Using this syndrome information, the error can be corrected with a classically controlled X gate. . 63

4.3 Quantum Circuit required to project an arbitrary state, ψ into | iI a 1 eigenstate of the Hermitian operator, U = U †. The measurement± result of the ancilla determines which eigenstate ψ is | iI projectedto...... 69

4.4 Quantum circuit to prepare the [[7, 1, 3]] logical 0 state. The 7 | i input state 0 ⊗ is projected into an eigenstate of each of the X stabilizers shown| i in Eq. 4.16. After each ancilla measurement the classical results are used to apply a single qubit Z gate to qubit i =1M2 +2M3 +4M1 which converts the state from a 1 eigenstates of (K1,K2,K3)to+1eigenstates...... − ...... 70

4.5 Quantum circuit to to correct for a single X and/or Z error using the [[7, 1, 3]] code. Each of the 6 stabilizers are measured, with the ﬁrst three detecting and correcting for Z errors, while the last three detect and correct for X errors...... 71

4.6 Two circuits to implement the transformation 111 000 111 111 . a) shows a version where a single X error can| cascadei| i→| into fouri| i errors while b) shows an equivalent circuit where the error only propagatestoasecondqubit...... 77

4.7 Bit-wise application of single qubit gates in the [[7, 1, 3]] code. Log- ical X, Z H and P gates can trivially be applied by using seven single qubit gates, Fault-tolerantly. Note that the application of ¯ seven P gates results in the logical P † being applied and visa versa. 81

4.8 Bit-wise application of a CNOT gate between two logical qubits. Since each CNOT only couples corresponding qubits in each block, this operation is inherently Fault-tolerant...... 85

4.9 Three circuits which measure the stabilizer K1. Fig a) represents a generic operator measurement where a multi-qubit controlled gate is available. Fig. b) decomposes this into single- and two-qubit gates, but in a non-Fault-tolerant manner. Fig. c) introduces four ancilla such that each CNOT is controlled via a separate qubit. ThisensuresFault-tolerance...... 87

xxv LIST OF FIGURES

4.10 Circuit required to measure the stabilizer K1, Fault-tolerantly. A four qubit GHZ state is used as ancilla with the state requiring veriﬁcation against multiple X errors. After the state has passed veriﬁcation it is coupled to the data block and a syndrome is extracted...... 88

4.11 Circuit required to prepare the [[7, 1, 3]] logical 0 state Fault- tolerantly. Each of the X stabilizers are sequentially| i measured using the circuit in Fig. 4.10. To maintain Fault-tolerance, each stabilizer is measured 2-3 times with a majority vote taken. .... 89

4.12 Circuit required for the LNN implementation of Fault-tolerant stabilizer measurement. These elements are then integrated into a preparation circuit identical to the non-LNN version [Fig. 4.11]. Note that when using a single error correcting code, SWAP gates are assumed to not tointroduce correlatederrors...... 91

4.13 7 qubit circuits for non-Fault-tolerant encoding for a LNN array of qubits. Dotted boxes represent compound gates that can be constructed via the canonical decomposition [Mak02, KC01, ZVSW03], [Chapter 5.4]. The numbering of each qubit is shuﬄed in order to increase the speed of the circuit...... 92

4.14 7 qubit circuits for non-Fault-tolerant encoding using a non-LNN array of qubits. Dotted boxes represent compound gates that can be constructed via the canonical decomposition [Mak02, KC01, ZVSW03],[Chapter5.4]...... 93

4.15 Stability of LNN circuit for the [[7, 1, 3]] encoding. Each curve shows output ﬁdelity vs probability of discrete error for the Fault- tolerant encoding plus correction, and the non-Fault-tolerant encoding...... 94

4.16 Stability of the non-LNN circuit for the [[7, 1, 3]] encoding. Each curve shows output ﬁdelity vs probability of discrete error for the Fault-tolerant encoding plus correction, and the non-Fault-tolerant encoding...... 95

xxvi LIST OF FIGURES

4.17 (From [SDF+06]). Density matrix simulations for the non-LNN circuit for [[7, 1, 3]] state preparation. Since the density matrix formalism is able to calculate ensemble results, the stochastic nature of state vector simulations is eliminated. These simulations 5 found the crossover error rate as p 5.3 10− where the Fault- ≈ × tolerant circuit becomes more stable. Note that these results are examining the crossover between the two circuits, not the Fault- tolerant threshold. The crossover point is the physical error rate for which the Fault-tolerant encoding fails with a lower probability than the direct encoding. At the crossover, neither circuit is below the Fault-tolerant threshold as the logical error rate is higher than the physical error rate. Although simulations were not continued to very low error rates, the Fault-tolerant circuit appears to reach 5 the threshold at p 1 10− ...... 96 ≈ ×

5.1 Modulations in the Rabi oscillations of a three-level system driven by the Hamiltonians, Hm and Hn. Fig. b) provides clear evidence that this system is not a qubit, while Fig. a) appears to show perfect confinement. However the analysis in the following sections will show that the subspace confinement for the system in Fig. a) is also not sufficient for large-scale QIP applications...... 110 5.2 Upper and lower bounds on ǫ for the four-level trial system governed by the Hamiltonian (5.22), characterized by a static coupling between the qubit states and a variable coupling γ to two higher levels. As γ 0 the subspace leakage approaches 0 and the bounds for ǫ becomemoreaccurate.→ ...... 114

5.3 Convergence of ǫu as the number of ensemble measurements, Ne, is increased for a system governed by the Hamiltonian (5.24), characterized by perfect subspace conﬁnement. The solid line represents the actual value of ǫu(Ha) = 0. Note error bars should only extend to zero as ǫ (H ) 0...... 118 u a ≥ 5.4 Convergence of ǫu as the number of ensemble measurements, Ne, is increased for the imperfectly conﬁned system governed by the Hamiltonian (5.25). The solid line represents the actual value of 4 ǫu(Hb) 7 10− . Note error bars should only extend to zero as ǫ (H ) ≈ 0.× ...... 119 u b ≥ 5.5 Distribution of d(Ha) for 5000 separate simulations. The average of the error, 3δd(Ha) is also shown, with approximately 99.9% found within 3σ of d =0...... 120

5.6 Distribution of d(Hb) for 5000 separate simulations. The average of the error, 3δd(Hb) is also shown, with approximately 99.8% found within 3σ of d =0...... 121

xxvii LIST OF FIGURES

5.7 Number of ensemble measurements required to ascertain statistically significant subspace leakage (imperfect confinement) for the three-level system governed by (5.30) as a function of the (analytically calculated) confinement using the confinement equations (5.32) and by directly identifying the third transition peak...... 123 5.8 Number of ensemble measurements required to ascertain significant subspace leakage (imperfect confinement) for the four-level system governed by (5.22) [Fig. a] and the six-level system governed by (5.31) [Fig. b] using the confinement equations and identifying a thirdpeak...... 125 5.9 (From Ref. [CSG+05]). Geometric structure of single qubit dynamics (the Bloch sphere). Any given pure state can be parameterized by the vector ˜s =(θ,φ) where s = 1. If the state is not an eigen- | | state of the Hamiltonian, the evolution processes around the axis defined by the Hamiltonian, d˜...... 133 5.10 (From Ref. [CSG+05]). Experimental protocol for a standard qubit oscillation experiment. The system is repeatedly initialized in the state 0 , allowed to evolve for time ∆t and then measured. This is repeated| i many times to determine σ . This process is then h zi repeated at successively longer time intervals, until the oscillation signalismappedout...... 135 + 5.11 (From Ref. [CGO 06]). Damped oscillation signal caused by σz decoherence of a single qubit. Fig. a) illustrates the case where the driving Hamiltonian is pure σx while Fig. b) illustrates when θ = π/4 in Eq. 5.63. In both cases, the steady state solution is z( )=0...... 137 ∞ 5.12 (From Ref. [ZVSW03]) Subgroups of all possible two-qubit gates. Local gates induce isolated single qubit operations on each qubit. Non-Perfect entanglers (e.g. SWAP) cannot prepare a maximally entangled Bell pair from a product state, while Perfect entanglers (e.g. CNOT) are able to prepare Bell states from initial, product states...... 143 5.13 Basic cubical geometry of SU(4). For any given gate, the coupling triplet [c1,c2,c3] is periodic in π/2 and hence defines a 3-Torus. However, representing this geometric structure as cubical will more naturally lead to a method for characterizing Hamiltonian dynamics.147 5.14 Illustration of one of the 24 degenerate Weyl chambers located within each unit cell representing the geometry of SU(4). In each Weyl chamber, every individual point corresponds to a unique local equivalence class. By utilizing the local operations shown in Eq. 5.91, the triplet [c1,c2,c3], describing a gate, can be swapped between non-degenerate chambers...... 148

xxviii LIST OF FIGURES

5.15 The location of the two, non-entangling gates within the geometry of SU(4). These gates deﬁned a Body Centered Cubic (BCC) unit cell with eight degenerate points representing the Identity operation. Note that within each cell there is only one point corresponding to the local equivalence class [SWAP] at the center.. . 151

5.16 Locations of two common, maximally entangling, gates in the degenerate cubical geometry of SU(4). The [CNOT] equivalence class deﬁnes a Face Centered Cubit (FCC) unit cell geometry while the [√SWAP] equivalence class deﬁnes a small cubic geometry within thelargerstructure...... 152

5.17 Density plot of a single Weyl chamber within the larger geometry of all two-qubit interactions. The color coding represents the maximum concurrence for a specific unitary gate parameterized by the coupling triplet, [c1,c2,c3]. Note that all possible trajectories 2 defined by [c1,c2,c3] are periodic in C , for example traversing the trajectory generated by [c1,c2,c3] = [c1, 0, 0] starts at the Identity, passed through the [CNOT] equivalence class and returns to the Identity at c1 = π/2. In contrast, the trajectory generated by [c1,c2,c3] = [c,c,c] passes through the maximal entangling class [√SWAP], finally arriving at the zero-entanglement SWAP gate at c = π/4...... 154

5.18 Power Density spectrum for the ψ 2 input state. Note that the respective symmetries discussed in| i section 5.4.3. The values for eachpeakheightareshowninTable5.3...... 165

6.1 a) Standard four port 50:50 beamsplitter. A single photon in any one of the input modes will be placed into a superposition state via the operator U in Eq. 6.2. The reflective surface for the beamsplitter is on the front surface such that reflection from A1 to B2 picks up a eiπ = 1 phase shift. b) The two atom/cavity qubits − are placed such that each cavity mode couples to the B1 and B2 optical modes. Once the single photon pulse induces a bit flip on each atom it leaks back out into the respective optical mode and is re-mixed on the beamsplitter. Each atom/cavity system has local optical control, used for single qubit rotations, initialization and readout, while the modes A1 and A2 are contain single photon detectors and the initial single photon source...... 177

xxix LIST OF FIGURES

6.2 Four level atomic system required for the quantum bus. The atomic qubit is defined as the 1 and 3 states, with classical | i | i pumping fields defined by their Rabi frequencies, (Ω1, Ω2), used for single qubit operations. A single photon pulse is introduced to the cavity in a controlled manner using Q-switching [GSPH06]. The single photon pulse induces a light shift on the state 3 , with strength β2/∆. Provided the photon remains in the cavity| i long − enough, a π phase shift can be induced on the atom without destroying the photon. After the atom/photon interaction is complete, the photon is switched out of the cavity and back to the free spaceopticalmodes...... 190

6.3 (From Ref. [DGH07]) Schematic of the conﬁguration required to demonstrate the qutrit transport protocol using a spatially deﬁned bus qutrit. By utilizing the MRAP protocol, the central site, C is never occupied. Here the single lines correspond to controlled| i tunnelling matrix elements for the single control particle Hamilto- nian, and the double lines correspond to the controlled interactions between the Bob sites and the donor qubits. Once the protocol is complete, the spatial location of the electron is measured, using a single electron transistor. Depending on if the electron is localized at A , the data qubits Q and Q are projected into 1 eigenstates | i 1 2 ± of XX...... 193

6.4 (From Ref. [DGH07]) Pulsing sequence for tunneling matrix elements (top) and schematics showing system evolution through the MRAP protocol to realise two-qubit parity measurements. (I) The qutrit is initialised at A with the two qubits in some arbitrary | i state φ . (II) MRAP takes the qutrit to the state ( B + B )/√2, | i | 1i | 2i (III) the controlled unitaries (X1,X2) are performed between each Bob site and the qubits, conditional on the presence of the qutrit at the appropriate site, and the system is transformed to ( B X + | 1i 1 B2 X2) φ /√2. (IV) the transport is reversed and a projective |measurementi | i of the spatial qutrit at A performed (using, for ex- | i ample a SET charge detection). The results of this measurement projects the qubits into an eigenstate of the operator XX, and depending on the measurement result, a phase ﬂip at either B or | 1i B2 and further MRAP protocol can be used to return the qutrit |toAliceforreuse...... i 198

xxx LIST OF FIGURES

6.5 An ion trap node. A single ion trap contains enough ions for a single encoded qubit, ancillas and an interface ion. Conventional single and two-qubit operations are performed via axial phonon modes of the trap. The interface ion may be entangled with its counterpart in another identical trap via photon interference and path erasure. The resultant Bell link is used to perform inter-trap operations. The simplest node consists of a single optically coupled ion, five ions encoding a single logical qubit, and several ancilla for Fault-tolerant operations and singlet purification...... 200 6.6 Distributed Ion Trap Architecture. The whole computer consists of a set of identical nodes, each holding a few physical qubits encoding a logical qubit, and associated ancillas. The nodes are connected by optical fibre linking the interface ions in each node. A heralded probablistic procedure entangles pairs of ions in separate nodes via interference and path erasure. An optical multiplexer allows arbitrary pairs of nodes to be entangled, and parallel operation is achieved using multiple beam-splitters and detectors...... 202 6.7 Quantum circuit measuring the stabiliser K1 for the [[5,1,3]] quantum code. a) non-Fault-tolerant circuit b) basic Fault-tolerant circuit. The Fault-tolerant circuit first requires the preparation and verification of a four qubit GHZ state. If the verification measurement = 1, then the ancilla block is reset and prepared again. To protect against Z errors in the ancilla block, the circuit is repeated up to three times and a majority vote of the syndrome results is taken...... 203 6.8 Full quantum circuit for non-Fault-tolerant, non-local preparation of an encoded Bell state across two nodes. Five ions in each trap are first encoded into the 0 state after which local error correction | iL is repeated continuously (say N times) to protect against memory errors while the Bell link is created. Once the Bell link is created, each interface ion is used as a control qubit for a blockwise X gate on each trap. The interface qubits are then measured locally and a classical Z¯ gate is applied to the second trap if the measurement result has odd parity. The final state of the two traps is the encoded Bell state, ( 0 0 + 1 1 )/√2...... 207 | iL| iL | iL| iL 6.9 Circuit to prepare and verify the interface ancilla blocks for Fault- tolerant operator measurement on the [[5,1,3]] code. The ancilla state requires the preparation of two Bell links between the separate data traps. After the local CNOT gates the second Bell link is measured. If the measurement result has odd parity, the interface block is reset and re-prepared. Local error correction can be performed on each data block while waiting for a verified interface ancillablock...... 209

xxxi LIST OF FIGURES

6.10 CCD micro-trap structure [KMW02] for a single logical qubit using the [[7,1,3]] Steane code. Each chip houses 39 ions: 7 data ions, 28 ancilla ions (allowing for simultaneous preparation and verification of two ancilla blocks using Steane’s rapid correction method [Ste97, Ste02]) and 4 interface ions for coupling to other logical qubits. The interface state required for the [[7,1,3]] code is identical to the [[5,1,3]] code since each of the seven dimensional stabilizers for the Steane code has weight four, hence Wt(Z¯) = Wt(X¯)= 3. Each of these chip nodes can then be connected to the optical multiplexer, increasing the total size of the quantum computer as needed. . . . 214 6.11 Schematics showing the basic design of a photonic module in freespace and photonic crystals. a. Photonic module design for a polarization independent atom/photon interaction, requiring two single photon HWP and two PBS in free space. The polarization dependent interferometer ensures that only the vertical component of the single photon interacts with the atom/cavity qubit. b. A photonic bandgap structure for a polarization dependent atom/photon version of the photonic module, required to prepare two photon Bell states and higher order GHZ states. The initial two cavities represent a Q-switched single photon source [GSPH06]. Single photons are adiabatically switched from the source to the first Q-switch cavity and then switched into the waveguide containing a Quar- ter Wave Plates (QWP) which rotates σ . The second ± Q-switch cavity is then used to adiabatically|±i → load | thei photon into the module cavity which contains the atomic system with a differential coupling between σ photons. The final Q-switch cavity ± is then used to out-couple| i the photon back into the waveguide mode once the interaction is complete, where a second QWP rotates σ . In both schematics, the atom/cavity qubit has ± appropriate| i → laser |±i control such that it can be initialized and read out. The measurement result of the atom qubit (green and red) N determines which eigenstate of X⊗ the exiting photon train is projectedto...... 221

xxxii Chapter 1

Introduction

The micro-computer revolution of the late 20th century has arguably been of greater impact to the world that any other technological revolution in history. The advent of transistors, integrated circuits, and the modern microprocessor has spawned literally hundreds of devices from pocket calculators to the iPod, all now integrated through an extensive worldwide communications system. However, as we enter the 21st century, the rate at which computational power is increasing is driving us very quickly to the realm of quantum theory. The component size of individual transistors on modern microprocessors are becoming so small that quantum effects are beginning to dominate over classical electronic properties. Unfortunately, current designs for micro-electronics mean that quantum mechanical behavior will tend to result in unpredictable and unwanted behavior. Therefore, the natural progression is to the field of quantum information processing (QIP). In QIP, rather than attempt to compensate for quantum effects we instead exploit them. This leads to a paradigm shift in the way we view and process information and has lead to considerable interest from physicists, engineers, computer scientists and mathematicians. The counter-intuitive and strange rules of quantum physics offers enormous possibilities for information processing and the development of a large scale quantum computer is the holy grail of many groups worldwide. In the past ten years there has been extensive work, both theoretical and experimental, on quantum information science. However, the rate at which progress has been made has led to an extensive gap between the latest experimental achievements and the theoretical underpinnings of QIP. Early theoretical work demonstrating the power of quantum computation and the ability of such a

1 CHAPTER 1. INTRODUCTION

device to be built is substantially divorced from the experiential reality of quantum control and device construction. Hence, this thesis is concerned with aspects that I refer to as “Quantum Engineering”. Speciﬁcally, focusing on ideas such as characterizing manufactured quantum systems, studying the eﬀects of imperfec- tion on otherwise ideal quantum devices and the design of more experimentally appropriate quantum processors. Essentially I am attempting to link the abstract, and often unrealistic assumptions of large scale quantum computation to the more concrete restrictions of experimental system design and construction.

Layout and Presentation

This thesis is, in general, laid out in chronological order. However, each Chapter is largely self contained, with each section of work covering a slightly diﬀerent area of research. Section 3.2, which introduces a linear nearest neighbor circuit for Shor’s factoring algorithm consists of research carried out by Austin Fowler and myself, and was originally presented within his Ph.D thesis [Fow05]. Section 5.4, which contains work related to experimental protocols for two qubit characterization was carried out in collaboration with Jared Cole has appeared within his Ph.D thesis [Col07]. However, this discussion was restricted to a small subset of two- qubit coupling and in section 5.4 I present a more generalized protocol. Excluding the introduction (Chapter 2), the review of Quantum Error Correction and Fault- tolerance in Chapter 4, Section 5.3, and the introductory sections of each major topic, each chapter of this thesis contains original research. All work in this thesis has appeared (or is currently in review for publication) in international journals and conference proceedings. Chapter 2 introduces the ﬁeld of quantum information processing, its major theoretical developments and recent experimental achievements, including a brief introduction to circuit and cluster based methods for quantum computation. Chapter 3 then introduces work related to the most well known quantum algorithm, Shor’s factoring routine. I will begin with a detailed overview of the algorithm, after which I will introduce a quantum circuit that is used to implement the quantum component of the algorithm on a linear array of qubits restricted to nearest neighbor interactions. Such a restriction is required when implementing circuits on solid state quantum computers such as phosphorus in silicon, gallium arsenide quantum dots or systems utilizing super-conducting qubits.

2 Section 3.3 then examines the stability of the algorithm under coherent quantum errors, showing that regardless of the circuit implementation, Shor’s factoring algorithm is extremely sensitive to quantum errors. The material in this chapter was presented in references [FDH04, DFH06, DFH05a, FDH05]. Chapter 4 then introduces, in detail, the concept of Quantum Error Correc- tion (QEC) and Fault-tolerant quantum computation. The chapter is largely a review of the most important aspects of Fault-tolerant error correction as the remainder of the thesis utilizes both the stabilizer formalism of Gottesman and various concepts related to QEC and Fault-tolerance. However, section 4.7 presents new work, examining the effective implementation of logical state preparation. I compare the stability of Fault-tolerant circuit constructions to non-Fault-tolerant versions, demonstrating that the stability of Fault-tolerant preparation only becomes effective at extremely low error rates. The material for this chapter has appeared in references [DFH05a] and [SDF+06]. Chapter 5 covers the field of intrinsic characterization techniques. I begin in section 5.2 by introducing an experimental protocol that can be used to estimate how well a quantum system is confinement to the qubit subspace when under external control. Section 5.3 then reviews previously developed protocols required for full characterization of single qubit dynamics with and without decoherence. This section is included for completeness since each characterization protocol often requires information gained from lower order protocols. Section 5.4 then introduces a characterization protocol appropriate for the identification of two qubit coupling dynamics, including a discussion related to the group theoretical structure of two qubit interactions. The material in this chapter has appeared in references [CDH06, DCH06, DSO+06, DSO+07]. Chapter 6 is focused on several ideas related to quantum architecture design and small scale quantum technologies. Section 6.2 introduces a protocol for universal quantum computation using a quantum bus that only carries parity information. This allows for more distributed quantum processor designs without the drawback of losing information during transport. I introduce the quantum bus protocol from the standpoint of an optical network and also discuss a potential implementation in the solid state. In section 6.3 I present a large scale proposal for distributed ion-trap computing, utilizing the Parity gate ideas derived from the quantum bus. Finally, I introduce the design of a small photonic module which allows for the deterministic preparation of a large class

3 CHAPTER 1. INTRODUCTION

of entangled photonic states, utilizing a single atom/cavity qubit and standard linear optical elements. The material in this chapter can be found in references [GDH06b, DGH07, ODH06, GDH06a, DGI+07].

4 Chapter 2

Background

This chapter begins with an introduction to the ﬁeld of quantum information and computation in order to contextualize the material presented in this thesis. The vast quantity of work currently in existence prohibits a detailed overview of any one topic, however I will present general introductions to the main developments in quantum information theory and experiment. In addition to this background chapter, each individual topic begins with an introductory section which focuses the review to the speciﬁc work covered in the chapter.

2.1 Emergence of the Quantum from the Clas- sical.

The emergence of quantum information science from the more established ﬁeld of classical computer science can be attributed to two reasonably disjoint problems. The ﬁrst is epitomized by Moore’s Law, a remarkably accurate empirical prediction make by Intel co-founder Gordon Moore in 1965. Moore hypothesized that “The Complexity for minimum component costs has increased at a rate of roughly a factor of two per year. Certainly over the short term this rate can be expected to continue, if not to increase” [Moo65] and since then, the component density of integrated circuits has followed this prediction extremely well [Dia]. In the late 20th century it was recognized by many people that if component size continued to follow Moore’s law into the 21st century, then the density of integrated circuits will reach the level where each individual transistor is ap-

5 CHAPTER 2. BACKGROUND

proximately the size of a single atom and each electronic pulse would be carried by single electrons. At this scale, quantum mechanics dominates, bringing with it all the counter-intuitive and spooky effects that characterizes this remarkable theory. This is disastrous for classical electronics, since quantum phenomena will lead to inoperable components. One possible resolution to the problem is to try and redesign conventional semiconductors to compensate for quantum effects, the other is to actively exploit them through quantum information processing. The second problem spawning interest in quantum computing is the idea of quantum simulation. During the 1970’s, several abstract papers were published which are generally viewed and the first work on quantum information processing. Holevo [Hol73] was the first to show that you cannot extract more than N classical bits from N qubits, while Bennett [Ben73] demonstrated that computation can be done reversibly (an essential component for quantum processing). However, it wasn’t until the early eighties, when Richard Feynman [Fey82] pointed out that simulating quantum systems efficiently on a classical computer appeared impossible, did the idea of a engineered device which actively exploits quantum mechanics to process information became a serious avenue of research. During the 1980’s the fundamental work on quantum information was first published. Toffoli [Tof81] introduced the first reversible quantum gate which now bares his name and in 1985 David Deutsch at Oxford formulated the quantum version of the Turing machine [Deu85] which cemented the idea of a universal quantum computer, i.e. a machine which can simulate any unitary evolution of a quantum system. In 1992-94 quantum computation exploded with the introduction of the first quantum algorithms which demonstrably showed that quantum computers could solve several problems much more efficiently than any classical device. In 1992, Deutsch and Jozsa [DJ92] introduced an algorithm which can discriminate between constant binary functions (functions returning the same result for all in- puts, either all 1 or all 0) and balanced functions (functions which return 0 for half the input values and 1 for the other half), quicker than an equivalent classical algorithm. Subsequently Simon [Sim94] proposed an algorithm which, given a binary function f which has the property that f(x)= f(y) iff x = y a, where ⊕ represents binary addition, can determine a more efficiently than an equivalent ⊕ classical algorithm. However, the turning point for quantum computing came in 1994 when Peter Shor introduced his quantum factoring algorithm [Sho97],

6 2.1. EMERGENCE OF THE QUANTUM FROM THE CLASSICAL.

which demonstrated the extraordinary power of quantum computation. Shor showed that an appropriate quantum algorithm can be constructed to efficiently factor a large composite number into its prime factors. This had major implications beyond the scientific community since many public key crypto-systems such as RSA [RSA78] are based upon the classical intractability of factoring. Since the introduction of Shor’s algorithm, quantum information science has progressed rapidly. Immediately there were several proposals for physical systems that could be used for quantum computation, including the ion trap design of Cirac and Zoller [CZ95], Molecular qubits using Nuclear Magnetic Resonance (NMR) [CFH97, GC97], the Kane quantum computer using buried phosphorus donors in silicon [Kan98], qubits based on gallium arsenide quantum dots [LD98], linear optics utilizing photonic qubits [KLM01], super-conducting systems [MOL+99, NPT99] and many others [ARD04]. Secondly, in 1995-96 Andrew Steane, Peter Shor and Robert Calderbank introduced the concept of Quantum Error Correction (QEC) [Sho95, Ste96a, CS96]. This was one of the final pieces of theory required to show that quantum information processing was theoretically possible. Prior to the introduction of QEC it was thought that it would not be possible to combat the inevitable degradation of quantum coherence caused by the coupling of qubits to a larger environment (termed decoherence). However, Shor, Steane and Calderbank demonstrated that active QEC can be used to not only maintain quantum coherence between qubits, but can do so without attaining any information regarding the state of the qubits (an essential property to quantum computing). In the past ten years, enormous progress has been made. Coherent control of single qubits has been experimentally demonstrated in several systems [WMI+98, NPT99, CNHM03, GHW05], controlled interactions between separate qubits has been achieved [JM98, MMK+95, OPW+03, YPA+03] and successful implementation of basic quantum algorithms [CGK98, VSB+01], quantum error correction [CPM+98, CLS+04] and multi-qubit entanglement [HHR+05, SKK+00] has been demonstrated (predominantly using ion trap and NMR quantum computers). Presently, the quantum computing community is engaged in a vast variety of research either directly linked to the construction of a functional quantum computer or related to the more fundamental aspects of information theory, atom/optics, condensed matter physics and fundamental quantum mechanics. It

7 CHAPTER 2. BACKGROUND

is clear that if the next ten years progress as quickly as the previous, quantum computation could soon become a reality.

2.2 Quantum Information Processing: Compu- tational Models.

As a basic introduction to the principles of quantum information processing, this section will brieﬂy detail the two most common models for quantum information processing, namely the quantum circuit model [Deu89, Yao93, NC00] and Cluster State computation [BR01, RB01, Nie06]. I speciﬁcally detail these two models since they are the two relevant schemes utilized throughout this thesis.

2.2.1 The Quantum Circuit Model.

The circuit model for quantum computation was the initial framework for quantum information processing. The circuit model is analogous to the classical circuit model of Boolean logic where a sequence of gate operations are performed sequentially to realize an algorithm. The fundamental unit of quantum computation is the qubit, which unlike classical bits can exist in a coherent superposition of two binary states, 0 , 1 . These basis states can be photonic polarization, {| i | i} atomic spin, electronic states of an ion or charge states in superconducting systems [ARD04]. Therefore, an arbitrary state of an individual qubit, φ , can be | i written as, φ = α 0 + β 1 , (2.1) | i | i | i where normalization requires,

φ φ 2 = α 2 + β 2 =1. (2.2) |h | i| | | | |

Quantum gate operations are represented by unitary matrices acting over the Hilbert space of the qubit register. Unlike classical information processing, conservation of probability for quantum states require that all operations be reversible, this enforces the condition that all gate operations are represented by unitary matrices. As arbitrary states of N qubits are completely speciﬁed by 2N complex co-eﬃcients, a gate operation over N qubits is represented by a 2N 2N ×

8 2.2. QUANTUM INFORMATION PROCESSING: COMPUTATIONAL MODELS.

matrix, G, such that G†G = I, where is the conjugate transpose operator and † I is the 2N 2N identity matrix. × To illustrate, we can ﬁrst consider some useful, single qubit, quantum gates. Fig. 2.1 details a quantum circuit which will enact the following transformation,

1 0 ( 0 + 1 ). (2.3) | i→ √2 | i | i i.e. the initial state 0 has been rotated to an equal superposition of 0 and 1 . | i | i | i In the circuit representation, time runs from left to right and the horizontal line represents the evolution of the qubit. The single qubit gate, H, is known as the

Figure 2.1: Circuit representation for a single qubit gate. Time runs left to right and the solid line represents the time evolution of the qubit, φ = 0 . The gate H is represented by a 2 2 unitary matrix and after the operation| i | i the output × state is given by H 0 | i

Hadamard transformation and has the following matrix representation,

1 1 1 H . (2.4) ≡ √2 1 1! − If an arbitrary state of the qubit, φ , is represented by the 2 1 column vector, | i × (α, β)T , then the gate operation is simply given by the matrix multiplication,

1 1 1 α 1 α + β H φ = . (2.5) | i ≡ √2 1 1! β! √2 α β! − − Hence for α = 1 and β = 0, the final state will be given by α = β =1/√2. Other interesting single qubit gates are the bit flip gate, X, the phase flip

9 CHAPTER 2. BACKGROUND

gate, Z, and the rotations, R (θ), R (γ) , given by the matrices, { z x } 0 1 α β X φ = , | i ≡ 1 0! β! α! 1 0 α α Z φ = , | i ≡ 0 1! β! β! − − (2.6) 1 0 α α Rz(θ) φ = , | i ≡ 0 eiθ! β! eiθβ! cos(γ) i sin(γ) α α cos(γ)+ iβ sin(γ) Rx(γ) φ = = . | i i sin(γ) cos(γ) ! β! iα sin(γ)+ β cos(γ)!

Note that the bit-flip, X, and phase-flip, Z, are specific forms of the Rz(θ) and

Rx(γ) gates when θ = π and γ = π/2 (up to a global and unphysical phase factor). In order to successfully implement a full quantum algorithm, multi-qubit gates are also required which can be used to generate entanglement between individual qubits. The most well known and useful multi-qubit interaction is the two qubit controlled-NOT (CNOT) gate. Fig. 2.2 illustrates the quantum circuit and the truth table for this gate. A CNOT interaction is designed such that if the

Figure 2.2: Circuit representation and truth table for the two-qubit CNOT gate. If the control qubit is in the state 1 , the target is ﬂipped, while if the control is in the 0 state the gate does nothing.| i | i

control qubit, a , is in the state 1 , then the target qubit, b , is ﬂipped, while if | i | i | i the control is 0 the gate does nothing. The corresponding matrix representation | i

10 2.2. QUANTUM INFORMATION PROCESSING: COMPUTATIONAL MODELS.

is, 1000 α α 0100 β β CNOT ab = (2.7) | i ≡ 0001 γ δ       0010 δ γ             where ab is the combined state of two qubits, which can be written as a linear | i superposition of all possible binary states as,

ab = α 00 + β 01 + γ 10 + δ 11 . (2.8) | i | i | i | i | i

Now that we have access to single and two-qubit operations, the preparation of entangled qubit states is possible. Consider the general circuit shown in Fig. 2.3. Four qubits are initialized in the state 0 , after the sequence of single and two | i qubit gates, the ﬁnal state is a four qubit entangled state,

1 φ′ = ( 0000 + 1111 ). (2.9) | i √2 | i | i

This state is a four qubit generalization of a Bell state known as a Greenberger- Horne-Zeilinger state (GHZ) [GHZ89]. Another required component for any computational model of quantum computation is measurement. In circuit diagrams, measurement is represented by the element shown on the bottom qubit in Fig. 2.3 and represents a destructive collapse of the wave-function, resulting in a classical bit value (either 1 or 0). For example, after we prepare the state φ′ , we measure one of the qubits. As this | i state is a four qubit GHZ state, there is a 50% chance that the we measure a 0 | i or 1 , collapsing the entire state (as each qubit is highly entangled) to the state | i 0000 or 1111 . | i | i The physical implementation of the single qubit gates shown in Eq. 2.6 and the two-qubit CNOT gate has been shown to be a universal set for quantum computation [NC00]. i.e. any 2N 2N unitary operation over N qubits can be × decomposed in to a time ordered sequence of single qubit gates and the CNOT. This is not the only possible set of gates which form a universal set [NC00], however, the majority of physical system proposals implement universality via this speciﬁc combination.

11 CHAPTER 2. BACKGROUND

Figure 2.3: Quantum circuit to prepare a four qubit GHZ state. After the state has been prepared, we measure the bottom most qubit, collapsing the state to 0000 or 1111 with an equal probability of 50%. | i | i

2.2.2 Cluster State Model

The cluster state model for universal quantum computation is an extremely novel method, first proposed by Raussendorf and Briegel in 2001 [RB01, BR01]. They demonstrated that universal computation could be achieved through single qubit measurement and a generic resource of highly entangled qubits which are completely independent of the algorithm being implemented. The resource for cluster state computation is a specific multi-qubit entangled state which can be prepared via controlled-PHASE (CZ) rotations between two qubits. The CZ gate is defined over two qubits such that an arbitrary state ab | i transforms as,

CZ ab = CZ(α 00 +β 01 +γ 10 +δ 11 )= α 00 +β 01 +γ 10 δ 11 , (2.10) | i | i | i | i | i | i | i | i− | i and in circuit representation is illustrated in Fig 2.4. To prepare a universal cluster state, an array of qubits are initialized in the state + =( 0 + 1 )/√2 and a CZ gate is applied between any two qubits linked | i | i | i together in the cluster. The topological conﬁguration of a cluster state, suitable for universal computation, is a 2-dimensional grid, where each node represents a physical qubit and each edge deﬁnes a CZ link, Fig. 2.5 illustrates. Note, that this

12 2.2. QUANTUM INFORMATION PROCESSING: COMPUTATIONAL MODELS.

Figure 2.4: Circuit representation of a controlled-PHASE (CZ) gate. geometry is topological, i.e. the physical location of each qubit node is irrelevant.

Figure 2.5: Topological conﬁguration of a cluster state appropriate for universal computation. Each node represents a physical qubit which can by physically located anywhere within the qubit array. Initially, each qubit is prepared in the + state and CZ gates are applied between any two connected qubits. | i

The physical states representing a cluster are speciﬁed in Fig. 2.6 for two, three and four qubits. To understand how computation is achieved using the cluster model, we actually use the circuit formalism to model information ﬂow. As mentioned in Section 2.2.1, universality can be achieved via arbitrary single qubit rotations and the two-qubit CNOT gate, consequently I will only illustrate how these operations are performed using the cluster.

First, consider an arbitrary rotation of a single qubit, Rz(θ), as given in Eq. 2.6. The circuit given in Fig. 2.7 illustrates how measuring the cluster achieves this rotation. An arbitrary initial state, φ = α 0 + β 1 , is ﬁrst entangled with | i | i | i

13 CHAPTER 2. BACKGROUND

Figure 2.6: Topological geometries and state vectors for two, three and four qubit cluster states. Two and three qubit cluster states are actually equivalent to Bell states and 3-qubit GHZ states (up to local rotations), however as the cluster gets larger the total number of basis states gets exponentially large.

a cluster qubit (initialized in the state + ) via a CZ gate, giving, | i 1 CZ(α 0 + β 1 ) + = (α 0+ + β 1 ), (2.11) | i | i | i √2 | i | −i where =( 0 1 )/√2. Next, the original qubit is measured in the R (θ)H |±i | i±| i { z } basis. This is represented by performing the single qubit rotations Rz(θ) and H and then measuring the qubit in the 0 , 1 basis. Just prior to measurement, {| i | i}

14 2.2. QUANTUM INFORMATION PROCESSING: COMPUTATIONAL MODELS.

Figure 2.7: Quantum circuit representation for an Rz(θ) gate applied using the cluster state model. Initially an arbitrary qubit is linked to a cluster qubit via a CZ gate. Next the original qubit is measured in the Rz(θ)H basis, corresponding to single qubit R (θ) and H gates and measurement{ in} the 0 , 1 z {| i | i} basis. The classical measurement result is then fed-forward to the second qubit and used to classically control a Z ﬂip. After this classical correction, the second qubit is in the state R (θ) φ corresponding to the desired rotation. z | i the total state of the system is given by,

1 1 0 (α + + βeiθ )+ 1 (α + βeiθ ). (2.12) 2| i | i |−i 2| i | i − |−i

Therefore, if the measurement of the ﬁrst qubit returns a 0, then the second qubit is projected to the state, α + + βeiθ , (2.13) | i |−i while if the measurement result is 1, the second qubit is projected to,

α + βeiθ . (2.14) | i − |−i

Finally, a Hadamard gate is applied to the second qubit regardless of the measurement result, while if the ﬁrst qubit is measured 1 a classically controlled Z gate is applied to the second qubit. After this ﬁnal step, the second qubit is in the state, α 0 + βeiθ 1 = R (θ) φ , (2.15) | i | i z | i giving the required transformation. Therefore, by entangling the qubit into a two-qubit cluster and performing measurement in the appropriate single qubit basis, we are able to perform an arbitrary rotation Rz(θ). Note that the original qubit has been destroyed (via measurement) and the rotated state has been teleported to the second qubit.

15 CHAPTER 2. BACKGROUND

Hence as a cluster computation proceeds, information is teleported along the qubit chain.

In order to simulate the rotation Rx(θ), we note the following identity,

Rx(θ)= HRz(θ)H, (2.16) up to a global (any unphysical phase). Therefore, rather than directly synthe- sizing Rx(θ), removing the rotation Rz(θ) and replacing the classically controlled Z-flip with an X-flip in Fig. 2.7, we can realize the rotation, φ H φ and | i → | i combining these two cluster operations will allow for the transformation Rx(θ). Notice that in this model, single qubit gates are required to implement single qubit operations. At first glance this might seem contradictory, however this is not a problem. Cluster state computation assumes the ability to measure in arbitrary single qubit bases. The single qubit rotations required on the top qubit in Fig. 2.7 represents this choice of readout basis relative to the 0 , 1 computational {| i | i} basis. The single qubit correction gates on the lower qubit, which depend on the measurement result, can always be implemented in the next operational step by simply modifying the readout basis for the next measurement. However, it should be noted that any physical system implementing cluster state computation will generally not require cluster state methods for single qubit gates. As all current proposals for qubit systems only have physical readout in one basis, single qubit rotations will always be needed on qubits for Cluster state computation. This does not diminish the applicability of cluster states, as the primary advantage of this scheme is the algorithmic independent preparation of the initial cluster and the implementation of two-qubit operations. So far we have shown how arbitrary rotations on a single qubit can be performed by linking a qubit into a linear cluster chain and performing arbitrary single qubit basis measurements and classical feedforward. However, this is not sufficient for universality, as we require the ability to perform at least a two- qubit CNOT operation. In order to achieve this, we require a cluster geometry that is no longer one dimensional. Fig. 2.8a illustrates the four-qubit cluster state needed for a CNOT gate, where the qubit labeled C represents the control qubit (which remains fixed to one site) and the qubits Tin and Tout represent the target qubit which is teleported from the input to the output during the operation. Fig 2.8b shows the equivalent quantum circuit to model the transformation. It is straightforward to track this circuit through, for a given input state,

16 2.2. QUANTUM INFORMATION PROCESSING: COMPUTATIONAL MODELS.

Figure 2.8: Topological geometry (a) and equivalent quantum circuit (b) to realize a CNOT operation over the two qubit state CT in the cluster model. Unlike single qubit operations, the CNOT gate requires| i a two-dimensional topological geometry where the control qubit, C , remains at a ﬁxed physical location and the target qubit, T , is teleported| fromi site two to four. After all four qubits have been linked together,| i X basis measurements are performed on qubits two and three (corresponding to Hadamard gates and measurement in the 0 , 1 basis). The results are then fed-forward and used to classically correct{| thei ﬁnal| i} state.

CT = α 00 + β 01 + γ 10 + δ 11 . The state of the full system immediately | i | i | i | i | i after the CZ gates is,

1 α 00 ( 0+ + 1 )+ β 01 ( 0+ 1 ) √8 | i | i | −i | i | i−| −i (2.17) 1 + γ 10 ( 0+ 1 )+ δ 11 ( 0+ + 1 ) . √8 | i | i−| −i | i | i | −i where the labels [1, 2, 3, 4] are implied and the ordering is . . . After the | i12| i34 Hadamard rotations on qubits two and three, but prior to measurement, the state of the system is,

1 00 (α 00 + β 01 + γ 11 + δ 10 ) 4| i23 | i | i | i | i 14 1 + 01 23(α 01 + β 00 + γ 10 + δ 11 )14 4| i | i | i | i | i (2.18) 1 + 10 (α 00 β 01 + γ 11 δ 10 ) 4| i23 | i − | i | i − | i 14 1 + 11 (α 01 β 00 + γ 10 δ 11 ) . 4| i23 | i − | i | i − | i 14

Hence, the result of measuring qubit three dictates if a classically controlled X

17 CHAPTER 2. BACKGROUND

gate is applied to the target qubit, while the measurement result of qubit two dictates if a Z gate is applied to both control and target. After these corrections, a CNOT gate has been applied between control and target with the control qubit fixed at position one while the target is teleported from site two to four. The ability to simulate a CNOT gate between two qubits plus the single qubit rotations shown earlier gives us universality using this model. In order to implement a specific algorithm using the cluster method, a quantum circuit is constructed and then mapped on to a measurement pattern over the cluster array. Single qubit gates are applied by consuming cluster qubits in a linear direction while two qubit gates are achieved by consuming links between rows of cluster qubits. Therefore, the physical array of qubits comprising the cluster state not only represents the total number of qubits required for the algorithm, but also the total number of time steps. In the circuit model, each physical qubit always retains its information during gate operations, but for cluster state computation the information is continually teleported along the cluster array as each gate operation is performed. Hence, one of the downsides to cluster state computation is the increase in physical qubits required for a specific algorithm. Even with this disadvantage, cluster state computation has enormous benefits when applied to specific systems. As the initial cluster state is completely independent of the quantum algorithm being implemented, it can be prepared offline, before computation is initiated. Additionally, if the fundamental interactions between qubits is probabilistic, then the standard circuit model can never be used (as error rates are far too high) and cluster state computation allows for such systems to be used for computation.

2.3 Conclusion

This chapter has provided a brief introduction to the ﬁeld of quantum information processing. Although I have only provided a very brief overview of the major developments in quantum computation and a simplistic introduction to computational models, the introductory sections to each of the following chapters provides a much more subject speciﬁc overview of the relevant history and material.

18 Chapter 3

Shor’s Factoring Algorithm

Contents 3.1 Introduction ...... 20 3.1.1 From Factoring to Period Finding...... 21 3.1.2 The Quantum Period Finding subroutine ...... 22 3.1.3 Classical post-processing ...... 24 3.2 Linear Nearest Neighbor quantum circuit...... 27 3.2.1 Decomposing the QPF subroutine ...... 28 3.2.2 Conclusion ...... 40 3.3 Stability of Shor’s Algorithm under errors ...... 42

3.3.1 Errormodelsandanalysis ...... 44 3.3.2 Stability under a ﬁxed number of errors ...... 47 3.3.3 Conclusion ...... 55

19 CHAPTER 3. SHOR’S FACTORING ALGORITHM

3.1 Introduction

As the following chapter deals exclusively with Shor’s algorithm, I will first provide a detailed overview. Along with the original paper of Shor [Sho97], there are several excellent reviews dealing with the algorithm [NC00, LMP03]. Although Shor’s algorithm is considered a quantum algorithm, there is still a large amount of processing that occurs using classical techniques. I will examine a few of these classical steps for the sake of completeness. The discovery in 1994 by Shor of an efficient quantum algorithm for factorization was treated with enormous enthusiasm. It was also quite surprising that one of the first large scale quantum algorithms invented had such a clear “Real World” application and showed such a significant speed up over all classical algorithms designed to solve the same problem. The classical intractability of factorizing extremely large composite numbers forms the current basis for public key-cryptographic protocols, particularly RSA [RSA78] which is extensively used in internet communication and public network commercial transactions. The computational difficulty in breaking RSA encryption has even lead to significant cash prizes for individuals or groups committed to the challenge [Cha]. The advent of Shor’s algorithm, while still experimentally difficult due to the lack of a physical quantum computer, makes the problem of factoring computationally tractable and demonstrated clearly the power a quantum computer could have over a classical device.

Original Material

Section 3.2 introduces a quantum circuit to implement Shor’s algorithm ap- • propriate for a quantum computer restricted to a linear array of qubits with nearest neighbor couplings. This work was completed in collaboration with Austin Fowler and appeared in Ref. [FDH04], with a detailed discussion presented in Austin Fowler’s Ph.D thesis [Fow05].

Section 3.3 presents numerical simulations investigating the stability of • Shor’s algorithm under coherent quantum errors and represents original work published in Refs. [DFH06, DFH05a].

20 3.1. INTRODUCTION

3.1.1 From Factoring to Period Finding.

Consider a given composite number N = N1N2 with binary length L = log2(N), for simplicity we assume that N1 and N2 are prime. Shor’s algorithm allows the determination of N1 and N2 with qubit and time resources that grow polynomial with L. In its most basic form, Shor’s algorithm is extremely easy to understand and is essentially a Fourier analysis program for a quantum register.

k In order to determine the factors, N1 and N2, the function f(k)= a mod N is examined, where 1 1. The quantum part of Shor’s algorithm is designed to efficiently find r, since all other steps in the algorithm can be efficiently calculated using classical algorithms. If r can be found, then by definition f(r) = ar mod N = 1 and hence (ar 1) mod N = 0 and (ar/2 1)(ar/2 + 1) mod N = 0. The last expression − − implies that the product (ar/2 1)(ar/2 + 1) divides N with no remainder and − therefore must be an integer multiple of N, i.e. (ar/2 1)(ar/2 +1) = αN, where − α < N Z. By definition N = N N , hence (ar/2 1)(ar/2 +1) = αN N . Since, ∈ 1 2 − 1 2 in general, α is unknown we extract N and N by calculating gcd(ar/2 1, N) 1 2 − and gcd(ar/2 +1, N) which can be done classically. Factorizing N clearly requires r to be even (since ar/2 must be an integer) and it also requires that ar/2 = N 1, 6 − as this would imply that (ar/2 +1) = N. However, it has been shown that for an odd value of N and a randomly selected a which satisfies gcd(a, N) = 1, the probability that f(k) has a suitable value of r is at least 0.75 [NC00]. Thus, on average, two iterations of the algorithm is needed if the first value of a chosen results in an unsuitable value for r. The quantum component of Shor’s algorithm is designed purely to find r. Therefore it is common to refer to the quantum component of Shor’s algorithm as the Quantum Period Finding (QPF) subroutine. This quantum subroutine is identical to a traditional Fourier analysis program, however it is performed on quantum registers rather than classical data sets.

21 CHAPTER 3. SHOR’S FACTORING ALGORITHM

3.1.2 The Quantum Period Finding subroutine

Traditional Fourier analysis is well known to anyone even remotely involved with signals analysis. A classical signal is known to be periodic, however the frequency of oscillation is unknown. Hence the data set representing the signal is passed through a Discrete Fourier Transform (DFT) algorithm, resulting in a signal peak occurring at multiples of the fundamental oscillation frequency, which is then read out. The QPF component of Shor’s algorithm is no diﬀerent. The function f(k) is known to be periodic and we wish to determine the period of this oscillation, r. Therefore, in a suitable quantum register, the full data set k , f(k) is {| i | i} prepared, after which a quantum equivalent of the DFT is applied to the register and is ﬁnally read out.

In general, to factorise a number of binary length L = log2 N, 3L qubits are initialized to the state 0 0 . For clarity we have broken these 3L qubits | i2L| iL into a 2L qubit register, to store the values k, and L qubits to store the function evaluations, f(k) = akmodN. In a similar manner to classical Fourier analysis, the k-register has to be large enough such that several oscillations of the signal are simultaneously stored within the register, while the f-register only has to be as big as the maximum number ever evaluated, max(f(k)) = N 1. − After initialization, a Hadamard transform is performed on each of the 2L qubits, placing the k-register into an equal superposition of all binary numbers from 0 22L 1, → −

22L 1 1 − 0 0 k 0 . (3.1) | i2L| iL −→ 2L | i2L| iL Xk=0 Step three is to apply the function f(k) = akmodN on the L qubit register, conditional on the values k.

22L 1 22L 1 1 − 1 − k 0 k akmodN . (3.2) 2L | i2L| iL −→ 2L | i2L| iL Xk=0 Xk=0 The next step is to measure the L qubit register. This step can actually be omitted when implementing the algorithm, however it is introduced to show how

22 3.1. INTRODUCTION

the period, r, appears within the procedure.

2 22L 1 2 L/r 1 1 − √r − k f(k) k + nr f , (3.3) 2L | i2L| iL −→ 2L | 0 i2L| 0iL n=0 Xk=0 X where r is the period of f(k), f0 is the value actually measured and k0 is the smallest value of k such that f0 = f(k0). The k-register will now contain only the binary values of k for which f(k)= f0. A Quantum Fourier Transform (QFT) is now applied to the k-register. The QFT is deﬁned such that it takes the binary state k to, | i 22L 1 1 − 2πi k exp jk j . (3.4) | i2L → 2L 22L | i j=0 X which transforms the state of the register to,

(22L 1) (22L/r 1) √r − − 2iπ exp j(k + nr) j f . (3.5) 22L 22L 0 | i2L| 0iL j=0 n=0 X X Disregarding the state f , Eq. 3.5 has the form, | 0iL

22L 1 − c j . (3.6) j| i2L j=0 X Hence the probability that when measured, the k-register collapses to the value j is given by, 22L/r 1 √r − 2iπ 2 Pr(j, r, L)= exp jnr . (3.7) 22L 22L n=0 X

Eq. 3.7 is strongly peaked at certain values of j. There are two speciﬁc cases. If the period r perfectly divides 22L then Eq. 3.7 can be evaluated exactly, with the probability of observing j = c22L/r for 0 c

23 CHAPTER 3. SHOR’S FACTORING ALGORITHM

Pr j 0.125

a. j 0 32 64 96 128 160 192 224

Pr j 0.1

b. j 0 26 51 77 102 128 154 179 205 230

Figure 3.1: (From Ref. [Fow05]) Plot of Eq. 3.7 for the case, 22L = 256 with a) r = 8 and b) r = 10.

while the total time required to implement the unitary transformations will be shown in section 3.2 to also scale polynomially with L. Hence the QPF subroutine represents an eﬃcient algorithm for factoring on a quantum device.

3.1.3 Classical post-processing

Even after the application of the QPF subroutine there is no guarantee that the result will lead to successful factorization of N. Once a value, j, is measured then there are several post-processing steps which need to be performed using classical techniques. In the case where r is not a power of two, the measured value of j c22L/r, ≈ where 0 c < 1 Z. In order to determine r, a continued fractions method is ≤ ∈ employed. Ref. [NC00, LMP03] contain more detailed summaries of the continued fractions method, instead I will simply provide an example (which is also given in [Fow05]) to highlight the technique. Let us assume that we wish to factor N = 143, using a = 2, with a resulting output of the QPF subroutine, j = 31674. Therefore, j c22L/r 31674 c65536/r. We now use a continued fraction ≈ → ≈

24 3.1. INTRODUCTION

expansion for the approximation c/r 31674/65536, which is, ≈ 31674 1 1 1 = 32768 = 1094 = 1 . (3.8) 65536 2+ 2+ 1 15837 15837 14+ 1 2+ 1 10+ 52

Any continued fraction expansion of a number between 0 and 1 is completely specified by the set of denominators once the expansion has lead to a numerator equal to 1, here this expansion is specified by the set, 2, 14, 2, 10, 52 . The { } nth convergent of a continued expansion is defined by truncating at the nth denominator. For example, the 3rd convergent is,

1 29 2, 14, 2 = 1 = . (3.9) { } 2+ 1 60 14+ 2

To ﬁnd r we substitute each denominator of the nth convergence into f(k)=2k mod 143. We only need to examine denominators that are less than N = 143, since r < N [LMP03]. In this case, 60 is the largest denominator less than 143 (the 4th convergent gives 304/629), giving 260 mod 143 = 1, hence r = 60. The continued fractions method assumes that j c22L/r, however there is ≈ still a ﬁnite chance that the resultant measurement of j is completely useless, i.e. either j is far from any major probability peak or j = 0. Hence we denote a set of useful j values to contain c22L/r and c22L/r , where and denote ⌊ ⌋ ⌈ ⌉ ⌊⌋ ⌈⌉ rounding down and up respectively and 0

25 CHAPTER 3. SHOR’S FACTORING ALGORITHM

The classical steps of Shor’s algorithm does add overhead to the factorization of N, however the dominant step is still the implementation of the QFP subroutine. Each classical step may require a re-run of the QFP subroutine which we can summarize as follows,

1. Select 1

2. Use the QPF subroutine several times to ﬁnd multiple values of j c22L/r. ≈ 3. Use j to determine r. If the number of j values extracted from the QPF subroutine is insuﬃcient (for example, due to several j = 0 measurements) repeat the QPF subroutine.

4. Check that r is even and that f(r/2) = N 1. If this fails, the entire 6 − sequence of steps is repeated.

5. Use r to ﬁnd N1 and N2.

Steps 3. and 4. may cause the algorithm to loop itself several times, however the success of Step 3, for a reasonably small number of iterations of the QPF subroutine, and the success of Step 4 does not influence the scaling of the algorithm with L. The probability of the QPF subroutine to return a useful value of j [Eq. 3.10] is the relevant quantity when investigating the efficiency of the algorithm, with an average of O(1/s) calls to the QPF subroutine is needed to return a useful value of j. Note that in the algorithm, there are quite a few classical checks which are used to see if j leads to a value of r appropriate for factoring N. This is not a problem since checking N1N2 = N is efficient using classical techniques.

26 3.2. LINEAR NEAREST NEIGHBOR QUANTUM CIRCUIT.

3.2 Linear Nearest Neighbor quantum circuit.

Constructing a suitable system to implement quantum information processing is certainly a very difficult task, with several physical architectures currently proposed [ARD04]. However, it was generally assumed in circuit design that within a quantum computer, any pair of qubits could be interacted independently, regardless of their physical location in the array. While this is an appropriate assumption for proposals such as ion traps [CZ95] and for systems with extremely mobile qubits such as photons [KLM01], for many architectures this assumption is unrealistic. The quintessential example of what we call a “local” architecture occurs for solid state systems, such as phosphorus in silicon [Kan98], GaAs quantum dots [LD98] and superconducting qubits [MOL+99, NPT99]. In these cases, qubit-qubit coupling is achieved via short range interactions and the physical qubits are completely immobile. Additionally, each of these solid state proposals initially recommended that the qubit array be one dimensional (recently this has changed to multi-dimensional structures [HGFW06, TED+05] for engineering reasons). Hence these systems fall into the general category of Linear Nearest Neighbor architectures (LNN). Determining whether appropriate quantum circuits can be constructed for LNN architectures which are both practical and efficient is an extremely important question. In order to demonstrate that the LNN restriction is not fatal to the design of efficient quantum circuits, we have chosen to focus on the quantum period finding component of Shor’s algorithm. This choice was motivated by the significant interest in this particular algorithm and the complicated nature of its circuit implementation. This section will introduce an appropriate circuit to implement the QPF subroutine on a LNN architecture, utilizing the SWAP channel for data transport. We show that in order to successfully factorize a L-bit number, this LNN circuit can be implemented using 2L+4 qubits, 8L4+40L3+116 1 L2+4 1 L 2 2 2 − gates arranged into a total circuit depth of 32L3 +80L2 4L 2, where the circuit − − depth refers to the total number of time steps required assuming that multiple, independent pairs of qubits can be coupled simultaneously. There has been several proposed circuits to implement the QPF subroutine which all assume that arbitrary qubit-qubit coupling is possible. Some of these circuits are optimized for speed, some for the total number of qubits and some proposing a tradeoff between the two. These circuits are summarized in Table 3.1.

27 CHAPTER 3. SHOR’S FACTORING ALGORITHM

Circuit Qubits Depth Simplicity [VBE96] 5L O(L3) ∼ 2 Speed 1 [MI05] O(L2) O(L log L) Speed 2 [CW00] O(L3) O(log L) Qubits [Bea03] 2L 32L3 ∼ ∼ Tradeoff 1 [Zal98] 50L 219L1.2 Tradeoff 2 [Zal98] ∼ 5L ∼ 3000L2 ∼ ∼ Table 3.1: Number of qubits required and circuit depth of different implementations of the QPF subroutine. Where possible, figures are accurate to leading order in L. The circuit of Cleve and Watrous [CW00] introduced a fast, parallel implementation of the Quantum Fourier Transform but did not examine, in detail, the full circuit required for modular exponentiation.

The circuit introduced in this section is based on the Beauregard circuit [Bea03] which assumes arbitrary qubit coupling (non-LNN) and is still currently the circuit with the fewest number of qubits which is able to implement the QPF subroutine. To first order, the total gate count and depth of the original Beauregard circuit is 8L4 and 32L3 provided it is slightly modified (one extra qubit is added) to account for the elimination of three qubit gates. Hence our LNN circuit is as efficient as the equivalent non-LNN circuit, conclusively demonstrating that the restriction of linear nearest neighbor architectures is not fatal to quantum circuit design.

3.2.1 Decomposing the QPF subroutine

The introductory section to this chapter has already introduced the theoretical basics of Shor’s algorithm, including an abstract description of the Quantum Period Finding subroutine. We now need to detail how the QPF subroutine is decomposed such that it can be implemented using only single and two-qubit quantum gates. Recall that the four main steps in the subroutine are,

1. Hadamard transform.

2. Modular Exponentiation.

3. Quantum Fourier Transform (QFT).

4. Measurement.

28 3.2. LINEAR NEAREST NEIGHBOR QUANTUM CIRCUIT.

Steps 1. 3. and 4. are very simple to implement from the perspective of a quantum circuit. Step 2. is the bottleneck when circuits for the QPF subroutine are designed and is the only section of the circuit that needs further decomposition. First we will detail the circuit required for the QFT, since it is not only needed for the last step of the QPF subroutine, but also because it is a major component of the modular exponentiation.

The Quantum Fourier Transform

As shown in Eq. 3.5, the QFT enacts the following unitary transformation on a quantum register containing L qubits,

2L 1 1 − 2πijk b L exp j = φ(b) L. (3.11) | i → √ L 2L | i | i 2 j=0 X The circuit required to perform this transformation is reasonably straightforward, consisting of single qubit Hadamard gates and controlled phase rotations. Fig. 3.2 shows the circuit to implement the QFT on an array of four qubits allowing for arbitrary couplings, while Fig. 3.3 shows the equivalent circuit for a LNN array.

b3 H φ(b0 ) π b2 2 H φ(b1 ) π π b1 4 2 H φ(b2 ) π π π b0 8 4 2 H φ(b3 )

Figure 3.2: Quantum circuit required to implement the QFT on an array of four qubits which allow for arbitrary couplings. Extending this circuit to a larger number of qubits is trivial, since for each additional qubit the controlled phase rotations simply decrease by powers of two. Also note that for the standard circuit the order of the qubits swap.

Note that the non-LNN circuit inverts the order of the qubits, while the LNN version does not. In Fig. 3.3 each dotted box represents a compound two qubit gate which we assume takes identical time. Compound gates can be obtained through the canonical decomposition [Mak02, KC01, ZVSW03], [Section 5.4] and therefore we will assume that any gate of this form can be implemented in ap-

29 CHAPTER 3. SHOR’S FACTORING ALGORITHM

b3 H H H H φ(b3 ) π π π b2 2 2 2 φ(b2 ) π π b1 4 4 φ(b1 ) π b0 8 φ(b0 )

Figure 3.3: Quantum circuit required to implement the QFT on a LNN array of four qubits. Unlike the standard circuit, the qubit order remains the same. Each dotted box represents compound gates which can be obtained via the canonical decomposition and are counted as a single gate [section 5.4].

proximately the same amount of time. The ﬁnal depth and gate estimates for this circuit assumes that these compound gates count as one operation. In both cases, L(L 1)/2 gates are required to implement the QFT on L qubits and assuming − that gates acting on independent pairs can be applied simultaneously, the total time required is 2L 3. −

Modular Exponentiation

Within the QPF subroutine, the controlled modular exponentiation is the most complicated section of the circuit. Using the methods developed by Beauregard, we break up modular exponentiation into the following steps,

1. (Controlled) Addition.

2. Controlled Modular Addition.

3. Controlled Modular Multiplication.

4. Controlled Modular Exponentiation.

(Controlled) Addition

The ﬁrst step is the addition of two binary registers. Utilizing the Beauregard method, addition is quite simple. If the computational register is ﬁrst placed into Fourier space, then addition is performed through a series of classically controlled phase rotations. Fig. 3.4 illustrates the quantum circuit for four qubit binary

30 3.2. LINEAR NEAREST NEIGHBOR QUANTUM CIRCUIT.

addition. In order to add the classical number, a, to a quantum register already containing b , a QFT is ﬁrst performed on the register taking b φ(b) . If | i | i→| i the register is in Fourier space, then addition is simply a series of phase rotations with each angle calculated from a [Fig. 3.4a]. In the case of controlled addition, each phase rotation becomes a controlled phase rotation + SWAP gate [Fig. 3.4b]. Finally the inverse QFT is applied, bringing everything back into the computational space, after which the control qubit has moved from one end of the register to the other.

Modular Addition

Now that a circuit for addition and controlled addition is available, we are able to construct a circuit to perform controlled modular addition, taking a register b (a + b)modN if the control qubits are in the state 1 , and b b if any | i→| i | i | i→| i of the control qubits are in the state 0 . Fig. 3.5 illustrates the required circuit. | i We have denoted the two control qubits as, k and x , which will be | ii | ji required when we utilize this circuit for the full modular exponentiation. The first block of gates is a two qubit decomposition of a Toffoli gate [Tof81, BBC+95], where we have introduced an ancilla, k = 0 , to avoid the need for doubly | x i controlled phase rotations later in the circuit. If the two control qubits are in the state 1 , then k is flipped to 1 , while if either of the control qubits are in | i | xi | i the state 0 , then k = 0 . An extra qubit is added to the register containing | i | xi | i b as a overflow qubit (initialized in the 0 state) and initially this L + 1 qubit | i | i register is placed into Fourier space b φ(b) . | i(L+1) →| i(L+1) The first gate performs a controlled addition of a and then an un-controlled subtraction of N, both of which are classical numbers (i.e. not stored within a quantum register), taking φ(b) φ(a + b N) if k = 1 and φ(b) | i→| − i | xi | i | i → φ(b N) if k = 0 . For arbitrary a < N and b < N, and assuming k = 1 , | − i | xi | i | xi | i there are two possibilities. First, take the case where (a + b) < N, this is the reason for the overflow qubit. If (a + b) < N, then (a + b N) < 0 and the − overflow qubit will flip. If (a + b) N, then (a + b N) 0 and the overflow ≥ − ≥ qubit remains in the state 0 . An inverse QFT is applied, bringing the register | i back into the computational space and a second initialized ancllia, MS = 0 , is | i | i then coupled, via a CNOT gate, to the overflow qubit. The register a + b N | − i is placed back into Fourier space and MS is used as a control for a subsequent | i addition of N. If(a + b N) < 0, the overflow qubit, and hence MS , is flipped − | i

31 CHAPTER 3. SHOR’S FACTORING ALGORITHM

−π φ φ a3a2a1a0 (b)3 0 (a+b)3 0 =RZ( 24 ) a.) −π φ φ a3a2a100 (b)2 1 (a+b)2 1 =RZ( 24 ) −π φ φ a3a20100 (b)1 2 (a+b)1 2 =RZ( 24 ) −π φ φ a3020100 (b)0 3 (a+b)0 3 =RZ( 24 )

Cont φ(a+b)3 b.) φ(b)3 0 φ(a+b)2

φ(b)2 1 φ(a+b)1

φ(b)1 2 φ(a+b)0

φ(b)0 3 Cont

Cont φ(a+b)3

φ(b)3 Cont + a φ(a+b)2

φ(b)2 φ(a+b)1

φ(b)1 φ(a+b)0

φ(b)0 Cont

Figure 3.4: Quantum circuit required to implement a) addition and b) controlled addition on a four qubit binary register. The initial state b is ﬁrst placed into | i Fourier space b φ(b) through the use of a QFT. Secondly, single qubit phase rotations are performed| i→| toi add a, with each angle classically calculated such that φ(b) φ(b + a) . If a controlled addition is desired, then each single qubit | i→| i phase rotation becomes a controlled phase rotation + SWAP. Finally, an inverse QFT is performed, taking φ(b + a) b + a , after which the control qubit has | i→| i been swapped from one end of the register to the other.

and N is re-added to the register, taking a + b N a + b . If a + b N 0, | − i→| i | − i ≥ then the overﬂow qubit does not ﬂip and hence the second controlled addition of N does not occur, leaving the register in the state a+b N . After this controlled | − i operation, the modular addition has actually been performed, i.e. b a + b | i→| i if a + b < N and b a + b N if a + b N. However the circuit is not | i→| − i ≥ yet complete since the ancilla, MS , has not been reset and measurement of this | i qubit (either deliberately or environmentally) could lead to possible knowledge

32 3.2. LINEAR NEAREST NEIGHBOR QUANTUM CIRCUIT.

of the input state, collapsing the computational wavefunction. In order to reset the ancilla, we first subtract a if k = 1 and then perform another inverse | xi | i QFT to return the register to the computational space. If (a + b) N, then ≥ the ancilla, MS was not originally flipped and the computational state contains | i (a + b N a) < 0, for b < N. Therefore the overflow qubit will flip to 1 . − − | i Performing a bit flip on the overflow qubit, a second CNOT between the overflow qubit and MS , and a second bit flip on the overflow qubit will leave MS = 0 , | i | i | i after which the register is placed back into Fourier space and a is re-added to the register. In the case where (a + b) < N, the ancilla, MS was flipped and | i the computational state (after the subtraction of a) will contain (a + b a) 0. − ≥ Here the overflow qubit will not flip and the bit flip-CNOT-bit flip between the overflow and MS will reset MS to 0 , after which a is re-added to the register. | i | i | i The final section of the circuit reverses the original Toffoli gate and then places all qubits back to their original starting positions, except for the qubits xj and xj+1 which are re-positioned in anticipation of the next time the circuit has to be applied for the modular exponentiation. Therefore the circuit in Fig. 3.5 reversibly performs the transformation,

b (a + b)modN , iﬀ ki = xj =1 | i→| i (3.12) b b , otherwise. | i→| i

The total gate count for the LNN circuit, for an L-bit register which is deﬁned via the size of b , is2L2 +8L +22 with a total depth of 8L + 16. This compares | i very favorably with the non-LNN version which requires 2L2 +6L + 14 gates arranged in a depth of 8L+13. The LNN modular addition circuit diﬀers slightly from the Beauregard circuit since the original doubly controlled addition gates utilized in [Bea03] are decomposed into two qubit operations and an extra qubit, k . Using the modular addition, we can now construct a circuit for modular | xi multiplication.

33 CHAPTER 3. SHOR’S FACTORING ALGORITHM 1 0 L j j+1 i (c) (c) (c) x x k φ φ φ MS kx

kx + a mod N 1 0 L = j+1 j i (b) (b) (b) R x x k φ φ φ MS kx

nd nd L L 0 1 j j+1 i (c) (c) (c) x x MS k φ φ φ kx R R

od od

a m a

+ + kx kx kx QFT

swap -1

QFT kx

If another modular addition follows then = null, else a MS

od od

a m a N

- -

kx kx /4 π MS + + MS π i -i e e MS /4 QFT /4 π π i -i e e swap = -1

kx QFT N /4 /4 π π

od N, - i -i

a m e e + kx /4 /4 π π i -i e e = L 0 L 1 j+1 j i (b) (b) (b) x x k φ φ φ MS kx

Figure 3.5: Quantum circuit required to implement controlled modular addition, for a L-bit quantum register. The control qubits ki and xj are required later, once this circuit element has been integrated into the full modular exponentiation. The diagonal circuit elements labeled swap simply represent a chain of two qubit SWAP gates, shuttling a qubit from one end of a block to the other. The circuit introduces three ancilla qubits, k stores the control information to avoid doubly | xi controlled phase rotations in the addition elements, MS is used as a control qubit | i if addition overﬂow occurs and the extra qubit contained in the φ(b) L+1 register is used to indicate if overﬂow has occurred during any addition.| i

34 3.2. LINEAR NEAREST NEIGHBOR QUANTUM CIRCUIT.

Modular Multiplication

The circuit for modular addition can now be combined to construct a circuit performing controlled modular multiplication for L-bit numbers. This is achieved through the following relation,

L 1 − j (ax)modN = (a2 xj)modN. (3.13) j=0 X

Where xj represent the jth-bit of x, used as a control in the modular addition circuit and a is again classical. Note that this circuit needs to be globally controlled by the qubit k , which again will be utilized later on. To perform controlled | ii modular multiplication we simply apply L controlled modular addition circuits. However, using a series of controlled modular additions is not quite appropriate for this task since it will result in the transformation,

x, 0 x, (ax)modN if ki =1, | i→| i (3.14) x, 0 x, 0 if k =0, | i→| i i when we want the transformation,

x, 0 (ax)modN, 0 if ki =1, | i→| i (3.15) x, 0 x, 0 if k =0, | i→| i i in order to keep the ﬁnal data within the same set of qubits as the initial data. To do this, we ﬁrst use a series of modular additions to take,

x, 0 x, (ax)modN , (3.16) | i→| i next the two registers are swapped if the control ki = 1,

x, (ax)modN (ax)modN, x . (3.17) | i→| i

1 1 Finally, a− is calculated classically. Note that a− is still an integer value but 1 1 has the property that (aa− ) = cN, where c Z, therefore (aa− )modN = 0. ∈ 1 a− is subtracted from the second register (this can be done by simply running the unitary inverse if the addition circuit) modularly multiplied by the ﬁrst if the

35 CHAPTER 3. SHOR’S FACTORING ALGORITHM

control ki = 1, giving,

1 (ax)modN, x (ax)modN, x (a− ax)modN = (ax)modN, 0 (3.18) | i→| − i | i

Therefore, if k = 0 none of the gates are applied and the state x, 0 x, 0 i | i→| i while if k = 1, x, 0 (ax)modN, 0 . Also, since the second register always i | i→| i starts and ﬁnishes in the state 0 , then although information is transfered to | i and from this register during the gate it is essentially scratch space and can be considered as internal ancilla qubits. The second step of this larger circuit requires a SWAP of the two registers if ki = 1. This is a more complicated procedure in a LNN system since the two registers need to be meshed together to allow equivalent qubits to be directly swapped. Fig. 3.6 illustrates the required circuit to mesh together two, four qubit registers, which can be trivially extended to more qubits. Fig. 3.7a gives the three qubit circuit required to perform a SWAP between two qubits controlled by a third and Fig. 3.7b shows the complete circuit required to perform a control SWAP between the two registers. After the SWAP, the control qubit has been shunted to the other side of the two registers.

a3 a3 a3 a3

a2 b3 a2 b3

a1 a2 a1 a2 mesh a0 b2 a0 b2

b3 a1 b3 a1

b2 b1 b2 b1

b1 a0 b1 a0

b0 b0 b0 b0

Figure 3.6: Quantum circuit required to mesh together two quantum registers. Once meshed together, performing two qubit SWAP gates between each pair and then applying the reverse of the mesh circuit will SWAP both registers. This circuit element is not needed in the non-LNN version since it is assumed that any pairs of qubits can be interacted directly.

Now we have all components to perform the required modular multiplication,

36 3.2. LINEAR NEAREST NEIGHBOR QUANTUM CIRCUIT.

c a’ c a’ a.) cswap a b’ a b’

b c b c

c a’3 b.) cswap

a 3 a’2

a 2 a’1 cswap mesh a 1 a’0 mesh a 0 b’3 cswap

b3 b’2

b2 b’1 cswap

b1 b’0

b0 c

Figure 3.7: a) Circuit element required to perform a controlled-SWAP between two qubits ( a , b ), conditional on a third ( c ). The control-(+) and control-( ) gates are identical| i | i to the gates shown in Fig.| i 3.5. b) The full circuit to perform− a controlled SWAP of two LNN quantum registers. First the qubits are meshed, three qubit control-SWAP gates are applied in a ladder structure and ﬁnally the two registers are un-meshed. After the operation the control qubit has swapped position.

Fig. 3.8 illustrates the full circuit, with L modular additions then a controlled register SWAP and ﬁnally L modular subtractions. The total gate count for the multiplier is 4L3 +20L2 +58L 2 in a total depth of 16L2 +40L 7, compared − − to the non-LNN gate count of 4L3 +13L2 +35L +4 in a depth of 16L2 +33L 6. −

37 CHAPTER 3. SHOR’S FACTORING ALGORITHM 4 3 2 1 0 3 2 1 0 i (0) (0) (0) (0) (0) k ax ax ax ax φ φ φ φ φ MS kx

ki a mod N 4 3 2 1 0 3 2 1 0 i (0) (0) (0) (0) (0) x x x x k φ φ φ φ φ MS kx 4 3 2 1 0 3 2 1 0 i (0) (0) (0) (0) (0) ax ax ax ax k φ φ φ φ φ MS kx

3 -1 x3ki - 2 ()a mod N -1 2 () x2ki - 2 a mod N

1 -1 x1ki - 2 ()a mod N

0 -1 x0ki - 2 ()a mod N i 3 2 1 0 kx x x x k x MS QFT

mesh

cswap cswap

cswap cswap MS i kx k 3 mesh x 2 x 1 x 0 -1 x QFT

3 x3ki + 2 a mod N

2 x2ki + 2 a mod N

1 x1ki + 2 a mod N

0 x0ki + 2 a mod N 4 3 2 1 0 3 2 1 0 i (0) (0) (0) (0) (0) x x x x k φ φ φ φ φ MS kx

Figure 3.8: Circuit to implement a controlled modular multiplication, requiring L modular additions, a controlled register swap and ﬁnally L modular subtractions.

38 3.2. LINEAR NEAREST NEIGHBOR QUANTUM CIRCUIT.

Modular Exponentiation

The ﬁnal step is to construct controlled modular exponentiation out of a series of controlled modular multiplications. We make use of the identity,

2L 1 − i f(k)= (a2 ki modN) (3.19) i=0 Y

Here, each subsequent multiplication is controlled by ki, which as you recall from section 3.1 comprises the k-register. Fig. 3.9 illustrates the full circuit required to implement Shor’s algorithm. This circuit is substantially different from the general idea of Shor’s algorithm as presented in the introduction. In this case the entire k-register is no longer 2L qubits, but instead can be replaced by a single Master Control qubit. This is known as the “one qubit trick”, which was introduced by Griffiths and Niu [GN96] and incorporated into the Beauregard circuit [Bea03]. Instead of performing a series of Hadamard gates on 2L qubits, coupling them to the f-register and finally performing a 2L-qubit QFT, we can instead decompose these steps into a sequence of single qubit Hadamard gates, measurements and classically controlled phase rotations. This is due to the fact that the final Fourier transform is immediately proceeded by measurement of the entire k-register and any controlled unitary proceeded by measurement can be rearranged such that measurement occurs first and the controlled unitary becomes a classically controlled unitary. This is extremely beneficial since it replaces a 2L-qubit register with a single qubit which is measured a total of 2L times in order to build up the classical result in a bit-wise fashion. Utilizing the one qubit trick, we can perform a final gate, qubit and depth count for the entire Quantum Period Finding subroutine. For L-bit factoring, L + 1 qubits are needed (the modular addition circuit to store φ(b) ), an additional L qubits are used to evaluate (ax)modN and three | i additional ancilla ( k , MS and the Master Control qubit) are needed. This | xi | i gives a total of 2L + 4 qubits to factor an L-bit number. The controlled multiplier requires 4L3 +20L2 +58L 2 gates in a total depth − of 16L + 40L 7. Performing the exponentiation requires 2L multiplications, − interlaced with five single qubit gates on the master control (required for the one qubit trick), plus a single L + 1 qubit QFT, which places the lower half of the

39 CHAPTER 3. SHOR’S FACTORING ALGORITHM

computer into Fourier space in anticipation of addition circuits. Hence the total count for the entire circuit is 2L + 4 qubits, using 8L4 +40L3 +116 1 L2 +4 1 L 2 2 2 − gates arranged into a depth of 32L3+80L2 4L 2, this compared to the non-LNN − − circuit of Beauregard, which requires 2L+4 qubits, 8L4 +26L3 +70 1 L2 +8 1 L 1 2 2 − gates arranged into a depth of 32L3 + 66L2 2L 1. − −

3.2.2 Conclusion

This section has introduced a quantum circuit that is appropriate to implement the quantum component of Shor’s period finding subroutine on a Linear Nearest Neighbor array of qubits. This demonstrates that provided care is taken, the restriction of LNN qubit arrays does not make circuit efficiency significantly lower than equivalent non-LNN circuits. In fact, to first order, the number of gates and the circuit depth is identical to the non-LNN version this circuit is based on. It should be clear that although both the LNN and non-LNN circuits for the Quantum Period Finding subroutine are efficient in the technical sense, they are still extremely complicated circuits to implement. The issues relating to Quantum Error Correction, Fault-tolerant gate design, transport in qubit systems and the effect of systematic errors when implementing quantum gate operations all need to be considered when designing these circuits. The next section will focus on one of these issues, presenting numerical simulations which were performed on both the LNN circuit introduced here and the non-LNN circuit of Beauregard, showing the extreme sensitivity of both circuits under coherent quantum errors.

40 3.2. LINEAR NEAREST NEIGHBOR QUANTUM CIRCUIT. 4 3 2 1 0 3 2 1 0 8 (0) (0) (0) (0) (0) x x x x k φ φ φ φ φ MS kx 8 M H 1 Z 27 k7 a mod N H R 7 M H 6 Z 6 2 k6 a mod N H R 3 M H 2 Z 2 2 1 k2 a mod N 3 /2 1 M H 2 M 3 R M 2 π i 0 M e H 1 Z 1 1 1 2 k1 a mod N 0 0 H R 1 0 0 HM 0 2 Reset: Phase: k0 a mod N ith measurement i 3 R Z M H QFT 0 0 0 0 1 0 0 0 0 0 0 0

Figure 3.9: Final circuit to implement the Quantum Period Finding subroutine. The entire 2L, k-register is replaced by a single master control qubit, which is used to control each multiplication gate in the modular exponentiation, interleaved with classically controlled phase gates and measurement. This decomposition removes the need for a large k-register and for controlled phase rotations to implement the ﬁnal QFT. Without these classically controlled phase rotations and measurement, this circuit would represent simple modular exponentiation.

41 CHAPTER 3. SHOR’S FACTORING ALGORITHM

3.3 Stability of Shor’s Algorithm under errors

Since Shor’s discovery, the construction of a large scale quantum computer (QC) has been an area of intense research. Currently there are many different proposals for constructing such a device [NC00, ARD04], but despite significant progress, the issue of decoherence and imperfect gate design leads to the question of whether such a large and complex algorithm can be experimentally realized beyond trivial problem sizes. The development of quantum error correction (QEC) [Ste96a, Sho95, CS96] and Fault-tolerant quantum computation [DS96, Got98a, Sho96] has shown, theoretically, how large scale algorithms can be implemented on imperfect devices. However, without a working QC, detailed classical simulations of QEC and quantum algorithms constitute the only method for reliable information regarding the behavior of such schemes and the best way they can be implemented on physical systems. The issue of appropriate use of QEC and the construction of arbitrary Fault-tolerant gates [Fow04] still requires detailed knowledge of the behavior of the underlying algorithm in order to tailor these schemes appropriately. For large scale quantum algorithms, the general method of analysis is to assume that all components within the algorithm must have a precision of 1/n , ≈ p where np = KQ represents the number of locations where an error can occur during an algorithm utilising Q qubits and K elementary steps (depth of the circuit). This estimate implies that a single error anywhere during calculation will result in failure. For small quantum circuits, this approximation is not an obstacle in component design. However, for more complex circuits, where qubits may be coupled in highly non-trivial ways, it is not obvious that such a naive estimate cannot be improved. In fact, our results show that they are not. In our analysis we examine the Quantum Period Finding (QPF) subroutine, which lies at the heart of Shor’s algorithm, in the presence of coherent, discrete errors. Quantum circuits to factor large integers, for example a 128-bit number, require n of the order 107 1010 depending on the specific circuit used [see p − 7 10 Table 3.1]. Engineering quantum gates with failure rates of 10− 10− is cur- − rently far from being experimentally realized in any of the numerous architectures currently proposed. Our simulations show that the 1/np precision requirement is not strictly required. We find evidence for a required precision of P (L)/np, where P (L) is a monotonically increasing function of L, the binary length of the

42 3.3. STABILITY OF SHOR’S ALGORITHM UNDER ERRORS

composite number, which is at least linear. This slower scaling increases the error rate at which quantum processing (as opposed to classical randomness) can be observed. Several authors have previously examined the effects of errors on Shor’s algorithm [FH04, WLHN05, Bra02]. These simulations are often limited to specific sections of the entire circuit, or to other sources of error such as phase drifts on idle qubits, imperfect gate operations or aspects relating to quantum chaos. Chuang et al [CLSZ95] was one of the first to look at the error stability of Shor’s algorithm, analytically, under the effects of environmental coupling. Miquel et al [MPP96] examined the stability of Shor’s algorithm using an identical error model to that used in this investigation. However, the stability of the algorithm was only investigated for a single problem size and did not investigate how the stability changes as the problem size increases. Several architectures, most notably solid state models, are restricted to a single line of qubits with nearest neighbour interactions only. The issue of appropriate quantum circuits for LNN architectures was introduced in the previous section, however the comparable stability of the LNN circuit also needs to be investigated to ensure that there is no significant difference compared to circuits designed for architectures that can interact arbitrary pairs of qubits (non-LNN).

We ﬁnd that if LNN circuits can be designed with comparable values of np, the stability is similar. In this section we examine speciﬁc circuits for both LNN and non-LNN architectures in the presence of a discrete error model, in order to determine:

The degree to which the ﬁnal required state of the computer is aﬀected by • small changes in the computational trajectory caused by these errors.

The impact of a LNN circuit as described in section 3.2 on the reliability of • the QPF subroutine.

If the 1/n bound for component precision remains absolute for various • p problem sizes.

The introduction of this chapter has already provided an overview to the major steps in Shor’s algorithm, including the QPF subroutine, subsequent deﬁ- nitions remain consistent with the introductory section.

43 CHAPTER 3. SHOR’S FACTORING ALGORITHM

3.3.1 Error models and analysis

In our simulations, errors were simulated using the discrete model in which a single qubit φ = α 0 + β 1 can experience a bit flip X φ = α 1 + β 0 , a | i | i | i | i | i | i phase flip Z φ = α 0 β 1 , or both at the same time XZ φ = α 1 β 0 . | i | i − | i | i | i − | i These discrete error operators are then applied to each qubit, after each operational time step with probability p/3 (i.e each error has identical probability of occurrence, with the total probability of error given by p). The operational time for all two qubit gates is assumed to be identical and all single qubit gates are combined with neighbouring two qubit gates via the canonical decomposition [Mak02, KC01, ZVSW03], [Section 5.4]. This simplifies the analysis since each compound gate counts as a single time step and errors are applied after these compound operations rather than after the individual component gates. The discrete error model represents the most common error model used within QEC analysis and oversimplifies error effects within a quantum computer in several ways.

The error model used is uncorrelated and random. Some architectures • may be more vulnerable to dephasing errors (Z operations), symmetric emission-absorption errors (X and XZ operations) or loss of qubits (this is particularly relevant in linear optical systems [KLM01]).

This model does not examine the eﬀect of systematic errors due to in- • accurate gate design. Inaccurate two qubit gates will generally produce correlated errors over pairs of interacting qubits.

This speciﬁc error model treats memory errors and gate errors identically, • which may not be realistic given a speciﬁc physical architecture.

Although this model represents a simplification of the many diverse effects that can cause errors within quantum computers, our interest in LNN architectures and their close adherence to this model make it appropriate. Furthermore, general continuous errors are equivalent to this discrete model when considering the noise digitization affect of QEC syndrome extraction. Syndrome extraction acts to project encoded qubits onto a state that is perturbed from an error free state by discrete X and/or Z gates, digitising continuous errors to a discrete set. Please refer to Chapter 4 for a review of Quantum Error Correction.

44 3.3. STABILITY OF SHOR’S ALGORITHM UNDER ERRORS

Using this error model, we can analytically describe the behaviour of the QPF subroutine in the presence of severe errors. Referring to the original quantum circuit used in these simulations [Section 3.2], [Bea03], j is obtained bit-by-bit via a series of measurements on a master control qubit. This master qubit simulates the entire 2L qubit k-register described in section 3.1. The Quantum Fourier Transform (QFT) on this single qubit required by Eq. 3.5 is performed through a series of Hadamard gates and classically controlled single qubit rotations. In a general analysis we can therefore model the entire computer as two registers, a single master qubit and the rest of the computer. Consider the state of the computer at a point just before the application of a controlled modular multiplication gate. At this point the master control qubit is in an equal superposition of 0 and 1 and the rest of the computer is some | i | i unknown superposition,

22L 1 1 − φ = ( 0 + 1 ) α m . (3.20) | i √ | imaster | imaster m| icomputer 2 m=0 X Now apply the modular multiplication gate, which will return a new superposition state for the m register (when the master qubit is in the 1 state). This | icomputer | i new superposition is denoted through the coeﬃcients, β , { m}

22L 1 22L 1 1 − 1 − φ = 0 α m + 1 β m . (3.21) | i √ | i m| i √ | i m| i 2 m=0 2 m=0 X X Prior to measurement, a classically controlled rotation (θ) and a second Hadamard gate is applied to the master control qubit. The value of θ is dependent on the result of all previous measurements on this qubit. Hence the state just before measurement is,

22L 1 22L 1 1 − 1 − φ = 0 (α + eiθβ ) m + 1 (α eiθβ ) m . (3.22) | i 2| i m m | i 2| i m − m | i m=0 m=0 X X With the probability of measuring a 1 or 0 is given by,

22L 1 − 1 1 1 1 iθ iθ Pr( )= (e α∗ β + e− α β∗ ), (3.23) 2 ∓ 2 2 ± 4 m m m m m=0 X

45 CHAPTER 3. SHOR’S FACTORING ALGORITHM

using the fact that total probability must be conserved, and hence,

22L 1 22L 1 − − α 2 = β 2 =1. (3.24) | m| | m| m=0 m=0 X X Errors cause the summation in Eq. 3.23 to asymptote to 0 resulting in an equal probability Pr = (0.5)2L of each j being observed. The period of the function, r, dictates the number of non-zero coefficients α ,β and the specific value of j simply changes the sequence of 1’s and 0’s { m m} measured at each step. Since errors act to randomly perturb these sets of co- efficients, considering different values of r and/or j will have no effect on the stability of the QPF subroutine. Hence to simplify simulations we will fix both r and j and simply examine the stability of the QPF subroutine as a function of problem size (L). The simulated QPF circuit is extremely complex and hence requires a large amount of classical simulation time. Ideally, simulations would proceed by applying a predetermined number of discrete error gates to every possible location within the circuit and averaging the probability of success, s, over all possible locations (where s is defined via Eq. 3.10 in section 3.1). For example, Fig. 3.10 shows the effect of a single X error on the QPF success probability, s, for the first modular multiplication gate in the LNN, L = 5, circuit.

Figure 3.10: Map showing how the location of a single bit flip error plays a major role in the final output success of the LNN circuit. This image is for L = 5 (14 qubits), and shows the first modular multiplication section of the circuit. Each horizontal block represents one of the 14 qubits while each vertical slice represents a single time step. Darker areas represent successively lower values for s.

From Fig. 3.10 we can see that the spatio-temporal location of an error plays a major role in the ﬁnal value of s calculated, with various sections invariant to the bit ﬂip error (colored in white). In order to analyse the behaviour of the

46 3.3. STABILITY OF SHOR’S ALGORITHM UNDER ERRORS

QPF subroutine we take an ensemble average over all possible error locations. For example, in Fig. 3.10, the average value of s over all possible locations for a single error iss ¯ =0.34. Most circuits are far too large to map out this topology eﬃciently: we are limited by computational resources to 50 statistical runs to obtain an approximate average value of s for these circuits. However, the results show that there is still suﬃcient data to observe trends in the results.

3.3.2 Stability under a ﬁxed number of errors

The classical simulation algorithm employed used a state vector representation. Matrix operations were performed to simulate both quantum gates and error operations. In Figs. 3.11 and 3.12 we plot the success of the QPF subroutine as a function of the total number of discrete errors. We plot the results for 2L +4= 14, 16, 18, 20, representing factorization of composite numbers from N = 27 to N = 247. Simulations examined functions that each had a period r = 6. Table 3.2 show the functions f(k) used for each value of L. These simulations aim to

2L +4 f(k)= akmodN, with r = 6. 14 8kmod27 16 31kmod63 18 10kmod77 20 27kmod247

Table 3.2: Functions used for various values of L. Note that for 2L +4=14, 16, the values of N used are not products of two primes. However, since we are only investigating the reliability of the QPF subroutine, this is not relevant to our analysis. investigate the behaviour of the QPF subroutine for high component precision, close to the 1/np bound. Simulations were performed in a half-stochastic, half- deterministic manner: The type and spatio-temporal location of discrete errors occur at random, however we specify exactly how many errors can occur within a given run of the subroutine. Simulations examine the probability of obtaining the speciﬁc useful value j = 22L/6 . Figs. 3.11 and 3.12 show the results for the non-LNN [Bea03] and LNN ⌊ ⌋ [Section 3.2] circuits respectively. For clarity, we have suppressed the statistical errors on these log plots, the complete data sets are given in Figs. 3.13, 3.14, 3.15 and 3.16. The deﬁnition of success for the QPF subroutine, given in section 3.1.3

47 CHAPTER 3. SHOR’S FACTORING ALGORITHM

L = 5

L = 6

L = 7

L = 8

Figure 3.11: Plot showing the relative probability of measuring j = 22L/6 as ⌊ ⌋ a function of the specific number of errors for the non-LNN circuit. The curves represent L =5 to L = 8. The horizontal lines show the point of random output for each successive value of L. takes into account that many different values of j may be used to determine r. However, for the sake of this analysis, we are only concerned if the QPF subroutine returns with high probability a value of j that is theoretically predicted. Therefore in each plot we normalise such that an error free calculation returns j = 22L/6 ⌊ ⌋ with probability one and Shor’s algorithm succeeds with a single call to the QPF subroutine. As the number of errors increase, the probability of measuring j = 22L/6 decreases until it reaches the point of random output, at this stage the ⌊ ⌋ QPF subroutine performs no better than randomly choosing a value of j in the range j =0 j =22L 1. → − Figs. 3.11 and 3.12 clearly shows how the quantum speed up of the QPF subroutine, and hence Shor’s algorithm, diminishes to a point where it is no different to randomly choosing a value from the j register, as the number of

48 3.3. STABILITY OF SHOR’S ALGORITHM UNDER ERRORS

L = 5

L = 6

L = 7

L = 8

Figure 3.12: Plot showing the relative probability of measuring j = 22L/6 as a function of the specific number of errors for the LNN circuit. The curves⌊ represent⌋ L =5 to L = 8. The horizontal lines show the point of random output for each successive value of L. errors increases (represented by the horizontal lines). At this point, any quantum processing can no longer be identified from the probability spectrum for j. Accurate curve fits are extremely difficult to obtain from the limited amount of data available due to long computation times. Each point represents 50 separate simulations where the total number of errors occur randomly within the QPF circuits. In order to get sufficient data to extract meaningful fits for each of these curves, one would expect the number of statistical runs should be the same order as the number of possible error locations (or error combinations). For example, in the L = 5 circuit, for one error, the number of possible error locations and types is 18000. Hence, it is quite surprising that even 50 statistical runs provides ≈ enough data to obtain a qualitative picture of how the QPF subroutine behaves

49 CHAPTER 3. SHOR’S FACTORING ALGORITHM

Figure 3.13: Plot showing the stability of the non-LNN (right plot) and LNN (left plot) QPF circuit for L = 5, included on this plot are errors associated with the stochastic nature of simulations, representing the variance on the 50 run data set used in simulations.

Figure 3.14: Plot showing the stability of the non-LNN (right plot) and LNN (left plot) QPF circuit for L = 6, included on this plot are errors associated with the stochastic nature of simulations, representing the variance on the 50 run data set used in simulations. for various values of L. To reduce the statistical errors and obtain accurate curve ﬁts for these plots, further simulations are required, preferably using the density matrix formalism. However, from this data we can still draw qualitative conclusions about the average robustness of the QPF routine as a function of increasing number of errors. To verify that a quantum computer implementing the QPF routine is processing in the quantum regime, it would be suﬃcient to observe peaks within the probability spectrum for j. The sharper the peaks, the fewer repetitions of the

50 3.3. STABILITY OF SHOR’S ALGORITHM UNDER ERRORS

Figure 3.15: Plot showing the stability of the non-LNN (right plot) and LNN (left plot) QPF circuit for L = 7, included on this plot are errors associated with the stochastic nature of simulations, representing the variance on the 50 run data set used in simulations.

Figure 3.16: Plot showing the stability of the non-LNN (right plot) and LNN (left plot) QPF circuit for L = 8, included on this plot are errors associated with the stochastic nature of simulations, representing the variance on the 50 run data set used in simulations.

QPF subroutine required and the more practical the computation. For very low visibility peaks, the number of repetitions of QPF scales exponentially with L, nullifying the advantages of the quantum algorithm over its classical version.

These simulations show that a maximum error rate of 1/np for all problem sizes is not required to obtain better performance than classically searching through the j values. By inspection of Figs 3.11 and 3.12, an estimate can be made regarding the number of errors (as a function of L) before quantum processing in the QPF cannot be identiﬁed [Fig. 3.17]. Fig. 3.17 represents only a preliminary estimate from Figs. 3.11 and 3.12, additional data is required to

51 CHAPTER 3. SHOR’S FACTORING ALGORITHM

perform an accurate curve fit. The purpose of Fig. 3.17 is simply to demonstrate that when attempting to observe quantum processing experimentally, more than one error can be tolerated, and the tolerance increases with L. When attempting to realise the full potential of the QPF routine, the probability of useful output should be kept as high as possible. It can be seen from Figs. 3.11 and 3.12 that even a single error significantly reduces this probability. Therefore, our simulations support the view that for large scale implementation of the QPF routine, ideally no errors should occur in the circuit. This would, of course, be achieved though quantum error correction, with work by Steane [Ste96b, Ste02, Ste03] already examining effective logical qubit error rates, given a specific physical error rate, for various error correcting codes.

Figure 3.17: Estimate on the maximum number of errors possible for each value of L before the LNN circuit becomes equivalent to a classical random search. non-LNN circuit, LNN circuit. × ≡ ≡

The error behaviour for the LNN and non-LNN circuits are largely indistin- guishable from each other. However, there is a slight diﬀerence in the error sensi-

52 3.3. STABILITY OF SHOR’S ALGORITHM UNDER ERRORS

tivity of the two circuits. We attribute this to a minor increase in the LNN circuit depth. As expected, the overall area (np) of the circuit is the dominating factor in its sensitivity. The mesh circuit required in the LNN design [Section 3.2] is the major diﬀerence between the LNN and non-LNN circuits. This section of the LNN circuit acts to slightly increase the overall depth, from 32L3 +66L2 2L 1 − − for the non-LNN circuit to 32L3 + 80L2 4L 2 for the LNN design [Tab. 3.3]. − − Hence the sensitivity of the LNN circuit increases slightly compared with the non-LNN circuit.

L LNN Circuit non-LNN Circuit 5 5978 5639 6 9766 9275 7 14866 14195 8 21470 20591

Table 3.3: Total circuit depths (K) for the LNN and non-LNN circuits, for L =5 to L =8

The scaling in the QPF subroutine shown by our simulations can be utilized when testing such a complex quantum circuit for evidence of quantum processing. As mentioned previously, peaks within the probability spectrum of j are indicative of quantum processing and our simulations have shown that such peaks will be present even when component precision is not bounded by 1/np. Although we are unable fit a specific curve to the data shown in Figs. 3.11 and 3.12, we can consider several different cases for the scaling of the tolerable number of errors. Since n O(L4) for the quantum circuit used in the simu- p ≈ lations, and restricting our analysis to integer powers of L, we can safely bound the tolerable error scaling as at most O(L3). If this scaling were O(L4) then the required component precision would scale as O(L4)/n constant, implying that p ≈ as the quantum circuit increases size, the required component precision remains constant. This argument also is valid for scaling faster than O(L4). If such a fast scaling occurred, increasing the circuit size would lead to a decrease in required component precision. Fig. 3.18 examines the required physical component precision required to observe quantum processing in the QPF, for potential scalings of 1/n ( ), L/n ( ), L2/n ( ) and L3/n (). As a probability spectrum p × p ⋄ p △ p of j is needed in all cases, the number of total QPF routine iterations needed is approximately of order 22L. Therefore, only L = 5 to L = 10 is shown, since for higher L the total number of circuit iterations become prohibitively large.

53 CHAPTER 3. SHOR’S FACTORING ALGORITHM

Figure 3.18: Required component precision in order to observe quantum processing for small values of L in the QPF subroutine. Each curve represents a separate type of additional scaling indicated by our simulations, 1/np, L/np, L2/n and L3/n . × ≡ ⋄ ≡ △ ≡ p ≡ p

Table 3.4 examines the minimum physical component precision required to observe quantum processing for L = 5 to L = 10 for a component precision of

1/np and for linear, quadratic and cubic scaling. Depending on the exact nature of this scaling, quantum processing can be observed for L = 5 to L = 10 with a physical component precision between one and three orders of magnitude lower than the 1/np bound. Even though the computational resources required to eﬀectively identify the functional form of this scaling are unavailable, we can con- clude that some non-constant scaling of component precision with L is present, leading to a reduction in the physical component precision required to observe quantum processing for small values of L. This result would reduce the pressure on experimental fabrication, in the short term, by potentially removing the need to implement complicated error correction protocols to simply observe quantum processing in small instances of the QPF subroutine.

54 3.3. STABILITY OF SHOR’S ALGORITHM UNDER ERRORS

required component scaling component precision required 6 1/np 1.3 10− × 5 L/np 1.3 10− 2 × 4 L /np 1.3 10− 3 × 3 L /n 1.3 10− p ×

Table 3.4: Minimum component precision required to apply the QPF subroutine to at least L = 10 for various scalings in component precision.

3.3.3 Conclusion

We have simulated the quantum part of Shor’s algorithm, the quantum period finding subroutine (QPF). Our simulations have shown that the structure of this quantum circuit leads to a robustness above the naive 1/np approximation for component precision, if one only wishes to demonstrate the existence of quantum processing. Depending on the functional form of this scaling, quantum processing can still be observed for error rates significantly higher that the 1/np bound, for small instances of the QPF subroutine. This additional robustness is advantageous in the short term since introducing complicated QEC protocols to simply observe quantum processing will be difficult. However, the results of our simulations suggest that for large practical problem sizes, extensive use of error correction will be required to ensure error free calculations. The restriction to a linear nearest neighbour design does not significantly alter the sensitivity of the subroutine, provided that appropriate LNN circuits can be designed roughly equivalent in depth and qubit numbers as non-LNN circuits.

55 CHAPTER 3. SHOR’S FACTORING ALGORITHM

56 Chapter 4

Quantum Error Correction and Fault-tolerance.

Contents 4.1 Introduction ...... 59

4.2 Quantum Error Correction ...... 61

4.2.1 The3-qubitcode ...... 62

4.2.2 Stabilizer Formalism ...... 64

4.2.3 QEC with stabilizer codes ...... 66 4.3 Digitization of Quantum Errors ...... 72

4.3.1 Systematic gate errors ...... 72

4.3.2 Environmental decoherence ...... 73 4.4 Fault-tolerant Quantum Error Correction and the thresholdtheorem...... 76

4.4.1 Fault-tolerance ...... 76

4.4.2 ThresholdTheorem ...... 77 4.5 Fault-tolerant operations on encoded data ...... 80

4.5.1 Single Qubit Operations ...... 80

4.5.2 Two-qubit gate...... 82 4.6 Fault-tolerant circuit design for logical state preparation ...... 86

4.7 Simulations of Logical State Preparation ...... 90

57 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

4.7.1 LNN circuit for logical encoding ...... 90 4.7.2 Results ...... 93 4.7.3 Conclusions...... 94

58 4.1. INTRODUCTION

4.1 Introduction

While the advent of Shor’s algorithm certainly spawned great interest in quantum information processing and demonstrated that the utilization of a quantum computer could lead to algorithms far more efficient than those used in classical computing, there was a great deal of debate surrounding the practicality of build- ing a large scale, controllable, quantum system. It was well known even before the introduction of quantum information that coherent quantum states were extremely fragile and many believed that to maintain large, multi-qubit, coherent quantum states for a long enough time to complete any quantum algorithm was unrealistic. However, shortly after the introduction of Shor’s algorithm Quantum Error Correction (QEC) was formulated. Starting in 1995, several papers proposed codes which were appropriate to perform error correction on quantum data [Sho95, Ste96a, CS96, LMPZ96]. This was the last theoretical aspect needed to convince the general community that quantum computation was indeed a possibility. Since this initial introduction there has been a extensive amount of work developing new codes [Ste96b, CG97, CRSS98, KLP05, Bac06], formulating efficient correction circuits [Ste97, Ste02, DA07] and developing the ideas of Fault-tolerant quantum computation [Sho96, DS96, Got98a], which leads directly to the threshold theorem for concatenated correction [KLZ96, ABO97]. This Chapter deals exclusively with the concepts of Quantum Error Correc- tion and Fault-tolerant quantum computation. As the rest of this thesis utilizes concepts from Fault-tolerant computation, the stabilizer formalism and operator (parity) measurements, this chapter represents a large review of the main concepts of QEC, while section 4.7 presents original work, examining the stability of Fault-tolerant and non-Fault-tolerant state preparation using similar numerical techniques to the simulations presented in Chapter 3. To cater for a large audi- ence, I have attempted to introduce these concepts through examples, specifically using the [[7, 1, 3]] Steane code [Ste96a]. For a more detailed discussion of error correction, Fault-tolerance, decoherence and the stabilizer formalism I encourage the reader to refer to [NC00, Got97, Zur01] and references therein. Hopefully this discussion will represent an effective introduction to the most important aspects of Fault-tolerant quantum error correction. Section 4.2 introduces quantum error correction through the traditional ex-

59 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

ample of the 3-qubit code, after which I continue the review of QEC via the stabilizer formalism [Got97], demonstrating how QEC circuits are synthesized once the structure of the code is know. Section 4.3 then brieﬂy relates the abstract analysis of QEC, where errors are assumed to be discrete and probabilistic, to some of the physical mechanisms which cause errors. Sections 4.4 and 4.5 introduces the concept of Fault-tolerant error correction, the threshold theorem and how logical gate operations can be applied directly to quantum data. Finally sections 4.6 and 4.7 present a basic Fault-tolerant circuit for logical state preparation using the [[7, 1, 3]] Steane code and introduces numerical simulations that were performed, comparing the stability of Fault-tolerant and non-Fault-tolerant state preparation.

Original Material

The majority of this chapter represents a review of the fundamental con- • cepts of Fault-tolerant quantum error correction.

The numerical results presented in section 4.7 represents original work which • ﬁrst appeared in Ref. [DFH05b], with subsequent simulations [Fig. 4.17] performed in collaboration with Ashley Stephens which ﬁrst appeared in Ref. [SDF+06].

60 4.2. QUANTUM ERROR CORRECTION

4.2 Quantum Error Correction

The concept of Quantum Error Correction (QEC) is fundamental to the large scale viability of quantum information processing. Although the ﬁeld is largely based on classical coding theory, there are several issues that need to be considered when transferring classical error correction to the quantum regime. First, coding based on data-copying, which is extensively used in classical error correction cannot be used, due to the no-cloning theorem of quantum mechanics [WZ82]. This result implies that there exists no transformation resulting in the following mapping,

U φ ψ = φ φ (4.1) | i⊗| i | i⊗| i i.e. it is impossible to perfectly copy an unknown quantum state. This means that quantum data cannot be protected from errors by simply making multiple copies. Secondly, direct measurement cannot be used to effectively protect against errors, since this will act to destroy any quantum superposition that is being used for computation. Error correction protocols must therefore be employed which can detect and correct errors without determining any information regarding the qubit state. Finally, unlike classical information, qubits can experience both bit errors, 0 1 and/or phase errors 1 1 , hence any error correcting | i↔| i | i ↔ −| i procedure needs to be able to simultaneously correct for both. QEC utilizes the idea of redundant encoding, where quantum data is protected by extending the size of the Hilbert space for a single, logically encoded qubit. This way, errors only perturb codeword states by small amounts which can then be detected and corrected, without directly measuring the state of any qubit. First I will detail the traditional example of the 3-qubit bit flip code which is always a good introduction to the principals of QEC. Afterwards I will continue with the concepts of QEC from the standpoint of stabilizer codes [Got97], rather than the state based approach used when QEC was first developed. The stabilizer formalism non only gives an extremely elegant description of QEC and circuit design, but I also utilize the formalism extensively in Chapter 6 when discussing large scale quantum architecture design.

61 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

4.2.1 The 3-qubit code

The 3-qubit bit-flip code is traditionally used as a basic introduction to the concept of Quantum Error Correction. However, it should be emphasized that the 3-qubit code does not represent a full quantum code. This is due to the fact that the code cannot simultaneously correct for both bit and phase flips. This code is a standard repetition code which was extended by Shor [Sho95] to the full 9-qubit quantum code which was the first demonstration that QEC was possible. The 3-qubit code encodes a single logical qubit into three physical qubits with the property that it can correct for a single σ X error. The two logical x ≡ basis states 0 and 1 are defined as, | iL | iL

0 = 000 , 1 = 111 , (4.2) | iL | i | iL | i such that an arbitrary single qubit state ψ = α 0 + β 1 is mapped to, | i | i | i

α 0 + β 1 α 0 + β 1 = α 000 + β 111 = ψ . (4.3) | i | i→ | iL | iL | i | i | iL

Fig. 4.1 illustrates the quantum circuit required to encode a single logical qubit via the initialization of two ancilla qubits and two CNOT gates. The reason why

Figure 4.1: Quantum Circuit to prepare the 0 state for the 3-qubit code where | iL an arbitrary single qubit state, ψ is coupled to two freshly initialized ancilla qubits via CNOT gates to prepare| iψ . | iL this code is able to correct for a single bit flip error is the binary distance between the two codeword states. Notice that three individual bit flips are required to take 0 1 , hence if we assume ψ = 0 , a single bit flip on any qubit | iL ↔ | iL | i | iL leaves the final state closer to 0 than 1 . The distance between two codeword | iL | iL states, d, defines the number of errors that can be corrected, t, as, t =(d 1)/2. − In this case, d = 3, hence t = 1.

62 4.2. QUANTUM ERROR CORRECTION

How are we able to correct errors using this code without directly measuring or obtaining information about the logical state? Two additional ancilla qubits are introduced, which are used to extract syndrome information (information regarding possible errors) from the data block without discriminating the exact state of any qubit, Fig. 4.2 illustrates. For the sake of simplicity I assume that all Correct Error

Figure 4.2: Circuit required to encode and correct for a single X-error. I assume that after encoding a single bit-flip occurs on one of the three qubits (or no error occurs). Two initialized ancilla are then coupled to the data block which only checks the parity between qubits. These ancilla are then measured, with the measurement result indicating where (or if) an error has occurred, without directly measuring any of the data qubits. Using this syndrome information, the error can be corrected with a classically controlled X gate. gate operations are perfect and the only place where the qubits are susceptible to error is the region between encoding and correction. I will return to this issue in section 4.4 when I discuss Fault-tolerance. I also assume that at most, a single error occurs on one of the three data qubits. Correction proceeds by introducing two ancilla qubits and performing a sequence of CNOT gates, which checks the parity of the three qubits. Table 4.1 summarizes the state of the whole system, for each possible error, just prior to measurement. For each possible situation, either no error or a single bit-flip error, the ancilla qubits are flipped to a unique state based on the parity of the data block. These qubits are then measured to obtain the classical syndrome result. The result of the measurement will then dictate if an X correction gate needs to be applied to a specific qubit. Provided that only a single error has occurred, the data block is completely restored. Notice that at no point during correction do we gain any information regarding the co-efficients α and β, hence the computational wave-function will remain intact during correction. This code will only work if one (and only one) error occurs. If two X errors

63 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

Error Location Final State, data ancilla | i| i No Error α 000 00 + β 111 00 Qubit 1 α|100i|11i + β|011i|11i Qubit 2 α|010i|10i + β|101i|10i | i| i | i| i Qubit 3 α 001 01 + β 110 01 | i| i | i| i Table 4.1: Final state of the ﬁve qubit system prior to the syndrome measurement for no error or a single X error on one of the qubits. The last two qubits represent the state of the ancilla. Note that each possible error will result in a unique measurement result (syndrome) of the ancilla qubits. This allows for a X correction gate to be applied to the data block which is classically controlled from the syndrome result. At no point during correction do we learn anything about α or β.

occur, then by tracking the circuit through you will see that the syndrome result becomes ambiguous. For example, if an X error occurs on both qubits one and two, then the syndrome result will be 01 . This will cause us to mis-correct by | i applying an X gate to qubit 3. Therefore, two errors will induce a logical bit ﬂip and causes the code to fail, as expected. This example shows the basic principle of error correction. As mentioned earlier, the 3-qubit code does not represent a full quantum code and for the moment I have neglected issues such as gate errors and the possibility of errors during state preparation and correction. However, in the coming section I will introduce the concept of full QEC using stabilizer codes and this will lead on to a description of full Fault-tolerant Quantum Error Correction.

4.2.2 Stabilizer Formalism

The stabilizer formalism which was ﬁrst introduced by Daniel Gottesman [Got97] uses essentially the Heisenberg representation for quantum mechanics which describes quantum states in terms of operators rather that the basis states themselves. An arbitrary state ψ is deﬁned to be stabilized by some operator, K, if | i it is a +1 eigenstate of K, i.e.

K ψ = ψ . (4.4) | i | i

Deﬁning multi-qubit states with respect to this formalism relies on the group structure of multi-qubit operators.

64 4.2. QUANTUM ERROR CORRECTION

It is well known that any dynamical operation on a single qubit, G, is a 1 member of the group U(2), which consists of all 2 2 unitary (G† = G− ) matrices. × Up to a global (and unphysical) phase, any single qubit operation can be expressed as a linear combination of the generators of SU(2) as,

G = cIσI + cxσx + cyσy + czσz, (4.5) where, 0 1 0 i 1 0 σx = , σy = − , σz = , (4.6) 1 0! i 0 ! 0 1! − are the Pauli matrices and σ is the 2 2 identity matrix. Within the group of all I × possible, single qubit operators, there exists a subgoup, denoted the Pauli group, , which contains the following elements, P

= σ , iσ , σ , iσ , σ , iσ , σ , iσ . (4.7) P {± I ± I ± x ± x ± y ± y ± z ± z}

It is easy to check that these matrices form a group through the commutation and anti-commutation rules for the Pauli set, σ = σ , σ , σ , { i} { x y z}

[σ , σ ]=2iǫ σ , σ , σ =2δ , (4.8) i j ijk k { i j} ij where ǫijk is the Levi-Civita symbol and δij is the Dirac delta function. For N qubits, the Pauli group extends by simply taking the N fold tensor product of , P i.e.

N N = ⊗ = σ , iσ , σ , iσ , σ , iσ , σ , iσ ⊗ . (4.9) PN P {± I ± I ± x ± x ± y ± y ± z ± z}

An N qubit stabilizer state, ψ is then speciﬁed through an N element Abelian | iN subgroup, , of the N qubit Pauli group, in which ψ is a +1 eigenstate of each G | iN element, = G G ψ = ψ , [G ,G ]=0 (i, j) . (4.10) G { i | i| i | i i j ∀ } ∈ PN Many extremely useful multi-qubit states are stabilizer states, including two- qubit Bell states, Greenberger-Horne-Zeilinger (GHZ) states [GHZ89, GHSZ90], Cluster states [BR01, RB01] and codeword states for QEC. As an example, con-

65 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

sider a three qubit GHZ state, deﬁned as,

000 + 111 GHZ 3 = | i | i. (4.11) | i √2

This state can be expressed via the three independent generators of the GHZ | i3 stabilizer group, G = σ σ σ XXX, 1 x ⊗ x ⊗ x ≡ G = σ σ σ ZZI, (4.12) 2 z ⊗ z ⊗ I ≡ G = σ σ σ IZZ. 3 I ⊗ z ⊗ z ≡ where the right-hand side of each equation is the short-hand representation of stabilizers which I will use for the remainder of the thesis. Note that these three operators form an Abelian group [Eq. 4.10] as,

[G ,G ] ψ = G G ψ G G ψ = ψ ψ =0, [i, j, ψ ]. (4.13) i j | i i j| i − j i| i | i−| i ∀ | i

Similarly, the four orthogonal Bell states,

00 11 Φ± = | i±| i, | i √2 (4.14) 01 10 Ψ± = | i±| i, | i √2 are stabilized by the operators, G = ( 1)aXX, and G = ( 1)bZZ. Where 1 − 2 − [a, b] 0, 1 and each of the four Bell states correspond to the four unique pairs, ∈{ } + + Φ , Ψ , Φ−, Ψ− = [0, 0], [0, 1], [1, 0], [1, 1] . { } { }

4.2.3 QEC with stabilizer codes

The use of the stabilizer formalism to describe quantum error correction codes is extremely useful since it allows for easy synthesis of correction circuits and also clearly shows how logical operations can be performed directly on encoded data. As an introduction I will focus on arguably the most well known quantum code, the 7-qubit Steane code, ﬁrst proposed in 1996 [Ste96a]. The 7-qubit code represents a full quantum code that encodes seven physical qubits into one logical qubit, with the ability to correct for a single X and/or Z

66 4.2. QUANTUM ERROR CORRECTION

error. The 0 and 1 basis states are deﬁned as, | iL | iL 1 0 L = ( 0000000 + 1010101 + 0110011 + 1100110 + | i √8 | i | i | i | i 0001111 + 1011010 + 0111100 + 1101001 ), | i | i | i | i (4.15) 1 1 L = ( 1111111 + 0101010 + 1001100 + 0011001 + | i √8 | i | i | i | i 1110000 + 0100101 + 1000011 + 0010110 ). | i | i | i | i The stabilizer set for the 7-qubit code is fully speciﬁed by the six operators,

K1 = IIIXXXX, K2 = XIXIXIX, K3 = IXXIIXX, (4.16) K4 = IIIZZZZ, K5 = ZIZIZIZ, K6 = IZZIIZZ.

As the 7-qubit codeword states are specified by only six stabilizers, the code contains two basis states, which are the logical states. With a final operator, 7 7 7 K = ZZZZZZZ = Z⊗ fixing the state to one of the codewords, K 0 = 0 | iL | iL and K7 1 = 1 . The 7-qubit code is defined as a [[n,k,d]] = [[7, 1, 3]] | iL −| iL quantum code, where n = 7 physical qubits encode k = 1 logical qubit with a distance between basis states d = 3, correcting t = (d 1)/2 = 1 error. Notice − that the stabilizer set separates into X and Z sectors which defines the code as a Calderbank-Shor-Steane (CSS) code. CSS codes are extreamly useful since they allow for straightforward logical gate operations to be applied directly to the encoded data [Section 4.5] and are easy to derive from classical codes. Although the 7-qubit code is the most well known Stabilizer code, there are literally dozens of stabilizer codes in existence which can encode multiple logical qubits and correct for more errors. The downside to these lager codes is that they require more physical qubits and more complicated error correction circuits. Tables 4.2 and 4.3 shows the stabilizer structure of two other well known codes, the 5-qubit code [LMPZ96] which represents the smallest possible quantum code and the original 9-qubit code of Shor [Sho95].

State Preparation and Error Correction

Using the stabilizer structure for QEC codes, the logical state preparation and error correcting procedure is straightforward. Recall that the codeword states are deﬁned as +1 eigenstates of the stabilizer set. In order to prepare a logical state

67 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

K1 X Z Z X I K2 I X Z Z X K3 X I X Z Z K4 Z X I X Z

Table 4.2: The Four Stabilizers for the [[5,1,3]] quantum code, encoding ﬁve physical qubits into one logical to correct for a single X and/or Z error. Unlike the 7- and 9-qubit codes, the [[5,1,3]] code is a non-CSS code, since the stabilizer set does not separate into X and Z sectors.

K1 Z Z I I I I I I I K2 Z I Z I I I I I I K3 I I I Z Z I I I I K4 I I I Z I Z I I I K5 I I I I I I Z Z I K6 I I I I I I Z I Z K7 X X X X X X I I I K8 X X X I I I X X X

Table 4.3: The eight Stabilizers for the [[9,1,3]] quantum code, encoding nine physical qubits into one logical to correct for a single X and/or Z error. This code was the ﬁrst quantum code to be introduced and has since been shown to be a member of a much broader class of subsystem codes known as Bacon-Shor codes [Bac06].

from some arbitrary input, we need to forcibly project qubits into eigenstates of these operators. Consider the circuit shown in Fig. 4.3. For some arbitrary input state, ψ , | iI an ancilla which is initialized in the 0 state is used as a control qubit for a | i Hermitian operation on ψ (U † = U). After the second Hadamard gate is | iI performed, the state of the system is,

1 1 ψ = ( ψ + U ψ ) 0 + ( ψ U ψ ) 1 . (4.17) | iF 2 | iI | iI | i 2 | iI − | iI | i

The ancilla qubit is then measured in the computational basis. If the result is 0 , the input state is projected to (neglecting normalization), | i

ψ = ψ + U ψ . (4.18) | iF | iI | iI

Since U is Hermitian, U ψ = ψ , hence ψ is a +1 eigenstate of U. If the | iF | iF | iF

68 4.2. QUANTUM ERROR CORRECTION

H H M

Figure 4.3: Quantum Circuit required to project an arbitrary state, ψ into a | iI 1 eigenstate of the Hermitian operator, U = U †. The measurement result of the± ancilla determines which eigenstate ψ is projected to. | iI

ancilla is measured to be 1 , then the input is projected to the state, | i

ψ = ψ U ψ , (4.19) | iF | iI − | iI which is the 1 eigenstate of U. Therefore, provided U is Hermitian, the general − circuit of Fig. 4.3 will project an arbitrary input state to a 1 eigenstate of U. ± This procedure is well known and is refered to as either a “parity” or “operator” measurement [NC00]. From this argument it should be clear how QEC state preparation proceeds. Taking the [[7, 1, 3]] code as an example, 7-qubits are first initialized in the 7 state 0 ⊗ , after which the circuit shown in Fig. 4.3 is applied three times with | i U = K1,K2,K3 , projecting the input state into a simultaneous 1 eigenstate { } ± of each X stabilizer describing the [[7, 1, 3]] code. The result of each operator measurement is then used to classically control a single qubit Z gate which is applied to one of the seven qubits at the end of the preparation, this single Z gate converts any 1 projected eigenstates into +1 eigenstates. Notice that the − 7 final three stabilizers do not need to be measured due to the input state, 0 ⊗ , al- | i ready being a +1 eigenstate of K4,K5,K6 . Fig. 4.4 illustrates the final circuit, { } where instead of one ancilla, three are utilized to speed up the state preparation by performing each operator measurement in parallel. Error correction using stabilizer codes is a straightforward extension of state preparation. Consider an arbitrary single qubit state that has been encoded,

α 0 + β 1 α 0 + β 1 = ψ . (4.20) | i | i→ | iL | iL | iL

69 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

M M i = 1 M 2 +2 3 + 4 1

0 H H M3

0 H H M2

0 H H M1

0 7 1 2 3 K K K Zi 0 1

Figure 4.4: Quantum circuit to prepare the [[7, 1, 3]] logical 0 state. The input 7 | i state 0 ⊗ is projected into an eigenstate of each of the X stabilizers shown in Eq. 4.16.| i After each ancilla measurement the classical results are used to apply a single qubit Z gate to qubit i =1M2 +2M3 +4M1 which converts the state from a 1 eigenstates of (K1,K2,K3) to +1 eigenstates. −

Now assume that an error occurs on one (or multiple) qubits which is described via the operator E, where E is a combination of X and/or Z errors over the N physical qubits of the logical state. By deﬁnition of stabilizer codes, Ki ψ = | iL ψ , i [1, .., N k], for a code encoding k logical qubits. Hence the erred state, | iL ∈ − E ψ , satisﬁes, | iL

KiE ψ =( 1)mEKi ψ =( 1)mE ψ . (4.21) | iL − | iL − | iL where m is defined as, m = 0, if [E,Ki] = 0 and m = 1, if E,Ki = 0. { } Therefore, if the error operator commutes with the stabilizer, the state remains a +1 eigenstate of Ki, if the error operator anti-commutes with the stabilizer then the logical state is converted from a +1 to a 1 eigenstate of Ki. − Hence the general procedure for error correction is identical to state preparation. Each of the code stabilizers are sequentially measured. Since a error free state is already a +1 eigenstate of all the stabilizers, any error which anti- commutes with a stabilizer will flip the eigenstate and consequently the Operator measurement will return a result of 1 . | i Taking the [[7, 1, 3]] code as an example, you can see that if the error operator is E = Xi, where i = [1, ..., 7], representing a bit-flip on any one of the 7 physical

70 4.2. QUANTUM ERROR CORRECTION

qubits, then regardless of the location, E will anti-commute with a unique combination of K4,K5,K6 . Hence the classical results of measuring these three { } operators will indicate if and where a single X error has occurred. Similarly, if

E = Zi, then the error operator will anti-commute with a unique combination of, K1,K2,K3 . Consequently, the ﬁrst three stabilizers for the [[7, 1, 3]] code { } correspond to Z sector correction while the second three stabilizers correspond to X sector correction. Note, that correction for Pauli Y errors are also taken care of by correcting in the X and Z sector since a Y error on a single qubit is equivalent to both an X and Z error on the same qubit, i.e. Y = iXZ. Fig. 4.5 illustrates the circuit for full error correction with the [[7, 1, 3, ]] code. As you can see it is simply an extension of the preparation circuit [Fig. 4.4] where all six stabilizers are measured across the data block. Even though we have speciﬁcally

0 H H M3 0 H H M6

0 H H M2 0 H H M5

0 H H M1 0 H H M4

7 1 2 3 4 5 6 K K K Zi K K K Xi 1

M M M M M i = 1 M 2 +2 3 + 4 1 i = 1 5 +2 6 + 4 4

Figure 4.5: Quantum circuit to to correct for a single X and/or Z error using the [[7, 1, 3]] code. Each of the 6 stabilizers are measured, with the ﬁrst three detecting and correcting for Z errors, while the last three detect and correct for X errors. used the [[7, 1, 3]] code as an example, the procedure for error correction and state preparation is identical for all stabilizer codes allowing for full correction for both bit and phase errors without obtaining any information regarding the state of the logical qubit.

71 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

4.3 Digitization of Quantum Errors

Up until now we have remained fairly abstract regarding the analysis of quantum errors. Specifically, we have examined QEC from the standpoint of a discrete set of Pauli errors occurring at certain locations within a larger quantum circuit. In this section we examine how this analysis of errors relates to more realistic processes such as environmental decoherence and systematic gate errors. Digitization of quantum noise is often assumed when people examine the stability of quantum circuit design or attempt to calculate thresholds for concatenated error correction. However, the equivalence of discrete Pauli errors to more general, continuous, noise only makes sense when we consider the stabilizer nature of the correction procedure. Recall from section 4.2 that correction is performed by re-projecting a potentially corrupt data block into +1 eigenstates of the stabilizer set. It is this process that acts to digitize quantum noise, since a general continuous mapping from a “clean” codeword state to a corrupt one will not satisfy the stabilizer conditions. I will first introduce how a coherent systematic error, caused by imperfect implementation of quantum gates, are digitized during correction, after which I will briefly discuss environmental decoherence from the standpoint of the Markovian master equation.

4.3.1 Systematic gate errors

consider an N qubit unitary operation, UN , which is valid on encoded data. As- sume that UN is applied inaccurately such that the resultant operation is actually

U ′ . Given a general encoded state ψ , the ﬁnal state can be expressed as, N | iN

U ′ ψ = U U ψ = α E ψ′ , (4.22) N | iL E N | iL j j| iL j X where ψ′ = U ψ is the perfectly applied N qubit gate and U is a coherent | iL N | iL E error operator which is expanded in terms of the N qubit Pauli Group, E = { j} N X Z I,X,Y,Z ⊗ . Now append two ancilla blocks, A and A , which are { } | 0i | 0i all initialized and are used for X and Z sector correction, then run a full error correction cycle, which we represent by the unitary operator, UQEC. It will be assumed that ψ is encoded with a hypothetical QEC code which can correct | iL for N errors (both X and/or Z), hence there is a one-to-one mapping between

72 4.3. DIGITIZATION OF QUANTUM ERRORS

the error operators, Ej, and the orthogonal basis states of the ancilla blocks,

X Z X Z U U ′ ψ A A = U α E ψ′ A A QEC N | iL| 0i | 0i QEC j j| iL| 0i | 0i j X (4.23) X Z = α E ψ′ A A . j j| iL| ji | ji j X The ancilla blocks are then measured, projecting the data blocks into the state 2 E ψ′ with probability α , after which the correction E† is applied based on j| iL | j| j the syndrome result.

For very small systematic inaccuracies, the expansion co-efficient, α0, which N corresponds to E0 = I⊗ will be very close to 1, with all other co-efficients small. Hence during correction there will be a very high probability that no error is detected. This is the digitization effect of quantum error correction. Since codeword states are specific eigenstates of the stabilizers, then the re-projection of the state when each stabilizer is measured forces any continuous noise operator to collapse to the discrete Pauli set, with the magnitude of the error dictating the probability that the data block is projected into discrete perturbation of a “clean” state.

4.3.2 Environmental decoherence

A complete analysis of environmental decoherence in relation to quantum information is a lengthy topic. Instead of a detailed review, I will instead simply present a specific example to highlight how QEC relates to environmental effects. The Lindblad formalism [Gar91, NC00, DWM03] provides an elegant method for analyzing the effect of decoherence on open quantum systems. This model does have several assumptions, most notably that the environmental bath couples weakly to the system (Born approximation) and that each qubit experiences un-corrolated noise (Markovian approximation). While these assumptions are utilized for a variety of systems [BHPC03, BM03, BKD04], it is known that they may not hold in some cases [HMCS00, MCM+05, APN+05, ALKH02]. Particu- larly in superconducting systems where decoherence is caused by small numbers of fluctuating charges. In this case more specific decoherence models need to be considered. Using this formalism, the evolution of the density matrix can be

73 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

written as, i ∂ ρ = [H, ρ]+ Γ [ρ]. (4.24) t −~ kL Xk Where H is the Hamiltonian, representing coherent, dynamical evolution of the system and [ρ]=([L , ρL† ] + [L ρ, L† ])/2 represents the incoherent evolution. Lk k k k k The operators Lk are known as the Lindblad quantum jump operators and are used to model speciﬁc decoherence channels, with each operator parameterized by some rate Γ 0. This diﬀerential equation is known as the quantum louiville k ≥ equation or more generally, the density matrix master equation. To link Markovian decoherence to QEC, consider a special set of decoherence channels that help to simplify the calculation, representing a single qubit undergoing dephasing, spontaneous emission and spontaneous absorption. Dephasing of a single qubit is modelled by the Lindblad operator L1 = Z while spontaneous emission/absorption are modelled by the operators L = 0 1 and L = 1 0 2 | ih | 3 | ih | respectively. For the sake of simplicity I assume that absorption/emission occur at the same rate, Γ. Consequently, the density matrix evolution is given by,

i Γ ∂ ρ = [H, ρ]+Γ (ZρZ ρ)+ (XρX + Y ρY 2ρ). (4.25) t −~ Z − 2 −

If it is assumed that the qubit is not undergoing any coherent evolution (H = 0), i.e. a memory stage within a quantum algorithm, then Eq. 4.25 can be solved by re-expressing the density matrix in the Bloch formalism. Set ρ(t) = I/2+ x(t)X + y(t)Y + z(t)Z, then Eq. 4.25, with H = 0, reduces to, ∂tS(t) = AS(t), with, S(t)=(x(t),y(t), z(t))T and,

(Γ+2Γ ) 0 0 − z A = 0 (Γ+2Γ ) 0 . (4.26)  − z  0 0 2Γ  −    This diﬀerential is easy to solve, leading to,

ρ(t)=[1 p(t)]ρ(0) + p (t)Xρ(0)X + p (t)Y ρ(0)Y + p (t)Zρ(0)Z, (4.27) − x y z where,

1 2Γt 1 2Γt (Γ+2Γz)t px(t)= py(t)= (1 e− ), pz(t)= (1 + e− 2e− ), 4 − 4 − (4.28) p(t)= px(t)+ py(t)+ pz(t).

74 4.3. DIGITIZATION OF QUANTUM ERRORS

The solution to the master equation leads to an operator sum expansion (OSR) of the density matrix. If this single qubit is part of a QEC encoded data block, then each term represents a single error on the qubit experiencing decoherence. Two blocks of initialized ancilla qubits are added to the system and the error correction protocol run. Once the ancilla qubits are measured, the state will collapse to no error, with probability 1 p(t), or a single X,Y or Z error, with − probabilities px(t),py(t) and pz(t). Consequently, in standard QEC analysis it is assumed that after each elementary gate operation, measurement, initialization and memory step, a hypothetical error correction cycle is run. This cycle digitizes all continuous errors (either systematic or environmental) into either an X and/or Z error on each qubit. This cycle is assumed to be error free and take zero time. In this way error correction can be analyzed by assuming perfect gate operations and discrete, probabilistic errors. The probability of each error occuring can then be independently calculated via a systematic gate analysis or through the evolution of the master equation.

75 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

4.4 Fault-tolerant Quantum Error Correction and the threshold theorem.

Section 4.2 detailed the protocols required to correct for quantum errors, however this implementation of QEC assumed the following,

1. Errors only occur during “memory” regions, i.e. when quantum operations or error correction are not being performed.

2. The quantum gates themselves do not induce any systematic errors within the logical data block.

Clearly these are two very unrealistic assumptions, therefore error correction procedures and logical gate operations need to be designed such that they can still correct for errors.

4.4.1 Fault-tolerance

The concept of Fault-tolerance in computation is not a new idea, it was actually first developed in relation to classical computing [Avi87]. However, in recent years the precise manufacturing of digital circuitry has made large scale error correction and Fault-tolerant circuits unnecessary. The basic principle of Fault-tolerance is that the circuits used for gate operations and error correction procedures should not cause errors to cascade. This can be seen clearly when we look at a simple CNOT operation between two qubits [Fig. 4.6]. In this circuit we are simply performing a sequence of three CNOT gates which act to take the state 111 000 111 111 . In the left circuit we | i| i→| i| i consider a single X error which occurs on the top most qubit prior to the first CNOT. This single error will cascade through each of the three gates such that the X error has now propagated to four qubits. the right hand circuit shows a slightly modified design that implements the same operation, but the single X error now only propagates to two of the six qubits. If we consider each block of three as a single logical qubit, then the staggered circuit will only induce a total of one error in each logical block, given a single X error occurred somewhere during the gate operations. This is the standard definition of Fault-tolerance. Fault-tolerant circuit element: A single error (either environmentally induced

76 4.4. FAULT-TOLERANT QUANTUM ERROR CORRECTION AND THE THRESHOLD THEOREM.

a) b) X X X

X X X X

Figure 4.6: Two circuits to implement the transformation 111 000 | i| i → 111 111 . a) shows a version where a single X error can cascade into four errors| i| whilei b) shows an equivalent circuit where the error only propagates to a second qubit. or caused by inaccurate gates) will cause at most one error in the output for each logical qubit block. It should be stressed that the idea of Fault-tolerance is a discrete definition, either a certain quantum operation is Fault-tolerant or it is not. What is defined to be Fault-tolerant can change depending on the error correction code used. For example, for a single error correcting code the above definition is the only one available (since any more than one error in a logical qubit will result in the error correction procedure failing). However, if the quantum code employed is able to correct multiple errors, then the definition of Fault-tolernace can be relaxed, i.e. if the code can correct three errors then circuits may be designed such that a single failure results in at most two errors in the output (which is then correctable). Section 4.7 will illustrate examples of a Fault-tolerant preparation circuit for the [[7, 1, 3]] code when comparing this large circuit to a smaller non-Fault- tolerant version, while section 4.5 illustrates how Fault-tolerant gates can be applied directly to encoded data.

4.4.2 Threshold Theorem

The threshold theorem is truly a remarkable result in quantum information and is a consequence of Fault-tolerant circuit design and the ability to perform dynamical error correction. Rather than present a detailed derivation of the theorem for

77 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

a variety of noise models, we will instead take a very simple case where we utilize a quantum code that can only correct for a single error, using a model that assumes uncorrolated, coherent errors on individual qubits. For more rigorous derivations of the theorem see [ABO97, Got97, Ali07]. Consider a quantum algorithm that requires Q qubits arranged into a circuit requiring K time steps and assume each physical qubit experiences either an X and/or Z error independently with probability p per time step. Furthermore, it is assumed that each logical gate operation and error correction circuit is designed according to the rules of Fault-tolerance and that a cycle of error correction is performed after each elementary logical gate operation. If an error occurs during a logical gate operation, then Fault-tolerance ensures this error will only propagate to at most one error in each block, after which a cycle of error correction will remove the error. Hence if the failure probability of un-encoded qubits per time step is p, then a single level of error correction will ensure that the logical step fails only when two (or more) errors occur. Hence the failure rate of each logical 1 2 1 operation, to leading order, is now pL = cp , where pL is the failure rate (per time step) of a 1st level logical qubit and c is the number of possible combinations where 2 errors can occur at a physical level within the circuit consisting of the gate operation + correction cycle. We now repeat the process, re-encoding the computer such that a level-2 logical qubit is formed, using the same quantum code, from n, level-1 encoded qubits. It is assumed that all error correcting procedures and gate operations at the 2nd level are self-similar to the level-1 operations (i.e. the circuit structures for the level-2 encoding are identical to the level-1 encoding). Therefore, if the level-1 failure rate per logical time step is 1 pL, then by the same argument, the failure rate of a 2-level operation is given 2 1 2 3 4 by, pL = c(pL) = c p . This iterative procedure is then repeated (referred to as concatenation) up to the kth level, such that the logical failure rate, per time step, of a k-level encoded qubit is given by,

(cp)2k pk = . (4.29) L c

Eq. 4.29 implies that for a ﬁnite physical error rate, p, per qubit, per time step, the failure rate of the kth-level encoded qubit can be made arbitrarily small by simply increasing k, dependent on cp < 1. This inequality deﬁnes the threshold.

The physical error rate experienced by each qubit per time step must be pth < 1/c to ensure that multiple levels of error correction reduce the failure rate of logical

78 4.4. FAULT-TOLERANT QUANTUM ERROR CORRECTION AND THE THRESHOLD THEOREM.

components. Hence, provided sufficient resources are available, an arbitrarily large quantum circuit can be successfully implemented, to arbitrary accuracy, once the physical error rate is below threshold. The calculation of thresholds is therefore an extremely important aspect to quantum architecture design. Initial estimates 4 at the threshold, which gave p 10− [Kit97, ABO97, Got97] did not suf- th ≈ ficiently model physical systems in an accurate way. Recent results [SFH07, SDT07, SBF+06, MCT+04, BKSO05], estimated for more realistic quantum processor architectures, suggests that thresholds will be significantly lower.

79 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

4.5 Fault-tolerant operations on encoded data

Sections 4.2 and 4.4 showed how Fault-tolerant QEC allows for any quantum algorithm to be run to arbitrary accuracy. However, the results of the threshold theorem assume that logical operations can be performed directly on the encoded data without the need for continual decoding and re-encoding. Using stabilizer codes, a large class of operations can be performed on logical data in an inherently Fault-tolerant way. If a given logical state, ψ , is stabilized by K, and the logical operation U | iL is applied, the new state, U ψ is stabilized by UKU †, i.e, | iL

UKU †U ψ = UK ψ = U ψ . (4.30) | iL | iL | iL

In order for the codeword states to remain valid, the stabilizer set for the code, , G must remain ﬁxed through every operation. Hence for U to be a valid operation on the data, U U † = . G G

4.5.1 Single Qubit Operations

The logical X¯ and Z¯ operations on a single encoded qubit are the ﬁrst examples of valid codeword operations. Taking the [[7, 1, 3]] code as an example, X¯ and Z¯ are given by,

7 7 X¯ = XXXXXXX X⊗ , Z¯ = ZZZZZZZ Z⊗ . (4.31) ≡ ≡

Since the single qubit Pauli operators satisfy XZX = Z and ZXZ = X − − then, XK¯ iX¯ = Ki and ZK¯ iZ¯ = Ki for each of the [[7, 1, 3]] stabilizers given in Eq. 4.16. The fact that each stabilizer has a a weight of four guarantees that

UKU † picks up an even number of 1 factors. Since the stabilizers remain fixed − the operations are valid. However, what transformations do Eq. 4.31 actually perform on encoded data? For a single qubit, a bit-flip operation X takes 0 1 . Recall that for a | i↔| i single qubit Z 0 = 0 and Z 1 = 1 , hence for X¯ to actually induce a logical | i | i | i −| i bit-flip it must take, 0 1 . For the [[7, 1, 3]] code, the final operator which | iL ↔| iL 7 7 7 7 fixes the logical state is K = Z⊗ , where K 0 = 0 and K 1 = 1 . | iL | iL | iL −| iL As XK¯ 7X¯ = K7, any state stabilized by K7 becomes stabilized by K7 (and − −

80 4.5. FAULT-TOLERANT OPERATIONS ON ENCODED DATA

vice-versa) after the operation of X¯. Therefore, X¯ represents a logical bit ﬂip. The same argument can be used for Z¯ by considering the stabilizer properties of the states = ( 0 1 )/√2. Hence, the logical bit- and phase-ﬂip gates |±i | i±| i can be applied directly to logical data by simply using seven single qubit X or Z gates, [Fig. 4.7].

7 7

Figure 4.7: Bit-wise application of single qubit gates in the [[7, 1, 3]] code. Logical X, Z H and P gates can trivially be applied by using seven single qubit gates, Fault-tolerantly. Note that the application of seven P gates results in the logical ¯ P † being applied and visa versa.

Two other useful gates which can be applied in this manner is the Hadamard rotation and phase gate,

1 1 1 1 0 H = , P = . (4.32) √2 1 1! 0 i! − These gates are useful since when combined with the two-qubit CNOT gate, they can generate a subgroup of all multi-qubit gates known as the Cliﬀord group

81 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

(gates which map Pauli group operators back to the Pauli group). Again, using the stabilizers of the [[7, 1, 3]] code and the fact that for single qubits,

HXH = Z, HZH = X, (4.33) PXP † = iXZ, PZP † = Z, a seven qubit bit-wise Hadamard gate will switch X with Z and therefore will simply ﬂip K1,K2,K3 with K4,K5,K6 , and is a valid operation. The bit- { } { } wise application of the P gate will leave any Z stabilizer invariant, but takes X iXZ. This is still valid since provided there are a multiple of four non- → identity operators for the stabilizer set, the factors of i will cancel. Hence seven bit-wise P gates is valid for the [[7, 1, 3]] code. What does H¯ and P¯ do to the logical state? For a single qubit, the Hadamard gate ﬂips any Z stabilized state to a X stabilized state, i.e 0, 1 +, . | i↔| −i 7 7 7 Looking at the transformation of K , HK¯ H¯ = X⊗ , the bit-wise Hadamard gate will invoke a logical Hadamard operation. The single qubit P gate leaves a Z stabilized state invariant, while an X eigenstate becomes stabilized by iXZ. ¯ 7 ¯ 7 ¯ Hence, P †(X⊗ )P = i(XZ)⊗ and the bit-wise gate, P †, represents a logical ¯ P gate on the data block. Similarly, bit-wise P gates enact a logical P † gate [Fig. 4.7]. Each of these Fault-tolerant operations on a logically encoded block are commonly refereed to as transversal operations, as a logical operation is obtained by a set of individual operations acting transversally on the physical qubits.

4.5.2 Two-qubit gate.

A two-qubit logical CNOT operation can also be applied in the same transversal way. For un-encoded qubits, a CNOT operation performs the following mapping on the two qubit stabilizer set,

X I X X, ⊗ → ⊗ I Z Z Z, ⊗ → ⊗ (4.34) Z I Z I, ⊗ → ⊗ I X I X. ⊗ → ⊗ Where the ﬁrst operator corresponds to the control qubit and the second operator corresponds to the target. Now consider the bit-wise application of seven CNOT

82 4.5. FAULT-TOLERANT OPERATIONS ON ENCODED DATA

gates between logically encoded blocks of data [Fig. 4.8]. First the stabilizer set must remain invariant, i.e,

= Ki Kj Ki Kj (i, j). (4.35) G { ⊗ }→{ ⊗ } ∀

Table 4.4 details the transformation for all the stabilizers under seven bit-wise CNOT gates, demonstrating that this operation is valid on the [[7, 1, 3]] code. The transformations in Eq. 4.34 are trivially extend to the logical space, showing that seven bit-wise CNOT gates invoke a logical CNOT operation.

X¯ I X¯ X,¯ ⊗ → ⊗ I Z¯ Z¯ Z,¯ ⊗ → ⊗ (4.36) Z¯ I Z¯ I, ⊗ → ⊗ I X¯ I X.¯ ⊗ → ⊗

The issue of Fault-tolerance with these logical operations should be clear. The X¯,Z¯, H¯ and P¯ gates are trivially Fault-tolerant since the logical operation is performed through seven bit-wise single qubit gates. The logical CNOT is also Fault-tolerant since each two-qubit gate only operates between counterpart qubits in each logical block. Hence if any gate is inaccurate, then at most a single error will be introduced in each block. While these subset of gates are useful for operating on quantum data, they do not represent a universal set for quantum computation. In fact it has been shown that by using the stabilizer formalism, these operations can be eﬃciently simulated on a classical device [Got98a, AG04]. In order to achieve universality one of the following gates must be added to the available set,

1 0 T = , (4.37) 0 eiπ/4! or the Toﬀoli gate [Tof81]. However, neither of these two gates are members of the Cliﬀord group and applying them in a similar way to the other gates will transform the stabilizers out the group and consequently does not represent a valid operation. Circuits implementing these two gates in a Fault-tolerant manner have been developed [NC00, GC99, SI05, SFH07], but at this stage the circuits are extremely complicated and resource intensive. This has practical implications

83 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE. ersnsavldoeaino h codespace. the on operation valid a represents h rnfrainde o aeaysaiie usd the outside stabilizer any take not does transformation the al .:Tasomtoso h [[7 the of Transformations 4.4: Table K i K K K K K K ⊗ 6 5 4 3 2 1 K j K K K K K 3 2 K ⊗ ⊗ 6 5 4 K 1 ⊗ ⊗ ⊗ K K ⊗ 1 K K K 3 1 I K K 1 1 1 1 2 K K , 1 K K K 3 1 , K ] tblzrstudrtegt operation gate the under set stabilizer 3]] ⊗ ⊗ 6 5 4 K 2 ⊗ ⊗ ⊗ K K ⊗ 2 K K K 3 1 I K K 2 2 2 2 2 K K K K K 2 1 K ⊗ ⊗ 6 5 4 K 3 ⊗ ⊗ ⊗ K K ⊗ 3 K K K 2 1 ru eeae by generated group I K K 3 3 3 3 3 K K K K K 3 2 1 K K K 6 5 I K K 4 4 4 ⊗ K ⊗ ⊗ ⊗ 4 4 K ⊗ ⊗ 4 K K K 4 K K 3 2 1 K K K 4 4 K 4 4 4 i ⊗ U K K K K K K 3 2 1 =CNOT K K K 6 4 I j K K 5 5 5 ( ⊗ K ,j i, ⊗ ⊗ ⊗ 5 5 K 5 ⊗ ⊗ K K K ) 5 K K ⊗ ∈ 3 2 1 K K K 7 5 5 [1 where , 5 5 5 .., , K K K ] therefore 6], K K 3 2 1 K K K 5 4 I → G K K 6 6 6 ⊗ K ⊗ ⊗ ⊗ 6 6 K 6 ⊗ ⊗ K K K U 6 K K 3 2 1 † K K K G 6 6 U U 6 6 6 oethat Note . =CNOT ⊗ 7

84 4.5. FAULT-TOLERANT OPERATIONS ON ENCODED DATA

Figure 4.8: Bit-wise application of a CNOT gate between two logical qubits. Since each CNOT only couples corresponding qubits in each block, this operation is inherently Fault-tolerant. to encoded operations. If universality is achieved by adding the T gate to the list, arbitrary single qubit rotations require long gate sequences (utilizing the Solovay- Kitaev theorem [Kit97, DN06]) to approximate arbitrary logical qubit rotations and these sequences often require many T gates [Fow04]. Finding more eﬃcient methods to achieve universality on encoded data is therefore still an active area of research.

85 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

4.6 Fault-tolerant circuit design for logical state preparation

Section 4.4 introduced the basic rules for Fault-tolerant circuit design and how these rules lead to the threshold theorem for concatenated error correction. How- ever, what does a full Fault-tolerant quantum circuit look like? In anticipation of numerical simulations I present in Section 4.7 I will introduce a full Fault- tolerant circuit to prepare the [[7, 1, 3]] logical 0 state. As the [[7, 1, 3]] code is | i a single error correcting code, we use the one-to-one definition of Fault-tolerance and therefore only need to consider the propagation of a single error during the preparation (any more that one error during correction represents a higher order effect and is ignored). As described in Section 4.2, logical state preparation can be done by initializing an appropriate number of physical qubits and measuring each of the X stabilizers that describe the code. Therefore, a circuit which allows the measurement of a Hermitian operator in a Fault-tolerant manner needs to be constructed. The general structure of the circuit used was first developed by Shor [Sho96], however it should be noted that several more recent methods for Fault-tolerant state preparation and correction now exist [Ste97, Ste02, DA07] which are more efficient than Shor’s original method. The circuits shown in Fig. 4.9a and 4.9b, which measure the stabilizer K1 = IIIXXXX are not Fault-tolerant, since a single ancilla is used to control each of the four CNOT gates. Instead, four ancilla qubits are used which are prepared in the state =( 0000 + 1111 )/√2. This can be done by initializing four qubits |Ai | i | i in the 0 state and applying a Hadamard then a sequence of CNOT gates. Each | i of these four ancilla are used to control a separate CNOT gate, after which the ancilla state is decoded and measured. By ensuring that each CNOT is controlled via a separate ancilla, any X error will only propagate to a single qubit in the data block. However, during the preparation of the ancilla state there is the possibility that a single X error can propagate to multiple ancilla, which are then fed forward into the data block. In order to combat this, the ancilla block needs to be verified against possible X errors. Tracking through all the possible locations where a single X error can occur during ancilla preparation leads to the

86 4.6. FAULT-TOLERANT CIRCUIT DESIGN FOR LOGICAL STATE PREPARATION

a) b) 0 H H M 0 H H M

7 7

1 K

1 1

c) 0 H H M 0 0 0

Figure 4.9: Three circuits which measure the stabilizer K1. Fig a) represents a generic operator measurement where a multi-qubit controlled gate is available. Fig. b) decomposes this into single- and two-qubit gates, but in a non-Fault- tolerant manner. Fig. c) introduces four ancilla such that each CNOT is controlled via a separate qubit. This ensures Fault-tolerance.

following unique states.

1 1 = ( 0000 + 1111 ), |Ai √2 | i | i 1 2 = ( 0001 + 1110 ), |Ai √2 | i | i (4.38) 1 3 = ( 0011 + 1100 ), |Ai √2 | i | i 1 4 = ( 0111 + 1000 ). |Ai √2 | i | i

From these possibilities, the last three states have a different parity between the first and forth qubit. Hence to verify this state, a fifth ancilla is added, initialized and used to perform a parity check on the ancilla block. This fifth ancilla is then measured. If the result is 0 , the ancilla block is clean and can | i

87 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

be coupled to the data. If the ancilla result is 1 , then either a single error has | i occured in the ancilla preparation or on this veriﬁcation qubit. In either case, the entire ancilla block is re-initialized and the ancilla prepared again. This is continued until the veriﬁcation qubit is measured to be 0 [Fig. 4.10]. The re- | i M Repeat Until M = 0 M

H H M 1

7 Measure K

Figure 4.10: Circuit required to measure the stabilizer K1, Fault-tolerantly. A four qubit GHZ state is used as ancilla with the state requiring verification against multiple X errors. After the state has passed verification it is coupled to the data block and a syndrome is extracted. preparation of the ancilla block protects against X errors, which can propagate forward through the CNOT gates. Z errors on the other hand, propagate in the other direction. Any Z error which occurs in the ancilla block will propagate straight through to the final measurement. This results in the measurement not corresponding to the eigenstate the data is projected to and can possibly result in mis-correction once all stabilizers have been measured. To protect against this, each stabilizer is measured 2-3 times and a majority vote of the measurement results taken. As any additional error represents a second order process, if the first or second measurement has been corrupted by an induced Z error, then the third measurement will only contain additional errors if a higher order error process has occurred. Therefore, we are free to ignore this possibility and assume that the third measurement is error free. The full circuit for [[7, 1, 3]] state preparation is shown in Fig. 4.11, where each stabilizer is measured 2-3 times. The total circuit requires a minimum of 12 qubits (7 data qubits and a 5 qubit ancilla block).

88 4.6. FAULT-TOLERANT CIRCUIT DESIGN FOR LOGICAL STATE PREPARATION 0 0 0 0 0 0 0 0 0 0 0 0

1 M Measure K 1

1 M Measure K 1 Apply dotted boxes only if first two results differ for each stabilizer.

1 M Measure K 1

2 M Measure K 2

3 M Measure K 3

3 M Measure K 3 Z 0 0 0 0 0 i

0 M M2M 3 1

L i = 1 +2 + 4

Figure 4.11: Circuit required to prepare the [[7, 1, 3]] logical 0 state Fault- tolerantly. Each of the X stabilizers are sequentially measured using| i the circuit in Fig. 4.10. To maintain Fault-tolerance, each stabilizer is measured 2-3 times with a majority vote taken.

89 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

4.7 Simulations of Logical State Preparation

The results that were presented in Chapter 3 demonstrated that the practical implementation of Shor’s algorithm will be heavily dependent on low effective error rates within any quantum computer. In particular, the sensitivity of Shor’s algorithm is such that any more than approximately 10-20 errors during the entire calculation, for modest problem sizes, leads to the algorithm failing to show any quantum processing. For large factoring problems involving hundreds of physical qubits and millions of gate operations, this sensitivity to qubit errors is of great concern. As shown in the previous sections, Quantum Error Correction and Fault-tolerant quantum computation provide a means to combat this problem. Adherence to Fault-tolerance inevitably leads to complicated circuit structures in order to prepare logically encoded qubits, to perform correction and to implement specific single and multi-qubit gates. Section 4.6 introduced a Fault- tolerant circuit needed to prepare the logical 0 state assuming a quantum archi- | i tecture which allows for arbitrary qubit couplings (non-LNN). However, quantum error correction is able to be implemented non-Fault-tolerantly in a much simpler manner. Circuit structures for non-Fault-tolerant computation are generally much quicker, simpler and require less resources. The drawback is that threshold theorems for non-Fault-tolerant circuits are unavailable and hence error correction to arbitrary accuracy is not guaranteed. This section will continue to examine the [[7, 1, 3]] Steane code and the practicality of using Fault-tolerant circuits compared with their non-Fault-tolerant counterparts. Numerical simulations investigate the output fidelity of preparing a logical 0 state with the [[7, 1, 3]] code for a given error probability using a | i discrete error model identical to the one utilized in section 3.3. As with the Shor simulations, I will present data examining circuits appropriate for both a linear nearest neighbour (LNN) array of qubits and also an array that allows arbitrary couplings.

4.7.1 LNN circuit for logical encoding

The circuit required for Fault-tolerant encoding assuming a non-LNN architecture has already been presented. The LNN version of this circuit is not diﬃcult to derive from the non-LNN version. Fig. 4.12 illustrates the circuit structure for a

90 4.7. SIMULATIONS OF LOGICAL STATE PREPARATION

LNN architecture in order to measure the stabilizer K1. As with the LNN circuit for Shor’s algorithm, the dotted boxes represent compound two qubit gates which are assumed to take identical time. This module is then used in the full circuit in an identical way to Fig. 4.11. It should be stressed that a major assumption is that systematic gate errors for SWAP gates are non-existant. It is well known that a true linear array of qubits cannot be used to build Fault-tolerant circuits when SWAP gates are used as transport. Instead, a small second dimension is needed to ensure errors in SWAP gates do not induce correlated errors within a single logical data block, this is achieved by swapping data qubits around each other [SFH07]. If larger error correcting codes are used, then true linear array can be utilized, provided the deﬁnition of Fault-tolernace is relaxed.

M Repeat Until M = 0 0 0 0

0 H H M 0 0 0 M 0 0 1 0 0 0 0 0 0 7 Measure K

Figure 4.12: Circuit required for the LNN implementation of Fault-tolerant stabilizer measurement. These elements are then integrated into a preparation circuit identical to the non-LNN version [Fig. 4.11]. Note that when using a single error correcting code, SWAP gates are assumed to not to introduce correlated errors.

Although section 4.2 presented state preparation through the measurement of the stabilizer set, non-Fault-tolerant circuits for [[7, 1, 3]] state preparation can be constructed in a simple way through a direct sequence of gates. Figs. 4.13 and 4.14 illustrates the non-Fault-tolerant circuits for both LNN and non-LNN architectures. Note that in the LNN version, the physical order of the qubits with respect to the stabilizer structure are shuﬄed to speed up the circuit. This is not an issue for further operations provided we know which qubit is which. Table 4.5 summarizes the resources required for each circuit, demonstrating

91 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

5 3 0 4 2 0 6 5 0 3 1 0 H 0 L 2 4 0 H 7 6 0 1 7 0 H

Figure 4.13: 7 qubit circuits for non-Fault-tolerant encoding for a LNN array of qubits. Dotted boxes represent compound gates that can be constructed via the canonical decomposition [Mak02, KC01, ZVSW03], [Chapter 5.4]. The numbering of each qubit is shuﬄed in order to increase the speed of the circuit.

that the non-Fault-tolerant circuits are signiﬁcantly quicker than the complicated Fault-tolerant versions. The error model that was used in these simulations is

Circuit type Qubits Depth Gates LNN, non-FT 7 4 10 LNN, FT 12 96∗ 244∗ non-LNN, non-FT 7 3 9 non-LNN, FT 12 60∗ 108∗

Table 4.5: Resources required for various types of circuits to prepare a logical 0 | i state. * indicates the minimum possible depth and gate counts for Fault-tolerant circuits, as these circuits can change depending on speciﬁc measurement results obtained during the calculation.

identical to that used in Chapter 3.3, in which a single qubit ψ = α 0 + β 1 | i | i | i can experience a bit flip X ψ , a phase flip Z ψ , or both at the same time | i | i Y ψ XZ ψ . Each of the three types of discrete error gates has an equal | i ≡ | i probability p/3 of occurring. For each circuit, p will be varied and output fidelity calculated. Output fidelity is defined as, F = φ 0 2 where φ is the output |h | iL| | i state from the simulations and 0 is the desired 7 qubit logical 0 state given | iL | i by Eq. 4.15. Within Fault-tolerant circuits, a single error should only cause at

92 4.7. SIMULATIONS OF LOGICAL STATE PREPARATION

0 0 L

0 H

Figure 4.14: 7 qubit circuits for non-Fault-tolerant encoding using a non-LNN array of qubits. Dotted boxes represent compound gates that can be constructed via the canonical decomposition [Mak02, KC01, ZVSW03], [Chapter 5.4]. most one error in the output, which is then correctable. Hence, given a single error probability of p, a Fault-tolerant circuit should fail with probability O(p2). This fact is clearly not true for the non-Fault-tolerant circuits. Consequently, simulations of the Fault-tolerant circuit consisted of a logical 0 preparation and | i then a correction cycle, which is an extension of the preparation circuit (refer back to section 4.2). The stability of this circuit was then compared with the non-Fault-tolerant preparation circuits shown in Figs. 4.13 and 4.14.

4.7.2 Results

The following figures show the results for both the LNN and non-LNN circuits. Figs. 4.15 and 4.16 shows output fidelity vs probability of discrete error for the LNN circuits and non-LNN circuit respectively. The simulations were conducted in a stochastic manner with 106 statistical runs performed for each data point. Each plot shows a direct comparison between the Fault-tolerant and non-Fault- tolerant circuits. From both plots it is clear that the non-Fault-tolerant circuit is generally more stable. Initially, the stochastic nature of the state vector approach did not allow for a detailed plot at extremely low error rates since computational time become prohibitively large to obtain sufficient statistics. However, we later

93 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

non-Fault-tolerant Fault-tolerant

Figure 4.15: Stability of LNN circuit for the [[7, 1, 3]] encoding. Each curve shows output ﬁdelity vs probability of discrete error for the Fault-tolerant encoding plus correction, and the non-Fault-tolerant encoding.

completed a density matrix simulator in which the stochastic nature of state vector simulations could be eliminated and work by Ashley Stephens and my- + 5 self [SDF 06] determined the crossover as p 5.3 10− [Fig. 4.17]. The ≈ × slight difference in the crossover point illustrated in Figs. 4.16 and 4.17 occurs because a slightly different (and more accurate) measurement metric was used in the density matrix simulations. In the state vector simulations [Fig. 4.16] we employed the standard measure of fidelity between the simulated state and the ideal state [NC00]. In the density matrix simulations [Fig. 4.17] we directly decomposed the simulated density matrix into terms corresponding to zero errors, one error, etc.. and then equated the co-efficients to the failure rate of the logical state.

4.7.3 Conclusions

Currently accepted threshold rates for Fault-tolerant computation lies in the re- 5 2 gion of p = 10− to p = 10− . This is the error range in which experimentalists

94 4.7. SIMULATIONS OF LOGICAL STATE PREPARATION

non-Fault-tolerant Fault-tolerant

Figure 4.16: Stability of the non-LNN circuit for the [[7, 1, 3]] encoding. Each curve shows output fidelity vs probability of discrete error for the Fault-tolerant encoding plus correction, and the non-Fault-tolerant encoding. are currently aiming for in fabricated devices, although recently is has become accepted that realistic thresholds will be lower [SFH07, SDT07, SBF+06, MCT+04, BKSO05]. This simulation data suggests that if we wish to use Fault-tolerant 5 circuits for the encoding of qubits, error rates must be at least p 10− , if ≈ not lower, in order for Fault-tolerant state preparation to be significantly more effective than quicker non-Fault-tolerant versions. These results are more significant when considering the efficient correction schemes introduced by Steane [Ste97, Ste02]. His scheme uses encoded 0 states | iL rather than =( 0000 + 1111 )/√2 states as ancilla blocks for QEC. The fact |Ai | i | i 5 that the non-Fault-tolerant version is more stable down to p 10− implies that ≈ non-Fault-tolerant circuits should always be used when preparing appropriate ancilla states for full Fault-tolerant error correction using Steane’s methods.

95 CHAPTER 4. QUANTUM ERROR CORRECTION AND FAULT-TOLERANCE.

Figure 4.17: (From [SDF+06]). Density matrix simulations for the non-LNN circuit for [[7, 1, 3]] state preparation. Since the density matrix formalism is able to calculate ensemble results, the stochastic nature of state vector simulations is 5 eliminated. These simulations found the crossover error rate as p 5.3 10− where the Fault-tolerant circuit becomes more stable. Note that these≈ results× are examining the crossover between the two circuits, not the Fault-tolerant threshold. The crossover point is the physical error rate for which the Fault-tolerant encoding fails with a lower probability than the direct encoding. At the crossover, neither circuit is below the Fault-tolerant threshold as the logical error rate is higher than the physical error rate. Although simulations were not continued to very low error rates, the Fault-tolerant circuit appears to reach the threshold at 5 p 1 10− ≈ ×

96 Chapter 5

Intrinsic Characterization of qubit systems

Contents 5.1 Introduction ...... 99

5.1.1 Spectroscopy ...... 100 5.1.2 State and Process Tomography ...... 101 5.2 Subspace Conﬁnement: How good is your qubit? . . 106

5.2.1 Motivation and Preliminaries ...... 107 5.2.2 Estimation of subspace leakage ...... 109 5.2.3 Finite Sampling Fourier analysis ...... 114 5.2.4 Eﬃciency of the protocol ...... 121 5.2.5 The eﬀect of Decoherence ...... 125 5.2.6 Conclusions ...... 130 5.3 Single qubit dynamics ...... 132

5.3.1 Characterization without decoherence ...... 132 5.3.2 Characterization in the Presence of decoherence . . . 135 5.3.3 Conclusions ...... 139 5.4 Characterization of two qubit dynamics ...... 141

5.4.1 Structure of two-qubit dynamics ...... 142 5.4.2 Periodic Entanglement and Two-qubit Hamiltonian Char- acterization...... 152

97 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

5.4.3 Characterization of a fully non-local interaction. . . . 156 5.4.4 Example: Spin-Orbit corrections in GaAs quantum dots.162 5.4.5 Conclusions ...... 166

98 5.1. INTRODUCTION

5.1 Introduction

System identification and characterization is instrumental to the entire field of quantum computing and more generally quantum control. In contrast to other fields in theoretical physics, the basic ideas related to quantum control can be taken directly from the non-relativistic quantum mechanics of Schrödinger [Sch26] and Heisenberg [Hei25]. In these systems, the dynamical evolution of a quantum state, ψ(t), is governed by the system Hamiltonian, H, through the non- relativistic Schrödinger equation,

∂ψ(t) i~ = Hψ(t). (5.1) ∂t

The main challenge of quantum control is to engineer and/or identify H. Within the language of quantum information processing, the engineering and characterization of the system Hamiltonian directly controls the interactions required for quantum gates and hence large scale algorithms. These control Hamil- tonians are deliberately constructed within the qubit system, such that classical control parameters directly or indirectly induce non-trivial Hamiltonian dynamics. For example, in the Kane solid-state system [Kan98], a classical potential is applied to surface gates (beneath which the spin qubit is buried), changing the hyperfine coupling between electron and nucleus. Here, the classical control does not induce a non-zero dynamical evolution but instead perturbs otherwise intrinsic system dynamics in a controlled fashion. As quantum information requires extremely low error rates to achieve successful computation, detailed knowledge of controllable system dynamics is crucial. This chapter addresses the issue of Hamiltonian characterization. I will introduce several experimental protocols which differ from standard techniques in that they are designed specifically for use in the context of large scale quantum computing. I will first briefly review the concepts related to more traditional characterization techniques, namely Spectroscopy and Tomography, and detail the reasons why these existing protocols may not be appropriate as a large scale “factory based” method for system identification. It should be noted that these two schemes are not the only methods for system characterization, protocols such as parameter estimation [Hel76, Hol82] and error detection techniques [ML06] have been introduced, which I will not review.

99 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

I will then introduce several schemes which fall under the heading of Intrinsic System Characterization. Each of these protocols are designed to characterize a small aspect of qubit systems, with each subsequent protocol “boot-strapped” from the results of the previous. Each of these methods are designed such that they only require the native machinery for standard qubit operations, can be highly automated and result in full characterization of the qubit dynamics and decoherence parameters. Sec- tion 5.2 introduces the ﬁrst experiment, designed to conﬁrm that single qubit control is isolated to a two-level system. Section 5.3 then reviews protocols that have been developed for single qubit Hamiltonian characterization, proposed in Refs. [SKO04, CSG+05, CGO+06]. Finally, in section 5.4, I discuss the interesting geometric structure of two qubit interactions and how this leads to a natural protocol for characterizing the dynamics between coupled systems.

5.1.1 Spectroscopy

Spectroscopy is the most well utilized method of identifying the Hamiltonian dynamics of quantum systems, and due to its maturity there exists an extensive body of work related to experimental methods and theoretical protocols [Cha70, SW76]. In a broad sense, the field of spectroscopy is based upon the idea of probing a quantum system with electromagnetic (EM) radiation and using photonic emission and/or absorption to determine the energy structure of the underlying system Hamiltonian. While a detailed description of spectroscopy is far too lengthy to include in this introduction, the basic idea is common throughout many techniques. The EM probe used to examine a given system can range from RF (utilized in Nuclear Magnetic Resonance), and infra-red [Lau99] to Gamma [GH95] and X-ray [Aga91] radiation, and photonic emission or absorption can be utilized depending on the material being probed. As this method is a mature technique in system identification, it has already been utilized for several quantum computing systems, including super-conducting cooper-pair boxes [NCT97] and quantum dots [OFvdW+98]. As a method for experimentalists to probe the Hamiltonian structure of small quantum systems, spectroscopy has enormous benefits. It is a reasonably simple experiment to conduct and if the probe field has a relatively large bandwidth, multiple transitions can be probed simultaneously. However, there are several

100 5.1. INTRODUCTION

disadvantages to spectroscopy that does not make the protocol amenable to automated characterization for the mass manufacturing of qubits. First, spectroscopy requires the EM probe field. Not only does this need to be generated and detected but the quantum system which is probed must be engineered such that it couples to this external field. For many systems, notably solid state qubits, this is undesirable as this external hardware is not required for standard qubit operations and incorporating these additional components simply for characterization would be difficult and ultimately unnecessary. Secondly, reconstruction of the Hamiltonian requires a very good theoretical model of the probed system. As spectroscopy only results in information related to the transition energies, the physical system has to be modeled, a trial Hamilto- nian constructed and analytical forms for the energy structure determined. This is not a potential problem if every manufactured quantum system is identical and consequently only one qubit needs to be characterized. However, in reality, mass qubit manufacturing will always suffer from imperfections and Hamiltonian dynamics will inevitably vary from qubit to qubit. Hence, not only would characterization hardware need to be separately incorporated into every manufactured qubit, but spectroscopic characterization would have to be performed on every possible two- (or multi-) qubit controlled interaction. Finally, coupling the qubit to some external EM field is an invasive process and there is no guarantee that during spectroscopic characterization, the probe field perturbs, or even damages, the normal operating dynamics of the qubit. Spectroscopic methods for system identification and characterization in quantum computing still remains the preferred method when examining prototype qubits. However, as quantum computing moves beyond the physics laboratory, characterization protocols developed specifically for mass manufacturing will inevitably take over in order to speed up the construction and characterization of controllable quantum systems.

5.1.2 State and Process Tomography

A second well known method for system identiﬁcation and characterization is state and process tomography [CN97, PCZ97, NC00, JKMW01, PR04]. As the name suggests, this method is analogous to similar techniques in medical imaging, where a 3-D image of a patient can be reconstructed by taking multiple 2-D

101 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

images from different spatial directions. For quantum computation, tomography is divided up into two separate schemes. State tomography can be traced back to Stokes [Sto52] who first determined four parameters that can be used to uniquely determine the polarization of light, which before the explosion of quantum computing was applied successfully to characterize photonic systems [SBRF93, RBdM94, WJMK01]. This general method has subsequently been refined in order to reconstruct the full density matrix of an arbitrary quantum state by examining measurement results in several different basis. The protocol requires preparing many copies of an identical quantum state and then performing repeated measurements in a “tomographically complete” set of measurement bases. In the case of an arbitrary single qubit state, three of the four Stokes parameters need to be identified, requiring the measurement in four (not necessarily orthogonal) basis. Ideally, this protocol can be used to fully reconstruct the density matrix of the system, given an arbitrary number of identically prepared states and error free measurements. In contrast, process tomography [CN97, PCZ97] is used to reconstruct the equivalent of a quantum gate. In this case, rather than identifying the density matrix of the system, the super-operator to the density matrix is reconstructed. Therefore, process tomography not only requires a tomographically complete set of measurement basis, but a tomographically complete set of input states. In general, for an N qubit system, d = 24N combinations of input/output settings are required [NC00], although this can be reduced (e.g. with ancilla assisted tomography [ABJ+03]). State and process tomography, like spectroscopy, has had significant experimental success. State tomography has been used to identify qubit states in small super-conducting systems [SAM+06], large multi-ion states [HHR+05] and color centers [HTW+06]. Process tomography has been experimentally utilized to characterize teleportation circuits and CNOT gates in NMR [NKL98, CCL01] and the linear optical CNOT gate [OPG+04]. The linear optics result was the first occasion where process tomography was utilized to characterize a two-qubit gate using pure states. Like spectroscopic methods, tomography is a useful method when attempting to characterize prototype qubit system but encounters problems when adapting the technique to an automated process for large scale manufacturing. Tomogra- phy certainly has substantial advantages over spectroscopy, no probe fields are

102 5.1. INTRODUCTION

required, the protocol is more adaptable when variations in qubit manufacturing are present and the protocol has been specifically adapted for use in quantum computation, with refinements such as ancilla assisted process tomography [ABJ+03] reducing the total number of required input states. However, there are still significant disadvantages in using tomographic techniques. Both state and process tomography require a huge amount of experimental data which scale exponentially when increasing the total number of qubits that are being characterized. Additionally, process tomography only results in reconstruction of the super-operator to the density matrix (corresponding to the actual unitary gate). This is problematic in several ways. There is no one-to-one correspondence between a unitary gate and the underlying Hamiltonian dynamics. Hence, determination of the super-operator cannot be used to reconstruct the system Hamiltonian, unless process tomography is performed for multiple unitary gates all derived from the same Hamiltonian. For certain qubit systems this is not a problem. In linear optical quantum computing the fundamental interaction is theoretically discrete (i.e. the KLM gate is designed to perform a discrete controlled phase rotation) and it is well known that a combination of these discrete gates can be utilized to construct any two-qubit interaction [DNBT02]. However, in the majority of systems, quantum gates are performed by applying a classical control pulse to the system. Here, the interaction is time-controlled directly from the Hamiltonian and therefore there is significant flexibility in arbitrary gate design. If the Hamiltonian structure is known, arbitrary two-qubit gates can be designed directly, rather than first designing and characterizing a discrete gate and using this to further construct arbitrary interactions. Additionally, more intelligent use of pulse shaping techniques [SPS05] will be impractical for quantum computing without direct knowledge of the interaction Hamiltonian. The results from process tomography do not provide any information regarding how to fix faulty gates. It was shown in [DNBT02] that when constructing arbitrary interactions from discrete gates, the total gate sequence becomes longer when the fundamental interaction generates less entanglement between qubits. Hence, if a CNOT gate is theoretically designed, but process tomography demonstrates that the experimental implementation is faulty, there is no way to determine which classical control fields should be “tweaked”. Therefore, in a similar way to spectroscopy, a good theoretical model is required in order to fix faulty

103 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

gates. A final issue when utilizing tomography is access to multiple input/output channels. The majority of qubit systems only have access to a single initialization/readout basis. In order to prepare appropriate input states and output projectors for tomography, single qubit rotations are required. This would clearly be difficult if the tomographic procedure is being utilized to characterize these dynamics. Again, this is not a problem for some systems. Linear optics achieve single qubit rotations through macroscopic wave plates which are already extremely accurate optical elements that were designed and characterized long before the advent of quantum computation. Similarly, ion traps can use spectroscopic techniques to characterize the behavior of a single qubit, after which the behavior of the entire array is essentially identical. This issue is more of a concern in the solid state where spectroscopy is not always practical and there is no guaranteed uniformity between the dynamics of every qubit. While Spectroscopy and Tomography have been extremely successful in the laboratory, the eventual commercial application of controllable quantum systems require characterization protocols that are highly automated, only requires the machinery put in place for standard qubit operations, directly reconstructs Hamil- tonian dynamics and does not require modification if manufacturing induces variations in qubit behavior. The following sections introduces several protocols which I believe satisfies these conditions.

Original Material

Section 5.2 represents original work, introducing an experimental proto- • col for single qubit conﬁnement which ﬁrst appeared in Ref. [DSO+06, DSO+07].

Section 5.3 reviews single qubit characterization techniques developed by • Cole et. al. which ﬁrst appeared in Refs. [SKO04, CSG+05, CGO+06].

Section 5.4 presents a review of the geometric structure of two-qubit inter- • action dynamics ﬁrst developed by Zhang et. al. [ZVSW03].

Section 5.4.2 presents original work illustrating the periodic nature of entan- • glement in this geometric picture and how this naturally leads to a characterization protocol for two-qubit dynamics. Initially I illustrate the protocol

104 5.1. INTRODUCTION

for a small subset of coupling Hamiltonians, speciﬁcally Ising and Heisen- berg interactions. This work was completed in collaboration with Jared Cole and ﬁrst appeared in Ref. [CDH06].

Section 5.4.3 consists of original work, introducing the generalized pro- • tocol for two-qubit Hamiltonian characterization, which ﬁrst appeared in Ref. [DCH06].

105 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

5.2 Subspace Conﬁnement: How good is your qubit?

The issue of subspace confinement for qubit systems is fundamental to the primary operating assumptions of quantum processors. The concepts of universality, quantum gate operations, algorithms, error correction and Fault-tolerant computation hinge on the precept that the fundamental quantum system is an isolated, controllable, two-dimensional system (qubit). It is well known that most of the physical realizations of qubits are in fact multi-level quantum systems, which can theoretically be confined to a two- dimensional (qubit) subspace. Important examples range from super-conducting qubits [MOL+99, NPT99, GPSW06] to atomic systems, such as cavity-coupled color centers [LH00, GOD+06, WKN01] and ion traps [CZ95]. In the former systems, a qubit is generally defined as the subspace (of the full Hilbert space) spanned by the two lowest energy states in an arbitrarily shaped potential such as the washboard potential of current-biased Josephson Junctions [CCD+88, BvDBZ03]. However, the potential number of valid quantum states within each well is not limited to two, and quantum gates, especially if sub-optimally implemented, may inadvertently populate other confined states. Similarly, in ion trap systems, a qubit is usually defined by two electronic states of an ion, either two hyperfine levels or a ground state and a meta-stable excited state. Once again there exist many other electronic levels that may be populated. Hence a more stringent definition of a qubit would consist of a two-level quantum system with classical control confined to the unitary group SU(2). The ability to initialize, operate and measure completely within the two-level subspace representing “the qubit” is vital to the successful operation of any large scale device constructed from such quantum systems. Standard quantum error correction protocols assume that the qubit system is precisely confined to the two- level subspace and that all quantum gates operate only on the qubit degrees of freedom. If poor control or environmental influences inadvertently result in non- zero population of higher levels then error correction protocols that can correct leakage errors are necessary. The issue of subspace leakage in quantum processing has been addressed in depth from the standpoint of error correction. Work by Lidar [WBL02, BLWZ05] examined the construction of Leakage Reduction Units (LRU’s), which use mod-

106 5.2. SUBSPACE CONFINEMENT: HOW GOOD IS YOUR QUBIT?

ified pulsing techniques to ensure that any unitary dynamics outside the qubit subspace can be compensated for. This technique has been adapted specifically for super-conducting systems [FPS99]. Another type of LRU uses quantum teleportation [Moc04] to map a multi-level quantum state back to a freshly initialized two-level qubit. Finally, active detection such as non-demolition measurements (which detect population in non-qubit states without discriminating between the qubit states) can be performed on the system [Pre98, GBP97, VWW05, KL03]. If an out-of-subspace detection event occurs, the leaked qubit is re-initialized or replaced. The inclusion of LRU’s based on teleportation has been investigated within the context of Fault-tolerant quantum computation [AT07] and shows that the inclusion of leakage protection does not adversely affect large scale concatenated error correction. Although these schemes are viable methods to detect and correct for improperly confined qubit dynamics, they can be cumbersome to implement and many systems admit, in principle, sufficiently confined Hamiltonian dynamics so that leakage could be expected to be heavily suppressed. For example, ion-trap qubits controlled by lasers, leakage to other ionic states can be made negligible by employing very finely tuned lasers and sufficiently long (and possibly optimally tailored) control pulses. Advances in qubit engineering may therefore allow us to eliminate or at least substantially reduce the need for laborious leakage detection/prevention schemes in many cases, provided that we can experimentally ascertain sufficiently high confinement of manufactured qubits under classically controlled Hamiltonian dynamics. This section introduces a simple generic protocol to estimate qubit confinement, or more precisely, establish bounds on the subspace leakage rates, for “quality control” purposes. The main goal is to allow us to empirically detect inferior qubits by using readily obtainable experimental data to derive tight bounds on the subspace leakage of the system. This protocol would represent one of the first steps towards full system characterization protocols, introduced throughout this chapter.

5.2.1 Motivation and Preliminaries

Estimation of qubit conﬁnement represents one of the ﬁrst major steps in qubit characterization. Therefore, the protocol should not be predicated on the avail-

107 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

ability of sophisticated measurements or control, and should be amenable to automation so that it could be used in conjunction with a potentially automated manufacturing process. The bounds on the subspace leakage will be based on the observable qubit evolution under an externally controlled driving Hamilto- nian. We assume that our classical control switches on the single qubit dynamics and that the governing Hamiltonian is piecewise constant in time. Hence the iHt Hamiltonian induces the unitary operator U(t)= e− , with ~ = 1. Although this assumption may not be applicable to all systems, e.g. systems subject to ultra-fast tailored control pulses, it is not as restrictive as it might appear. It is generally valid for systems such as quantum dots or Josephson junctions subject to external potentials created by voltage gates if the gate volt- ages are (approximately) piecewise constant. It is also a good approximation for systems subject to time-dependent fields such as laser pulses in a regime where the rotating wave approximation (RWA) is valid and the pulse envelopes can be approximated by square-waves. In this case, the Hamiltonian relevant for our purposes is the (piecewise constant) RWA Hamiltonian determined by the amplitudes, detunings and possibly phases of the control pulses. This model can even be valid for other pulse shapes if the Hamiltonian is taken to be an average Hamiltonian describing the effective dynamics on a certain time scale (beyond which we do not resolve the time-dependent dynamics). However, the main focus of this section is not when the dynamics of a system can be modeled in this way, but rather how to assess subspace confinement for systems where this model of the dynamics is valid. The driven system generally undergoes coherent oscillations, which are often referred to as Rabi oscillations, especially for optically driven systems in the RWA regime. Although our model is not limited to these systems, we shall use the terms coherent oscillations and Rabi oscillations interchangeably throughout this discussion. The measurement model assumed is crucial to the relevance of the protocol. Some standard measurement models in quantum computation assume the ability to detect both the 0 and 1 states independently (such as SET detectors in | i | i solid state charge qubits [DS00, CGJH05, AJW+01]). In this case, estimating subspace leakage is fairly straightforward and requires only repeated measurement of the system while undergoing evolution. The leakage is simply given by the deviation of the cumulative probability of measuring 1 or 0 from unity. | i | i

108 5.2. SUBSPACE CONFINEMENT: HOW GOOD IS YOUR QUBIT?

However, this measurement model is not realistic for the majority of proposed systems. Color centers and ionic qubits use externally pumped transitions to discriminate between a light state ( 0 ) and other “dark” states, while readout in ≡| i super-conducting systems [MNA02, WS05] involves lowering a potential barrier such that only one of the qubit states can leak to an external detection circuit. The measurement outcome of the indirectly probed state is inferred from the non-detection of the directly measured state and for such measurement models estimating confinement is more complicated. We utilize the latter model in order to quantify confinement. It should be noted that we are not considering the concept of weak measurement [Car99], in each case we assume that the measurement of the system causes a full POVM collapse of the wavefunction. Strong non-qubit transitions can still be identified directly via modulations in the Rabi oscillations data as shown in Fig. 5.1b for a three-state system evolving under the trial Hamiltonian,

0 10.5 Hm =  1 1 0  . (5.2) 0.5 0 1.5     However, the Rabi oscillation data for the modiﬁed three-state Hamiltonian,

0 10.01 Hn =  1 1 0  , (5.3) 0.01 0 1.5     depicted in Fig. 5.1a shows that an apparent lack of modulations in the oscillation data is not proof of perfect conﬁnement, and that quantitative measures of subspace leakage and experimental protocols are needed.

5.2.2 Estimation of subspace leakage

By deﬁning the projection operator onto a two dimensional subspace, Π = 0 0 + | ih | 1 1 , subspace leakage is given by, | ih |

ǫ =1 Tr[Πρ], (5.4) −

109 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

a) b) 1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6 0.5 0.5 0.4 0.4 Probability of measuring 0 0.3 Probability of measuring 0

0.3 0.2

0.2 0.1 0 10 20 30 40 50 0 10 20 30 40 50 time (Inverse Energy) time (Inverse Energy)

Figure 5.1: Modulations in the Rabi oscillations of a three-level system driven by the Hamiltonians, Hm and Hn. Fig. b) provides clear evidence that this system is not a qubit, while Fig. a) appears to show perfect confinement. However the analysis in the following sections will show that the subspace confinement for the system in Fig. a) is also not sufficient for large-scale QIP applications.

with ρ = U †(t) 0 0 U(t). Unfortunately, we cannot calculate ǫ directly with- | ih | out knowledge of the Hamiltonian. However, we can estimate subspace leakage experimentally from the oscillation data.

Perfect conﬁnement

Consider a general N-level system undergoing coherent evolution via a driving

Hamiltonian, HN , in the closed system case of no environmental decoherence. If conﬁnement under this Hamiltonian is perfect, HN has a direct sum decomposition,

HN = H2 2 H(N 2) (N 2), (5.5) × ⊕ − × − where H2 2 represents the control Hamiltonian conﬁned to the qubit subspace, × span 0 , 1 , the state 0 being deﬁned by the measurement, and the excited {| i | i} | i state 1 by the allowed transition. For our measurement model the observed | i Rabi oscillations have the functional form f(t) = 0 U (t) 0 2. As there is no |h | N | i| coupling between the state 0 and states outside the H2 2 subspace, we can | i × expand f(t) by diagonalizing U2 2(t) = exp( iH2 2t), × − ×

iλ0t iλ1t 2 2 iλ0t 2 iλ2t 2 f(t)= 0 A†diag e− , e− A 0 = c0 e− + c1 e− |h | { } | i| || | | | | (5.6) 4 4 2 2 iω01t iω01t = c + c + c c (e + e− ), | 0| | 1| | 0| | 1|

110 5.2. SUBSPACE CONFINEMENT: HOW GOOD IS YOUR QUBIT?

iHdt where U2 2(t)= A†e− A, A 0 = c0 0 + c1 1 , ω01 = λ0 λ1 and λj are the × | i | i | i − { } eigenvalues of H2 2. For perfect conﬁnement, H2 2 induces coherent oscillations × × between the two qubit levels at a Rabi frequency given by the diﬀerence in the eigenvalues. Taking the Fourier transform of f(t) gives,

4 4 F (ω) = FT[f(t)]=( c0 + c1 )δ(ω) | | | | (5.7) + c 2 c 2δ(ω ω )+ c 2 c 2δ(ω + ω ). | 0| | 1| − 01 | 0| | 1| 01 Conservation of probability (total population) thus implies ( c 2 + c 2)2 = c 4 + | 0| | 1| | 0| c 4 +2 c 2 c 2 = 1, and hence the heights of the two Fourier peaks for perfect | 1| | 0| | 1| conﬁnement will satisfy the relation h +2h = 1, where h = c 4 + c 4 and 0 0,1 0 | 0| | 1| h = c 2 c 2. 0,1 | 0| | 1|

Imperfect conﬁnement

If the system experiences leakage to states outside the qubit subspace then the corresponding control Hamiltonian HN can no longer be reduced to a direct sum representation [Eq. 5.5] but it can be diagonalized H = diag[ λ ], λ being d { j} { j} iHd the eigenvalues of HN , and the propagator UN (t) expressed as UN (t)= A†e− A. The Rabi data is now a linear superposition of multiple oscillations corresponding to diﬀerent transitions of the N-level system,

N 1 2 − iH t 2 2 iλat 2 2 i(λa λ )t f(t)= 0 A†e− d A 0 = c e− = c c e− − b . (5.8) |h | | i| | a| | a| | b| a=0 a,b X X and the corresponding peak heights in the Fourier spectrum can be expressed in N 1 terms of the expansion co-eﬃcients, A 0 = − c a , as, | i a=0 a| i

N 1 P − h = c 4, h = c 2 c 2, a = b. (5.9) 0 | a| a,b | a| | b| 6 a=0 X Conservation of probability leads to,

N 1 2 N 1 − − 1= c 2 = c 4 + c 2 c 2, = h + h . (5.10) | a| | a| | a| | b| 0 a,b a=0 a=0 a=b a=b X X X6 X6 and imperfect conﬁnement implies h0 +2h0,1 < 1. We see from this analysis that the subspace leakage, ǫ, is determined by the cumulative amplitudes of all

111 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

non-qubit states for a given eigenstate of HN , which can be calculated from all the peak heights in the Fourier spectrum,

h h ǫ = a,b a,c , b,c = a. (5.11) h 6 a s b,c X However, exact calculation of ǫ requires identiﬁcation of all peaks in the Fourier spectrum and knowledge of which peak corresponds to each trasition. It is therefore desirable to derive bounds on the subspace leakage that only involve a few dominant and thus easily identiﬁable Fourier peaks.

Bounds on subspace leakage

We can derive upper and lower bounds on ǫ using only the heights of the primary spectral peaks h0 and h0,1,

h +2h =( c 2 + c 2)2 + c 4 = Tr[Πρ]2 + c 4 0 0,1 | 0| | 1| | a| | a| a=0,1 a=0,1 X6 X6 (5.12) = (1 ǫ)2 + c 4. − | a| a=0,1 X6 4 Provided a=0,1 ca 1, i.e., subspace leakage is reasonably small, we obtain 6 | | ≪ a tight lowerP bound for ǫ as a function of only the two major peak heights:

h +2h (1 ǫ)2 ∴ ǫ 1 h +2h . (5.13) 0 0,1 ≥ − ≥ − 0 0,1 p The upper bound for ǫ can also be calculated quite easily. Recall that,

2 2 2 4 2 2 4 ǫ = ca = ca + ca cb ca . (5.14) | | | | | | | | ≥ | | a=0,1 a=0,1 a,b>1,a=b a=0,1 X6 X6 X6 X6 Comparison with (5.12) immediately yields,

h +2h (1 ǫ)2 + ǫ2 =1 2ǫ +2ǫ2, (5.15) 0 0,1 ≤ − − which can be solved for ǫ,

1 ǫ (1 2h +4h 1). (5.16) ≤ 2 − 0 0,1 − p 112 5.2. SUBSPACE CONFINEMENT: HOW GOOD IS YOUR QUBIT?

The other solution to Eq. (5.15) is invalid due to the asymptotic behavior of both the upper and lower bounds,

lim min(ǫ)=0, (h0+2h0,1) 1 → (5.17) lim max(ǫ)=0. (h0+2h0,1) 1 → Since the second term in (5.14) represents the heights of all the Fourier peaks not associated with the 0 1 , 0 a or 1 a transitions, for a = 1 , | i↔| i | i↔| i | i↔| i | i 6 | i for a well conﬁned system this is a very small correction to ǫ2, consequently the bound is again strong. Therefore, the subspace leakage is bounded above and below by, 1 1 h +2h ǫ (1 2h +4h 1). (5.18) − 0 0,1 ≤ ≤ 2 − 0 0,1 − Note that this doublep inequality involves onlyp the two main peaks in the Fourier spectrum, i.e., we can bound the subspace leakage without determining the heights of all peaks. For the trial Hamiltonians (5.2) and (5.3) we obtain the following bounds

0.0497 ǫHm 0.0511, ≤ ≤ (5.19) 4 4 3.9754 10− ǫ 3.9762 10− , × ≤ Hn ≤ × while the actual values of ǫHm and ǫHn are

2 4 ǫ =5.11 10− , ǫ =3.9762 10− . (5.20) Hm × Hn ×

In both cases the upper bound for ǫ equals the actual value of ǫ. This is due to the fact that both systems are of dimension three, and when estimating max(ǫ) we neglected terms of the form

c 2 c 2, (5.21) | a| | b| (a,b)=(0,1),a=b 6 X 6 which naturally vanish for a three-level system. Fig. 5.2 shows how the bounds (5.18) for ǫ converge as conﬁnement increases

113 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

(γ 0) for the test Hamiltonian, → 0 1 γ γ 11 0 0  H4 = . (5.22) γ 0 1.5 0    γ 0 0 1.7    

-3 0.16 x 10 6.86

0.14

0.12

tn 0.1 e m

e 6.79 nifn 0.08 0.0209 0.0210 o C

0.06

Upper 0.04 Actual

0.02 Lower

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 γ

Figure 5.2: Upper and lower bounds on ǫ for the four-level trial system governed by the Hamiltonian (5.22), characterized by a static coupling between the qubit states and a variable coupling γ to two higher levels. As γ 0 the subspace leakage approaches 0 and the bounds for ǫ become more accurate.→

5.2.3 Finite Sampling Fourier analysis

The previous section details how quantitative bounds on the subspace leakage can be obtained, in principle, from the Fourier spectrum of the Rabi data. However, to translate this method into a viable experimental protocol we need to consider

114 5.2. SUBSPACE CONFINEMENT: HOW GOOD IS YOUR QUBIT?

the effects of finite sampling and taking the discrete Fourier transform (DFT), which raises several issues. First, the Nyquist criterion for sampling [Bra00] must be satisfied, i.e., to avoid aliasing, some rough estimate of the Rabi period TRabi is needed to guarantee that at least two sample points are chosen per oscillation period, i.e., ∆t ≤ TRabi/2. The second issue that must be considered is the resolution of the Fourier spectrum. The frequency resolution ∆ω is given by ∆ω = 2π/tob, with tob the total observation time of the Rabi signal. If the control Hamiltonian induces a non-qubit transition with a frequency within ∆ω of the primary peak then the DFT will combine the amplitudes for qubit and non-qubit transitions in the same frequency channel thus leading to an overestimate of h0,1 and hence qubit confinement. To avoid such problems it is necessary to ensure that the total observation time tob is long enough such that non-qubit transitions are distinguishable from the primary transition. Thus, some estimates of the system parameters are required, although these do not need to be very accurate and will generally be known on theoretical or experimental grounds. Finally, the DFT has the property that a pure sinusoidal signal will approach a delta function only if there is zero phase difference between the start and the end of the observed signal. If this phase matching condition is not met then all frequency peaks will broaden. Phase matching for system identification has already been addressed for the identification of single qubit control Hamiltonians in [CSG+05] and we will follow the same approach, which essentially involves truncating the Rabi oscillation data at progressively greater values of tob such as to maximize the trial function

2F (ω ) F (ω 1) F (ω + 1) P (t )= p − p − − p , (5.23) ob F (ω 1) + F (ω + 1) p − p where F (ω) represents the amplitude of the Fourier Spectrum at frequency ω and

ωp represents the frequency of the maximum Fourier peak. The value of tob where

P (tob) is maximized represents the cut oﬀ time to the Rabi signal that produced the best phase matching for the DFT. To simulate real experiments we numerically propagate the initial state 0 , | i under the Hamiltonian H, by U(t ) = exp( it H) for discrete times t = k∆t k − k k where k = 0, 1,...,K and K∆t = tob. A single measurement at time tk is simulated by mapping the target state U(t ) 0 to 0 , 1 , where the probability k | i {| i | i}

115 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

of obtaining 0 is given by p = 0 U(t ) 0 2; the ensemble average at a single 0 |h | k | i| time tk is determined by dividing the number of zero results by the total number of repeat experiments, Ne. For the following numerical simulations we shall use the trial Hamiltonians,

010 00 110 00 H = 0 0 1.5 0 0 , (5.24) a     00 0 1.7 0   000 02     and, 0 10.01 0.005 0  110 00 H = 0.01 0 1.5 0 0 , (5.25) b     0.0050 0 1.7 0    000 02     where Ha represents a five-level system with a perfectly decoupled two-level subspace consisting of the two lowest energy states, while Hb represents a five-level system with weak coupling between the qubit sub-manifold and two of the upper levels. We only consider Hamiltonians that have couplings between the 0 state | i and higher levels, as this state is fixed by the measurement basis. We are therefore free to diagonalize the lower block of the Hamiltonian, which also helps to simplify the comparison between different systems.

The out-of-subspace coupling in Hb was chosen such that the leakage from 4 the qubit subspace, ǫ 7 10− , is small (too small to cause noticeable mod- ≈ × ulations in the Rabi oscillations) yet signiﬁcant (in fact above certain critical thresholds) for quantum computing applications. The part of the Hamiltonian governing the qubit dynamics was chosen arbitrarily and is common to all the Hamiltonians examined within this section to maintain consistency between different simulations. The accuracy of the protocol is not aﬀected by the choice of single qubit dynamics.

116 5.2. SUBSPACE CONFINEMENT: HOW GOOD IS YOUR QUBIT?

Estimating uncertainty in leakage bounds

Estimating uncertainties in the bounds for ǫ is crucial since for the majority of qubit systems it will be practically impossible to prove that the evolution of the system under a given Hamiltonian is completely confined to the SU(2) subspace, i.e., ǫ = 0. Instead, in practice, it is sufficient for quality control purposes to experimentally confirm that the leakage from the qubit subspace is below a threshold value where it can effectively be ignored, i.e., it is the upper bound max(ǫ) that is relevant. The accuracy of our estimate for max(ǫ) will be primarily limited by our ability to accurately determine the main peak heights h0 and h0,1 due to projection noise induced by the DFT. Quantifying this uncertainty is relatively straightforward. Defining the noise function, ν(ω), of the Fourier spectrum to be the amplitude of each Fourier channel excluding h0 = F (0) and h0,1 = F (ωp), the uncertainty in h0 and h0,1 is given by the standard deviation of the noise function δh = sd[ν(ω)]. From this we can derive the uncertainty associated with max(ǫ) ǫ as, ≡ u ∂ǫ 2 ∂ǫ 2 (δǫ )2 = u (δh )2 + u (δh )2 u ∂h 0 ∂h 0,1 0 01 (5.26) ∂ǫu ∂ǫu 3δh +2 δh0δh0,1 = . ∂h0 ∂h01 2√2h +4h 1 0 1 −

δǫu can be reduced by increasing the number of ensemble measurements, Ne, taken at each point in the Rabi cycle. Figures 5.3 and 5.4 show how the estimate for ǫu converges as Ne is increased for the Hamiltonians (5.24) and (5.25), respectively. It should be noted that ǫ 0, hence for each plot the lower error bars should u ≥ only extend to the zero point, but keeping the error bars symmetrical around the data point makes the convergence behavior clearer. For large values of Ne,

ǫu converges to zero for the perfectly conﬁned system governed by Ha but the 4 non-zero value 7 10− for the imperfectly conﬁned system described by H . ≈ × b The respective observation times for each Hamiltonian were chosen to be tob =

30TRabi to ensure that all peaks are resolved, i.e., there are no contributions from additional transitions present within ∆ω of the primary peak.

117 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

0.015

0.01

0.005 u 0

0 4 16 64 256 1024 4096 16384 Ne

Figure 5.3: Convergence of ǫu as the number of ensemble measurements, Ne, is increased for a system governed by the Hamiltonian (5.24), characterized by perfect subspace conﬁnement. The solid line represents the actual value of ǫu(Ha) = 0. Note error bars should only extend to zero as ǫ (H ) 0. u a ≥

Numerical tests of error bound accuracy

To test the overall accuracy of the uncertainty estimates for ǫu we can expect to obtain from realistic Rabi oscillation data, we calculated the distance between the simulated value, ǫu, and the analytical value, ǫu′ , calculated directly from the Hamiltonian using Eq. 5.16,

d(H )= ǫ (H ) ǫ′ (H ) , (5.27) k | u k − u k | where k [a, b] and δd(H ) is the error in d resulting from the error associated ∈ k with estimating ǫu(Hk). We ﬁrst calculated the distance d(Hk) and δd(Hk) for

5000 simulated runs of two known trial Hamiltonians (Ha and Hb) with ǫu′ (Ha)= 4 0 and ǫ′ (H ) 7 10− , respectively. The distributions of d(H ) for H and u b ≈ × k a Hb, with Ne = 1024 and tob as in Figs 5.3 and 5.4 are shown in Figs 5.5 and 5.6,

118 5.2. SUBSPACE CONFINEMENT: HOW GOOD IS YOUR QUBIT?

0.015

0.01

0.005 u 0

0 4 16 64 256 1024 4096 16384 Ne

Figure 5.4: Convergence of ǫu as the number of ensemble measurements, Ne, is increased for the imperfectly conﬁned system governed by the Hamiltonian (5.25). 4 The solid line represents the actual value of ǫu(Hb) 7 10− . Note error bars should only extend to zero as ǫ (H ) 0. ≈ × u b ≥ respectively. The average error 3δd(H ), k [a, b], was given by 3δd(H ) 4.92 k ∈ a ≈ × 4 4 10− , encompassing 99.9% of the data, and 3δd(H ) 5.02 10− , encompassing b ≈ × 99.8% of the data. Next we examined how the protocol behaves when simulating a large number of randomly selected multi-level Hamiltonians. For these simulations we choose N-level Hamiltonians of the form,

9 9 H = E k k + 0 1 + a 0 k + h.c. (5.28) N k| ih | | ih | k| ih | Xk=0 Xk=2 with E 0, 1, 1.5, 2, 2.4, 2.5, 2.9, 3, 3.3, 4 . The vector ~a = [a ,...,a ] was { k}≡{ } 2 9 then chosen at random in two stages. First the dimensionality of ~a is randomly selected, allowing the Hamiltonian to coherently drive any multi-level system, N [2, 3, ..., 10]. The non-zero coupling values were then randomly assigned ∈

119 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

400

350

300

250

quen 200 Fre 150 3δd(Ha)

100

0 0 1 2 3 4 5 6 −4 d(Ha) x 10

Figure 5.5: Distribution of d(Ha) for 5000 separate simulations. The average of the error, 3δd(Ha) is also shown, with approximately 99.9% found within 3σ of d = 0. such that each element of ~a was approximately two orders of magnitude less than the qubit coupling term to ensure that all of the multi-level systems had high conﬁnement. We randomly generated 5000 of these Hamiltonians and d(H )= ǫ (H ) k | u k − ǫ′ (H ) was calculated. The average (analytical) value of ǫ′ (H ) for these 5000 u k | u k 4 trial Hamiltonians was found to be ǫ (H )=1.68 10− . We then examined the u′ k × ratio, Num (d(H ) 3δd(H ) 0) R = { k − k ≤ }, (5.29) 5000 indicating the percentage of successful estimates of the subspace leakage within 3σ. This ratio was calculated to be R = 99.9%, with the conﬁnement estimates being outside the error bounds for only three of the randomly generated Hamil- tonians. These results are consistent with the expectation that approximately 99.7% of

120 5.2. SUBSPACE CONFINEMENT: HOW GOOD IS YOUR QUBIT?

400

350

300

250

quen 200 Fre 150 3δd(Hb)

100

0 0 1 2 3 4 5 6 −4 d(Hb) x 10

Figure 5.6: Distribution of d(Hb) for 5000 separate simulations. The average of the error, 3δd(Hb) is also shown, with approximately 99.8% found within 3σ of d = 0. the data should lie within 3σ of the mean and demonstrates that our methodology for characterizing subspace leakage can indeed be expected to yield accurate upper bounds on the subspace leakage in the vast majority of cases.

5.2.4 Eﬃciency of the protocol

The protocol presented in the previous section allows us to determine quantitative bounds on the subspace leakage for imperfect qubits by determining only the main peaks in the Fourier spectrum. An alternative strategy is to try to identify all peaks in the Fourier spectrum. The presence of any peaks in addition to the two main peaks is indicative of subspace leakage and a quantitative estimate of the leakage rate can be obtained by determining the heights of the additional peaks. Both approaches have potential advantages and disadvantages. The former approach requires only the identiﬁcation of the two main peaks but these need to be clearly resolved and the peak heights determined with high precision.

121 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

The latter approach does not require precise estimates of peak heights but relies on the detection of additional peaks, which for high confinement will be much smaller than the major peaks, and are likely to be difficult to discriminate from the noise floor. This raises the question which strategy is more efficient to decide if the subspace leakage for a given qubit is below a certain error threshold. To answer this question, we performed a series of numerical simulations comparing the total number of measurements required to ascertain that the lower bound on the leakage rate, ǫ = 1 h +2h > 0, within error, versus iden- l − 0 0,1 tifying a statistically significant thirdp peak in the Fourier spectrum. For the purpose of the simulations we consider the following trial Hamiltonians,

0 1 γ H3 = 11 0  , (5.30) γ 0 1.5     representing a system with a variable coupling γ to a third level, as well as the four-level system governed by the Hamiltonian (5.22) and a six-level system governed by, 0 1 γ γ γ γ 110 0 0 0  γ 0 1.50 0 0  H6 =   . (5.31) γ 0 0 1.7 0 0      γ 00 01.9 0    γ 00 0 02.2     These Hamiltonians represent systems with variable but equal coupling to between one and four out-of-subspace levels, respectively.

The lower bound, ǫl, is taken to be non-zero for a discrete data set, if the analytical value, ǫl′, of the lower bound calculated directly from the Hamiltonian exceeds six times the uncertainty, δ(ǫl), for the discrete data calculated from the simulated Fourier spectrum, i.e.,

3δh ǫl′ 6δ(ǫl) > 0, δ(ǫl)= . (5.32) − 2 h0 +2h0,1 p Six times the uncertainty in ǫl represents the total distance between the maximum and minimum possible value of ǫl (using a 3σ upper and lower conﬁdence bound) and this interval should be smaller than the analytical value, ǫl′.

122 5.2. SUBSPACE CONFINEMENT: HOW GOOD IS YOUR QUBIT?

A peak F (ω′) in the discrete Fourier spectrum is taken to be signiﬁcant if it is more than three standard deviations, δh = sd[ν(ω)], above the projection noise ﬂoor,ν ¯(ω), i.e.,

F (ω′) ν¯(ω) 3δh > 0. (5.33) − − This deﬁnition will underestimate the number of ensemble measurements required slightly as it only represents the point where the third peak is greater than at least 99.7% of the noise channels.

- - - - Using confinement equations Identifying a third peak.

2 10

1 10

.002 .0046 .0072 .0102 .0134

Figure 5.7: Number of ensemble measurements required to ascertain statistically significant subspace leakage (imperfect confinement) for the three-level system governed by (5.30) as a function of the (analytically calculated) confinement using the confinement equations (5.32) and by directly identifying the third transition peak.

For the simulations, a range of out-of-subspace coupling strengths, γ, were chosen for each of the trial Hamiltonians (5.30), (5.22) and (5.31), and the corresponding subspace leakage rate, ǫ, as well as the analytical lower bound, ǫl′, computed. For each of the Hamiltonians we then simulated experimental Rabi data and computed the discrete Fourier spectrum, the observation time in all

123 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

cases was 30 Rabi cycles. The number of ensemble measurements for the Rabi data simulations was gradually increased until a statistically signiﬁcant third peak was found or Eq. 5.32 was satisﬁed, respectively.

Fig. 5.7 shows the number of ensemble measurements, Ne, necessary to con- clude that the system is imperfect in the sense that leakage is statistically significant for the three-level system governed by (5.30) for both methods. The horizontal axis represents the analytical value of confinement ǫ′(γ). Both curves scale roughly 1/√Ne, which is consistent with the scaling of the projection noise, and hence the errors associated with estimating ǫl and detecting a statistically significant third peak. For the three-level system it is clear that confirming imperfect confinement by verifying (5.32) requires more ensemble measurements than detecting a third peak according to (5.33). This is not too surprising since for a three-level system there is only one additional transition 0 2 , and from the | i↔| i derivations of the confinement equations (5.10) we have,

N 1 2 N 1 − − 1= c 2 = c 4 + c 2 c 2 = h + h , (5.34) | a| | a| | a| | b| 0 a,b a=0 a=0 X X Xa,b Xa,b i.e., there is a conservation law for the cumulative sum of all the peak heights. Hence, if the number of possible additional peaks is small, then for a given level of confinement, the additional peaks will be greater, and thus easier to detect, than for a system with weak coupling to a large number of out-of-subspace levels, and hence many small transition peaks. We therefore conjecture that estimating subspace leakage using (5.32) will become preferable for a system with coupling to multiple out-of-subspace levels. The results of numerical simulations for the Hamiltonians (5.22) and (5.31), shown in Fig 5.8 support this conjecture. We observe the same general scaling behavior as for the three-level system. For the four-level system it is clear that although searching for the additional transition peak is still somewhat more efficient, the difference between both methods is small. For the six-level system, the curves have swapped position, i.e., using the confinement equations has become a more efficient way to ascertain statistically significant subspace leakage. From these simulations it is clear that searching for the third peak in the Fourier spectrum is only really beneficial for systems with at most one extra transition. Hence, the proposed method for estimating subspace leakage will be more efficient than obvious alternatives in most cases.

124 5.2. SUBSPACE CONFINEMENT: HOW GOOD IS YOUR QUBIT?

a) b)

3 10 - - - - Using confinement equations - - - - Using confinement equations Identifying a third peak. Identifying a third peak.

2 10

2 Ne 10 Ne

1 10 1 10

0.0025 0.0055 0.0097 0.0113 0.0147 0.002 0.005 0.0076 0.011 0.014

Figure 5.8: Number of ensemble measurements required to ascertain significant subspace leakage (imperfect confinement) for the four-level system governed by (5.22) [Fig. a] and the six-level system governed by (5.31) [Fig. b] using the confinement equations and identifying a third peak.

5.2.5 The eﬀect of Decoherence

It is well known that even if subspace leakage is theoretically suppressed for an arbitrary control field, it is unlikely that decoherence will also be suppressed. Hence, we need to examine if the proposed confinement protocol will still be effective in the open system case when a qubit is subject to decoherence, possibly of the same order, or greater, than subspace leakage. The study of arbitrary decoherence for N-level systems is a lengthy discussion, including Markovian and possible non-Markovian processes. Even for the simpler case of Markovian decoherence we would need to consider the complete N-level decoherence model with all the associated restrictions of completely positive maps [SS04]. Hence, we will instead only focus on a restricted case to show that, for a simple example, decoherence does not invalidate the protocol. It should be stressed that this only represents a preliminary analysis under a specific model of decoherence. Future work will involve investigating more complicated and system-specific decoherence effects such as N-level dephasing and spontaneous emission as well as possible system specific non-Markovian decoherence. However, due to the extremely complicated nature of such an analysis we will limit our discussion to a specific case. We consider a perfectly confined qubit which undergoes Markovian decoher-

125 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

ence and hence can be described by the quantum Liouville equation,

i 3 ∂ ρ = [H, ρ]+ Γ [ρ], (5.35) t −~ kLk Xk=1 where, [ρ] = ([L , ρL† ] + [L ρ, L† ])/2, H represents the single qubit control Lk k k k k Hamiltonian, and Lk are the Lindblad quantum jump operators, which describe the eﬀect of the environment on the system, each parameterized by some rate Γ 0. This form of the master equation corresponds to a diagonal form of the k ≥ more general equation [Bac03],

i ∂ ρ = [H′, ρ]+ a ([F , ρF †] + [F ρ, F †]), (5.36) t −~ i,j i j i j ij X where a is a positive, semi-deﬁnite, Hermitian matrix and F = σ , σ , σ . i,j { i} { x y z} The more well known equation [Eq. 5.35] is obtained by choosing a basis where ai,j is diagonal with eigenvalues Γk, corresponding to the decoherence rates for each Pauli operator. For a basic decoherence analysis we restrict the Lindblad operators to the Pauli set and consider a perfectly conﬁned, control Hamiltonian of the form,

d H = | |[d σ + d σ + d σ ]. (5.37) 2 x x y y z z

Where [d ,d ,d ] R3 and d2 + d2 + d2 = 1. This decoherence model is suﬃcient x y z ∈ x y z to describe pure dephasing as well as symmetric population relaxation processes in the diagonal Lindblad basis, but does not account for asymmetric relaxation processes. Including each Pauli Lindblad term with an associated decoherence rate is not altered by a change from the diagonal Lindblad basis, however it will act to perturb the Hamiltonian based on the basis rotation relating the computational basis to the basis where Eq. 5.36 is diagonal. We can solve the master equation under this model (in the diagonal Lindblad basis) by using the Bloch vector formalism. Expressing the density matrix as

ρ(t)= I/2+ x(t)σx + y(t)σy + z(t)σz , Eq. (5.35) takes the form ∂tS(t)= GS(t),

126 5.2. SUBSPACE CONFINEMENT: HOW GOOD IS YOUR QUBIT?

where S(t)=(x(t),y(t), z(t))T and

2(Γ +Γ ) d d d d − y z −| | z | | y G = d d 2(Γ +Γ ) d d . (5.38)  | | z − x z −| | x  d d d d 2(Γ +Γ )  −| | y | | x − x y    The Rabi oscillations under this evolution are described by the function f(t) =

Tr[AP A†ρ(t)], where P = 0 0 and A SU(2) is the rotation relating the 0 0 | ih | ∈ computational basis to the basis where the Lindblad equation is diagonal. The initial state of the system is therefore given by ρ(0) = A 0 0 A†. | ih | To solve this diﬀerential equation, we convert to Fourier space. Since the Fourier transform for a system governed by decoherence-induced semi-group dynamics is only deﬁned for t 0, we use the cosine and sine transforms, ≥

∞ [f(t); ω]= f(t) cos(ωt), C Z0 (5.39) ∞ [f(t); ω]= f(t) sin(ωt), S Z0 noting that,

∞ iωt [f(t); ω] i [f(t); ω]= f(t)e− = [f(t); ω]. (5.40) C − S F+ Z0

Taking the sine and cosine transforms of ∂tS(t)= GS(t), noting that,

[f˙(t); ω]= ω [f(t); ω] f(0), C S − (5.41) [f˙(t); ω]= ω [f(t); ω], S − S gives, ω [S(t); ω] S(0) = G [S(t); w], S − C (5.42) ω [S(t); ω]= G [S(t); w]. − C S Combining these equations and setting S(ω)= [S(t); ω] yields, F+

iωS(ω) S(0) = GS(ω), (5.43) − and hence 1 S(ω)= (G iωI )− S(0). (5.44) − − 3 where I is the 3 3 identity matrix. 3 ×

127 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

If the diagonal Lindblad basis is not the same as the computational basis then the Fourier Transform of the oscillation spectrum is given by,

1 1 f(t) = Tr[AP A†ρ]= Tr[AA†ρ]+ Tr[Aσ A†ρ] 0 2 2 z 1 1 1 1 = + n Tr[σ ρ]+ n Tr[σ ρ]+ n Tr[σ ρ] (5.45) 2 2 1 x 2 2 y 2 3 z 1 = + n x(t)+ n y(t)+ n z(t) 2 1 2 3 where we have used the fact that since σz is Hermitian and Unitary, any unitary transformation of σz must be of the form,

AσzA† = n1σx + n2σy + n3σz, (5.46) with [n , n , n ] R3 and n2 + n2 + n2 = 1. The initial condition, can be found 1 2 3 ∈ 1 2 3 as, A 0 0 A† A 1 1 A† = n σ + n σ + n σ , | ih | − | ih | 1 x 2 y 3 z A 0 0 A† + A 1 1 A† = I, | ih | | ih | (5.47) 1 n1 n2 n3 ∴ Aρ(0)A† = A 0 0 A† = I + σ + σ + σ , | ih | 2 2 x 2 y 2 z this gives, 1 1 Re[F (ω)] = δ(ω) Re[Q], (5.48) 2 − 2 where, 1 T Q =(n , n , n ).(G iωI )− .(n , n , n ) , (5.49) 1 2 3 − 3 1 2 3 and the first term corresponds to a global offset to the signal since we are measuring the observable P0. This expansion is far too lengthy to include here, however standard symbolic toolkits such as Mathematica can handle the result. The first step is to consider only the real component of Q. Next, the denominator is expanded to second order around ω = 0 or ω = d . After this, we expand ±| | the numerator and denominator, neglecting all terms of the form Γ / d and x,y,z | | smaller, assuming Γ d and being careful to note that for expansions x,y,z ≪ | | around ω = d we must keep terms of the form ωΓ / d . After simplifying ±| | x,y,z | | the expressions we find,

1 N 2 Γ h = + 3 α , 0 2 2 ω2 +Γ2 α (5.50) 1 N 2 Γ h = − 3 β , 0,1 4 (ω d )2 +Γ2 ±| | β

128 5.2. SUBSPACE CONFINEMENT: HOW GOOD IS YOUR QUBIT?

where, 2 2 2 Γα = 2[dz(Γx +Γy)+ dy(Γx +Γz)+ dx(Γy +Γz)], 2 2 2 Γβ = (1+ dx)Γx +(1+ dy)Γy +(1+ dz)Γz, (5.51)

N3 =(dxn1 + dyn2 + dzn3). Therefore, the solution to the master equation describes three Lorentzian curves centered on ω = 0 and ω = d . ±| | The mismatch between the computational basis and the diagonal Lindblad basis is interesting. The co-eﬃcients of each Lorenzian are related to the Hamil- tonian structure in the Computational basis. You can easily demonstrate this for an arbitrary rotation, A. If the Hamiltonian in the diagonal Lindblad basis is given by Eq. 5.37 and, AσzA† =(n1σx + n2σy + n3σz), then it is straightforward to show that the rotated Hamiltonian is given by,

d A†HA = | | [N σ + N σ + N σ ] , (5.52) 2 1 x 2 y 3 z where N3 =(dxn1+dyn2+dzn3). In contrast, the broadening of the Lorenzians, Γα and Γβ are related to the Hamiltonian parameters in the diagonal Lindblad basis. This is expected, since a basis rotation should not aﬀect the peak broadening caused by decoherence but should alter the respective peak heights in the Fourier spectrum. In order to describe how the maximum peak of each Lorentzian varies with

Γ we integrate h0 and h0,1 around an interval η of the peaks,

2 η η 2 N3 Γα 1 1 N3 η h0(η)= 2 2 dω + δ(ω)dω = + arctan , 2 η ω +Γα 2 η 2 π Γα Z− Z− 2 d +η 2 (5.53) 1 N3 | | Γβ 1 N3 η h0,1(η)= − 2 2 dω = − arctan . 4 d η (ω d ) +Γβ 2π Γβ Z| |− −| | Hence, under decoherence, the peak heights in the Fourier spectrum vary as a function of the integration window η and the decoherence rates Γα,β. This is consistent since as Γ 0, both arctan functions approach π/2 and h +2h = α,β → 0 0,1 1. The integration window η is analogous to frequency resolution of the Fourier transform ∆ω, while the total area of the Lorentzian is equal to the peak heights when Γx,y,z = 0. Hence for small Γx,y,z we can simply choose the resolution of the Fourier transform such that the entire Lorentzian is essentially contained within the data channel of the primary peak.

129 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

Consider the case where we wish to ensure that the subspace leakage does not exceed ζ. Using the upper bound for the subspace leakage (5.16) we have, assuming that the integration interval is approximately equal to the frequency resolution of the DFT (i.e. η ∆ω), ≈ 1 ζ = 1 2h (∆ω)+4h (∆ω) 1 , 2 − 0 0,1 − q (1 2ζ)2 +1 1 N 2 ∆ω 1 N 2 ∆ω − = + 3 arctan + − 3 arctan , (5.54) 2 2 π Γ π Γ α β π(1 2ζ)2 ∆ω − = arctan , 2 Γ where the last line assumes that Γ Γ = Γ. When the Rabi frequency is much α ≈ β greater than the inverse of the decoherence rate (as necessary for any qubit realis- tically considered for quantum information processing), then the entire Lorentzian broadening caused by decoherence will be contained within one frequency channel. Thus, Eq. (5.54) allows us to calculate the maximum frequency resolution of the Fourier transform for successful leakage estimation using our protocol. For 4 1 example, if Γ 10− s− and we wish to confirm that the subspace leakage is at ≈ 8 most ǫmax = 10− , then the resolution of the Fourier transform cannot exceed ∆f 250Hz if only the primary peak channels are used. Obviously, this re- ≈ striction on the frequency resolution can be lifted by including multiple channels around the central peak when estimating the peak area. Although the decoherence model considered is not the most general possible case for an imperfectly confined control Hamiltonian, this calculation demonstrates that the effect of decoherence does not void the protocol for estimating subspace leakage for a common decoherence model. A more detailed analysis considering a full N-level decoherence model, including the effect of spontaneous emission and absorption processes and the possibility of system-specific non-Markovian decoherence is desirable but at this stage is reserved for further investigation.

5.2.6 Conclusions

We have introduced an intrinsic protocol for quantifying the degree of subspace leakage for a realistic ‘qubit’ system. The protocol relies on very minimal theoretical assumptions regarding qubit structure and control, and utilizes a measurement

130 5.2. SUBSPACE CONFINEMENT: HOW GOOD IS YOUR QUBIT?

model that is restrictive but extremely common to a wide range of qubit systems. We have introduced a quantitative measure of subspace leakage, and shown that the discretization noise as a result of finite sampling does not limit the ability of the protocol to quantify (with appropriate error/confidence bounds) the subspace leakage for well-confined (near perfect) qubits. The ability to experimentally characterize subspace leakage to a high degree of accuracy using automated, system independent methods, which rely on the intrinsic control and measurement apparatus of the quantum device (required for standard quantum information processing) will be vital for the commercial success of quantum nano-technology. This protocol represents one of the first steps in a general library of characterization techniques that will be required as “quality control” protocols once mass manufacturing of qubit systems becomes common. Although, in this discussion, the qubit state 1 is only defined through the | i strongest transition it should be emphasized that if confinement estimates are made on multiple control fields (for example two separate Hamiltonians which induce orthogonal axis rotations), the computational 1 state must be common | i for both Hamiltonians. This is not a significant problem, since for well engineered qubits, the computational 1 state will be known on theoretical grounds. | i There are many open problems including subspace leakage estimates for systems undergoing a whole range of potential decoherence processes, quantifying confinement for multi-qubit control Hamiltonians and combining these schemes with other proposed methods for system characterization. Hopefully, in the near future, a complete set of characterization protocols will be developed which will augment large scale manufacturing techniques, allowing for efficient and speedy transition of quantum technology from the physics laboratory to the commercial sector. This confinement protocol represents the first step for intrinsic characterization. Once confinement has been confirmed we can begin characterizing single and two qubit interaction dynamics.

131 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

5.3 Single qubit dynamics

The next step required for intrinsic characterization of quantum systems is the ability to identify the controllable Hamiltonian dynamics for a single qubit. This section presents a brief review of single qubit characterization protocols which have been developed by Cole et. al. [CSG+05, CGO+06], with a detailed discussion presented in Jared Cole’s Ph.D thesis [Col07]. Section 5.3.1 examines the basic protocol for single qubit characterization, while section 5.3.2 looks at the required protocol modiﬁcations needed to incorporate Markovian decoherence. I will review these single qubit protocols without focusing on the issues related to ﬁnite sampling and omit discussions related to error analysis. Please refer to the above references for detailed calculations related to these issues.

5.3.1 Characterization without decoherence

First consider a completely general Hamiltonian, H, which is piece-wise constant in time, driving the evolution of a single qubit. This Hamiltonian can be written as a linear combination of the Pauli matrices,

d˜.σ d H = = | | [d σ + d σ + d σ ] , (5.55) 2 2 x x y y z z where [d ,d ,d ] R3 and d2 + d2 + d2 = 1. Also consider an arbitrary single x y z ∈ x y z qubit state, ψ , which can be expressed in Polar form, | i

ψ = cos(θ) 0 + eiφ sin(θ) 1 . (5.56) | i | i | i

Expressing the qubit state in this form explicitly links the state vector (˜s) on the Bloch sphere [NC00], [Fig. 5.9], parameterized by the angles θ and φ. A pure quantum state will always exist on the surface on the sphere, i.e. s = 1. The | | action of H on an arbitrary quantum state will act to rotate the Bloch vector around the axis deﬁned by (d ,d ,d )T at a rate given by d . Using polar co- x y z | | ordinates, we can re-express the Hamitonian as,

d˜ = d [d ,d ,d ]T = d [sin(θ) cos(φ), sin(θ) sin(φ), cos(θ)]T . (5.57) | | x y z | |

For a single, isolated qubit the phase, φ, is un-observable hence we are free

132 5.3. SINGLE QUBIT DYNAMICS

0 | i

s d

θ φ x

1 | i Figure 5.9: (From Ref. [CSG+05]). Geometric structure of single qubit dynamics (the Bloch sphere). Any given pure state can be parameterized by the vector ˜s =(θ,φ) where s = 1. If the state is not an eigenstate of the Hamiltonian, the evolution processes| | around the axis deﬁned by the Hamiltonian, d˜.

to set φ = 0 and align the Hamiltonian with the x-axis of the Bloch sphere. Therefore the Hamiltonian can be written as,

d H = | | [sin(θ)σ + cos(θ)σ ] . (5.58) 2 x z

Now consider the evolution of a qubit initialized in the state 0 . The mea- | i surement model utilized in this protocol assumes that the computational states, 0 and 1 are discriminable, hence the Rabi oscillations for the system evolution | i | i

133 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

are described by,

z(t) = Tr(σ ρ(t)), ρ(t)= U †(t) 0 0 U(t), (5.59) z | ih | where U(t) = exp( iHt). This can be evaluated directly, leading to, − ω t ω t z(t) = cos2( q ) + cos(2θ) sin2( q ) = cos(ω t) sin2(θ) + cos2(θ). (5.60) 2 2 q where ω = d . Therefore, determination of the oscillation frequency of z(t) will q | | give the magnitude of the Hamiltonian, while determination of cos2(θ) will give the components dx and dz. In order to fully reconstruct this Hamiltonian, we therefore need to map out the projection function z(t). This is achieved by repeatedly initializing the system in the state 0 , allowing the system to evolve for successively longer | i periods, and then measuring the qubit state in the 0 , 1 basis. As with {| i | i} qubit conﬁnement, repeated experiments will slowly build up the oscillation plot described by Eq. 5.60, with a total number of measurements given by, NT =

Netob/∆t, where Ne is the number of repeat experiments performed at each time point, tob is the total observation time for the Rabi signal and ∆t is the discrete time interval used for sampling. Fig. 5.10 illustrates the process. Note that this experiment is quite standard in quantum control and is the same data set that we use in order to quantify qubit conﬁnement [Section 5.2]. Once the Rabi data has been mapped out, we again take the data set into Fourier space, in the ideal case where we map out z(t) precicely, the Fourier Transform will have the following functional form,

π F (ω)= F T [z(t)] = √2π cos2(θ)δ(ω)+ sin2(θ)δ(ω ω ). (5.61) 2 ± q r The magnitude of the Hamiltonian is consequently given by the location of the 2 δ-function peak in Fourier space, while cos (θ) and hence (dx,dy) can be found from the height of either peak. i.e,

2F (ωq) F (0) d = ωq, dx = , dz = . (5.62) | | r π r 2π

This analysis is the basics of how the Hamiltonian dynamics of a single qubit can be estimated from the Rabi spectrum. There are clearly many practical issues

134 5.3. SINGLE QUBIT DYNAMICS

Init. H(∆t) Meas.

Init. H(2∆t) Meas.

Init. H(3∆t) Meas.

Init. H(4∆t) Meas.

+1 z(t)= σ h zi −1

Figure 5.10: (From Ref. [CSG+05]). Experimental protocol for a standard qubit oscillation experiment. The system is repeatedly initialized in the state 0 , allowed to evolve for time ∆t and then measured. This is repeated many times| i to determine σz . This process is then repeated at successively longer time intervals, until theh i oscillation signal is mapped out. related to implementing this protocol including projection noise introduced by the

Fourier Transform, the total number of ensemble measurements, Ne, needed to accurately estimate the Hamiltonian parameters and estimating the phase angle, φ which is set to 0 for an isolated qubit, but becomes important when considering the characterization of multiple qubits. However, I direct the interested reader to references [CSG+05, Col07] where these issues are covered in much more depth.

5.3.2 Characterization in the Presence of decoherence

As with the characterization of conﬁnement, we cannot neglect the possibility that an otherwise ideal qubit suﬀers from decoherence caused by environmental coupling. Once again, introducing decoherence does not void the protocol and can actually be used to not only estimate Hamiltonian dynamics, but also decoherence

135 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

rates. This section briefly reviews the general protocol introduced by Cole et. al. in Refs. [CGO+06, Col07] As with the confinement calculations introduced in section 5.2, we simply consider decoherence effects which can be modeled via the Lindblad master equation. Non-Markovian decoherence channels in specific systems will need to be treated in a more detailed way and will constitute the next generation in single qubit characterization protocols which we will hopefully develop in the near future. The Master equation for a single qubit undergoing Makovian decoherence was introduced in section 5.2 [Eq. 5.35] and takes the form,

i 3 ∂ ρ = [H, ρ]+ Γ [ρ]. (5.63) t −~ kLk Xk=1

Pure dephasing

We ﬁrst begin with environmental dephasing. This is generally acknowledged to be the most common form of noise for a variety of systems, especially solid-state qubits. To model dephasing, we use the Pauli set for the Lindblad operators, L = σ , σ , σ , with associated rates, Γ = Γ , Γ , Γ . { k} { x y z} { k} { x y z}

Substituting only Lz = σz into Eq. 5.63 assumes that the Lindblad equation [Eq. 5.63] is diagonal in the computational basis and acts to exponentially dampen the coherences (the oﬀ-diagonal elements in ρ) between the two basis states.

For the case of pure σz dephasing in the computational basis, we expect that the Rabi signal will decay exponentially. The speciﬁc eﬀect this has on the Rabi signal depends on the form of the driving Hamiltonian. Recall from the previous section that we can write a general single qubit Hamiltonian as,

d H = | | [sin(θ)σ + cos(θ)σ ] . (5.64) 2 x z

In the case where θ = π/2, the Hamiltonian drives a pure σx rotation on the Bloch sphere ( 1 z(t) 1) and the introduction of dephasing induces exponential − ≤ ≤ decay of the Rabi signal to the ﬁnal mixed state ρ( )=( 0 0 + 1 1 )/2. ∞ | ih | | ih | However, when θ = π/4 the system evolution will only oscillate between the points 0 z(t) 1. When characterizing both Hamiltonian evolution and decoherence ≤ ≤ these eﬀects need to be accounted for. Fig. 5.11 illustrates several numberical

136 5.3. SINGLE QUBIT DYNAMICS

simulations from Ref. [CGO+06], showing the exponential decay of the Rabi signal induced by only σz dephasing at several diﬀerent rates. Fig. 5.11a illustrates for H = d σ /2, while Fig. 5.11b illustrates when θ = π/4 in Eq. 5.63. | | x 1 Γ/d=1/1000 (a) Γ/d=1/100 0.5 Γ/d=1/10 0 z(t)

−0.5

−1 1 Γ/d=1/1000 (b) Γ/d=1/100 Γ/d=1/10 0.5 z(t)

0 0 5 10 15 time [units of 2π/d]

+ Figure 5.11: (From Ref. [CGO 06]). Damped oscillation signal caused by σz decoherence of a single qubit. Fig. a) illustrates the case where the driving Hamiltonian is pure σx while Fig. b) illustrates when θ = π/4 in Eq. 5.63. In both cases, the steady state solution is z( )=0. ∞ In section 5.2 we calculated the eﬀect of decoherence on the evolution of a perfectly conﬁned qubit using the each of the three Pauli Lindblads, each acting with independent rates. As expected, we found that the two δ-function peaks in the Fourier spectrum broadened into Lorenzians,

Γ sin2(θ) Γ F (0) cos2(θ) α , F ( d ) β , (5.65) → ω2 +Γ2 ±| | → 2 (ω d )2 +Γ2 α ±| | β where, 2 2 Γα = 2[cos (θ)(Γy +Γx) + sin (θ)(Γy +Γz)], (5.66) 2 2 Γβ =Γx[1 + sin (θ)]+Γy +Γz[1 + cos (θ)], and Γ , Γ , Γ are the decoherence rates associated with the three Pauli Lind- { x y z} blads σ , σ , σ . { x y z}

137 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

Note that these equation differ from those in section 5.2 in that we assume the diagonal Lindblad basis coincides with the computational basis and we are measuring the observable σz, hence there is no δ(ω)/2 offset to the zeroth order Fourier peak. The functional forms for the Fourier peaks now gives us a general procedure for estimating both the Hamiltonian parameters and decoherence parameters Γα and Γβ. The Fourier transform of the Rabi data is taken and the peak position will give us and approximate value for d /2. We then use the analytic forms of | | Eq. 5.65 and perform a non-linear fit on the data to obtain the parameters d /2, | | θ, Γα and Γβ. Unfortunately, characterization of a single driving Hamiltonian does not allow us to determine all three, independant, decoherence rates, Γx, Γy and Γz, especially if the Lindblad equation is not diagonal in the computational basis. However, from the standpoint of large scale qubit applications, the decoherence rates need only be estimated to ensure that they are below appropriate thresholds for concatenated error correction. Therefore, since Γα and Γβ are both linear in

Γx, Γy and Γz, estimating these terms is suﬃcient.

Spontaneous emission/absorbption

The characterization of single qubit dynamics under a more realistic model requires the addition of Lindblad terms that model spontaneous emission and absorption. These processes require the addition of the jump operators,

L = Γ σ , (5.67) ± ± ± p where σ = 0 1 is the lowering operator, corresponding to spontaneous emission + | ih | and σ = 1 0 is the raising operator, corresponding to spontaneous absorption. − | ih | In this general case, we write the non-unitary part of the master equation as,

j[ρ]=Γz z[ρ]+Γ+ +[ρ]+Γ [ρ]. (5.68) L L L −L− j X This model assumes that each qubit undergoes pure σz dephasing (in the computational basis) as well as spontaneous emission/absorption, each with potentially independent rates.

138 5.3. SINGLE QUBIT DYNAMICS

Incorporating this type of decoherence model into the characterization equations requires us to re-deﬁne the Bloch terms as, x′(t) = x(t) x( ), y′(t) = − ∞ y(t) y( ), z′(t) = z(t) z( ) and the initial conditions as, x′(0) = x( ), − ∞ − ∞ − ∞ y′(0) = y( ) and z′(0) = z( ). Solving the master equation gives (in the − ∞ − ∞ steady state limit), 2d cos(θ)y( ) x( )= ∞ , ∞ 4Γz +Γ+ +Γ − y( )= Kz( ), (5.69) ∞ ∞ Γ+ +Γ z( )= − , ∞ Γ+ +Γ + d sin(θ)K − where, 2d sin(θ)(4Γz +Γ+ +Γ ) − K = 2 2 2 . (5.70) 4d cos (θ)+(4Γz +Γ+ +Γ ) − The solution of the Bloch equations, in the Fourier domain, is given by,

(CF +Γ+)(1 z( )) Γ (1 + z( )) d sin(θ)[L(ω)+ L∗( ω)] Z′(ω)= − , − ∞ − ∞2d2M−sin2(θ) − iω +Γ +Γ+ + 2 2 2 − M +4d cos (θ) (5.71) where, (C iω)[y( )+ ix( )] dz( ) sin(θ) L(ω)= F − ∞ ∞ − ∞ , (5.72) M 2id cos(θ) − M =2iω + 4Γz +Γ+ +Γ , (5.73) −

L∗(ω) is the complex conjugate of L(ω) and CF is a constant of integration resulting from the fact that z′(0) = 0. As z( ) is constant Z′(ω > 0) = F T [z′(t)= 6 ∞ z(t) z( ),ω > 0] = Z(ω > 0) and Z′(0) = Z(0) + z( ) hence this algebraic − ∞ ∞ solution can be used as a fitting function for the oscillation spectrum. Depending on the specific decoherence channels, discriminating the cases of spontaneous emission/absorption and dephasing can be difficult in the Fourier domain (especially if dephasing is dominant). However there are techniques that can be used to simplify the curve fitting process for the above equations by appealing to the physics of the system [Col07, CGO+06].

5.3.3 Conclusions

This section has provided a simple introduction to single qubit characterization protocols with and without decoherence. As this section is a review of previous work I have not included details regarding ﬁnite Fourier analysis, the error analy-

139 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

sis of the characterization procedure and subtle issues relating to characterization in the presence of both dephasing and spontaneous emission/absorption. The in- tent of this section is to show that each characterization protocol is effectively bootstrapped from the previous protocol. In this case, qubit confinement is first estimated, then single qubit dynamics and decoherence. The next section details the protocol required to characterize two-qubit interaction dynamics assuming that single qubit dynamics have been characterized and hence we have full single qubit control.

140 5.4. CHARACTERIZATION OF TWO QUBIT DYNAMICS

5.4 Characterization of two qubit dynamics

In the context of quantum information processing, the system Hamiltonian is used to design quantum gate operations. A large amount of work has already been completed regarding the accurate construction of two qubit gates [BDD+02, DNBT02, VD04, ZW05, ZVSW03], this work can be split into two broad categories. The first assumes that the quantum computing system exhibits a discrete, well defined, two-qubit interaction. This occurs for architectures such as linear optics systems. These discrete two-qubit operations are traditionally charac- terised using state and process tomography where the interaction Hamiltonian is initially assumed on well founded theoretical and/or experimental grounds and once characterized, used to build arbitrary two-qubit gates. The second category is exemplified by solid state architectures. Here, two qubit interactions exhibit time dependent control where unitary gates, U(t), can be specified by the interaction Hamiltonian H, U(t) = exp( iHt), where ~ = 1. − State and process tomography can be performed, but in general a discrete gate has to be designed beforehand that can produce entangled states (for example a CNOT). However, it is much more efficient to design interactions directly, using the Hamiltonian. In this section, I present a generalized method for two-qubit characterization. It will be shown how mapping the entanglement of the two qubit system gives enough information to not only determine the entangling properties of a given interaction, but to fully reconstruct the Hamiltonian. I will provide a detailed review of the geometric structure of two-qubit interactions in section 5.4.1, showing how the Cartan decomposition of the group SU(4) leads to an elegant description of all two-qubit gates. In sections 5.4.2 and 5.4.3 I will demonstrate how the entanglement generated by two-qubit interactions can be used to fully reconstruct the Hamiltonian dynamics for a large class of systems. Finally, in section 5.4.4 I apply this general method to characterise a trial Hamiltonian describing anisotropic spin-orbit correction in GaAs quantum dots [SBD+03, BSD01, Kav01].

141 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

5.4.1 Structure of two-qubit dynamics

Considerable effort has been focused on understanding the non-local properties of two-qubit quantum states and operations. The general structure of two-qubit dynamics has been studied extensively, leading to an interesting structure to the group geometry of two-qubit interactions. Here I review the geometric structure of two-qubit operations and demonstrate how this structure leads to an intuitive way to experimentally characterize two-qubit control Hamiltonians. Makhlin [Mak02] was one of the first to examine the entangling properties of two-qubit interactions by deriving locally invariant quantities for any two-qubit gate, while Kraus and Cirac [KC01] focused on finding the best separable two- qubit state such that a given gate produces a maximally entangled state. Later Zhang et.al. [ZVSW03] introduced the geometric theory of non-local two-qubit interactions which gave an elegant geometric picture of all two-qubit gates. It is this interpretation that I will focus on since it leads to an intuitive method for characterizing two-qubit interaction Hamiltonians. The geometric picture of two-qubit interactions is based on the Cartan decomposition of su(4), the Lie algebra of the SU(4) special unitary group. More detailed descriptions of the Cartan decomposition can be found in references [Hsi98, Hel78]. We can restrict the study of all two-qubit gates to SU(4) since it is well known that all two-qubit interactions are members of the unitary group U(4), which is a product of the gate G SU(4) and a un-obsevable global phase shift, ∈ eiα. It is known that SU(4) can be partitioned into three separate subspaces. The first partition is between gates which only act locally on both qubits, i.e. G SU(2) SU(2) and gates which act coherently on both qubits, G 1 ∈ ⊗ 2 ∈ SU(4) SU(2) SU(2). Within the second subspace we can make a further par- \ ⊗ tition, into gates which can produce a maximally entangled Bell state from an initially un-entangled state (for example a CNOT gate), which are denoted perfect entanglers, and gates which cannot produce maximally entangled states (for example, the two-qubit SWAP), denoted imperfect entanglers. Fig. 5.12 illustrates the partitioning, which is taken from Ref. [ZVSW03]. Without detailing the group theory of Cartan decompositions, the Lie algebra

142 5.4. CHARACTERIZATION OF TWO QUBIT DYNAMICS

SWAP Local Gates SU(2) SU(2) ⊗ Non-Perfect Entanglers

Non-local Gates SU(4) SU(2) SU(2) \ ⊗

Perfect Entanglers

CNOT

Figure 5.12: (From Ref. [ZVSW03]) Subgroups of all possible two-qubit gates. Local gates induce isolated single qubit operations on each qubit. Non-Perfect entanglers (e.g. SWAP) cannot prepare a maximally entangled Bell pair from a product state, while Perfect entanglers (e.g. CNOT) are able to prepare Bell states from initial, product states.

of SU(4), g = su(4) has a direct sum decomposition, g = p l, where, ⊕ i l = span IX,IY,IZ,XI,YI,ZI , 2{ } (5.74) i p = span XX,YY,ZZ,XY,XZ,YX,YZ,ZX,ZY . 2{ } where X,Y,Z are the Pauli matrices, I is the 2 2 identity matrix and the { } × tenor product is implied. The subgroup l represents local dynamics while p represents the non-local couplings between the two qubits. The commutation relations between each subgroup is,

[l, l] l, [p, l] p, [p, p] l. (5.75) ∈ ∈ ∈ hence the decomposition g = p l represents a Cartan decomposition of su(4). ⊕

143 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

Within the non-local group, p, the largest Abelian subgroup is spanned by,

i a = span XX,YY,ZZ . (5.76) 2{ }

Therefore, in accordance with Proposition 1 in Ref. [ZVSW03], any element G ∈ SU(4) can be written as,

i G = K G K = K exp (c XX + c YY + c ZZ) K , (5.77) 1 d 2 1 {2 1 2 3 } 2 where l is the Lie algebra generating the group K ,K K and [c ,c ,c ] { 1 2}∈{ } 1 2 3 ∈ 3 R . Since l only contains local terms acting on a single qubit, the matrices K1 and K are elements of SU(2) SU(2). 2 ⊗ What the Cartan decomposition of su(4) implies is that any two-qubit interaction can be speciﬁed via the non-local coupling matrix, Gd, and therefore the vector [c1,c2,c3]. Any two gates G1,G2 that exhibit the same vector decomposition, i.e. G1(c1,c2,c3)= G2(c1,c2,c3) are deﬁned as locally equivalent, since they are related through only local operations to each qubit.

Calculating a given gate decomposition.

In order to decompose an arbitrary gate G SU(4) we ﬁrst perform a basis ∈ transformation taking,

Gb = Q†GQ = Q†K1QQ†GdQQ†K2Q, (5.78) where, 10 0 i 1 0 i 1 0  Q = = . (5.79) √ 2 0 i 1 0   −  10 0 i  −    Recall that the local terms are generated by the algebra,

i l = IX,IY,IZ,XI,YI,ZI , (5.80) 2{ } hence in the new basis, Q†KQ is speciﬁed by the algebra Q†(l)Q, which generates the group SO(4), consisting of all 4 4, orthogonal matrices with determinant × one. Hence Q†K Q = O SO(4) for i [1, 2]. Additionally, the matrix G is i i ∈ ∈ d

144 5.4. CHARACTERIZATION OF TWO QUBIT DYNAMICS

i generated by the algebra a = XX,YY,ZZ and therefore Q†G Q is generated 2 { } d by, i i Q† a Q = Q† XX,YY,ZZ Q = ZI, IZ,ZZ , (5.81) { } 2 { } 2{ − } giving,

i F = Q†G Q = exp (c ZI c IZ + c ZZ) = d {2 1 − 2 3 } i (c1 c2+c3) e 2 − 0 0 0 i (c1+c2 c3) (5.82)  0 e 2 − 0 0  i . (c1+c2+c3)  0 0 e− 2 0  i  ( c1+c2+c3)  0 0 0 e 2 −      T Next, we examine the matrix, m = Gb Gb which can be expressed as,

T T T 2 m = Gb Gb =(O1FO2) (O1FO2)= O2 F O2, (5.83) as, OT O = II, where II is the 4 4 identity matrix. Therefore, diagonalizing 1 1 × m will result in eigenvalues,

i(c1 c2+c3) i(c1+c2 c3) i(c1+c2+c3) i( c1+c2+c3) λ1 = e − , λ2 = e − , λ3 = e− , λ4 = e − , (5.84) T and eigenvectors which form the rows of O2. Next we deﬁne O1 = GbO2 F ∗, such T that O FO = G O F ∗FO = G . This ensures that O SO(4) as, 1 2 b 2 2 b 1 ∈

T T T 2 T O1 O1 = F ∗O2Gb GbO2F ∗ = F ∗O2O2 F O2O2 F ∗ = F ∗F F F ∗ = II. (5.85)

Therefore, for a given two-qubit gate, G, calculation of the local operations K1,K2 is straight-forward. Calculating the non-local properties, speciﬁed by the vector

[c1,c2,c3] is achieved by looking at the eigenvalues of m. Since det(Gb) = 1, det(m) = 1 and hence j λj = 1. Expressing the eigenvalues of m as λj = exp(iǫ ) sets the condition ǫ =2nπ, where n Z. j Q j j ∈

Recall that the eigenvaluesP of m are related to the vector [c1,c2,c3] as,

iǫ1 i(c1 c2+c3) e 0 0 0 e − 0 0 0

iǫ2 i(c1+c2 c3)  0 e 0 0   0 e − 0 0  = . iǫ3 i(c1+c2+c3)  0 0 e 0   0 0 e− 0   iǫ4   i( c1+c2+c3)  0 0 0 e   0 0 0 e −         (5.86)

145 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

Equating these two matices gives,

ǫ =(c c + c ), 1 1 − 2 3 ǫ2 =(c1 + c2 c3), − (5.87) ǫ =( c c c ), 3 − 1 − 2 − 3 ǫ =( c + c + c ). 4 − 1 2 3

Assuming the eigenvalues satisfy j ǫj = 0 (this can be enforced by subtracting 2π from the n largest eigenvalues, of F 2, if n = 0), these equations can be inverted P 6 to give, 1 c = (ǫ + ǫ ), 1 2 1 2 1 c = (ǫ + ǫ ), (5.88) 2 2 2 4 1 c = (ǫ + ǫ ). 3 2 1 4

Hence the decomposition, G = K1GdK2 is now complete by returning Gb =

O1FO2 to the computational basis.

Geometric structure of SU(4).

Now that we have the decomposition for an arbitrary two-qubit gate, it can be related to an elegant geometric structure. As we have seen, all two-qubit gates with the same vector [c1,c2,c3] are related to each other via local rotations. Hence the more interesting structure of SU(4) is related to the triplet, [c1,c2,c3]. The ﬁrst thing of note is that this triplet is locally periodic in each parameter. i.e, π G (c ,c ,c )= iXXG (c ,c ,c ), d 1 2 3 ∓ d 1 ± 2 2 3 π G (c ,c ,c )= iYYG (c ,c ,c ), (5.89) d 1 2 3 ∓ d 1 2 ± 2 3 π G (c ,c ,c )= iZZG (c ,c ,c ). d 1 2 3 ∓ d 1 2 3 ± 2

Hence, two-qubit gates that are related by π/2 shifts in [c1,c2,c3] are locally equivalent. Therefore the geometry deﬁnes a 3-Torus, which is periodic in all three coupling parameters. We can therefore immediately deﬁne a unit cell [Fig. 5.13], for the inequality 0 [c ,c ,c ] π/2. Each point within this cubical geometry ≤ 1 2 3 ≤ will correspond to a local-equivalence class. However, each point is not unique and consequently this cubical geometry is degenerate. This will become clearer

146 5.4. CHARACTERIZATION OF TWO QUBIT DYNAMICS

π/2

π/2 O c π/2 2

Figure 5.13: Basic cubical geometry of SU(4). For any given gate, the coupling triplet [c1,c2,c3] is periodic in π/2 and hence deﬁnes a 3-Torus. However, representing this geometric structure as cubical will more naturally lead to a method for characterizing Hamiltonian dynamics.

when we specify the location of several well known two-qubit gates in the next section. To restrict the geometry to a non-degenerate structure can be done by observing that the following local operations permute the triplet, [c1,c2,c3].

Gd(c2,c1,c3)= P12Gd(c1,c2,c3)P12† ,

Gd(c3,c2,c1)= P13Gd(c1,c2,c3)P13† ,

Gd(c1,c3,c2)= P23Gd(c1,c2,c3)P † , 23 (5.90) G ( c , c ,c )= M G (c ,c ,c )M † , d − 2 − 1 3 12 d 1 2 3 12 G ( c ,c , c )= M G (c ,c ,c )M † , d − 3 2 − 1 13 d 1 2 3 13 G (c , c , c )= M G (c ,c ,c )M † , d 1 − 3 − 2 23 d 1 2 3 23

147 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

where, i π Z i π Z P12 = e 4 e 4 , π ⊗ π P = ei 4 Y ei 4 Y , 13 ⊗ i π X i π X P23 = e 4 e 4 , ⊗ (5.91) i π Z i π Z M = e 4 e− 4 , 12 ⊗ i π Y i π Y M = e 4 e− 4 , 13 ⊗ i π X i π X M = e 4 e− 4 , 23 ⊗ represent purely local operations on the two qubits. These permutations in the triplet allow for the cubic structure shown in Fig. 5.13 to be split up into 24 equivalent chambers, commonly referred to as Weyl chambers. Each point, [c1,c2,c3], within the chamber now corresponds to a single, non-degenerate, local-equivalence class. While each non-degenerate Weyl chamber can be reached, from any other chamber, by performing the local operations in Eq. 5.91. Fig 5.14 illustrates the Weyl chamber corresponding to the conditions, c c c 0,c + c π/2 1 ≥ 2 ≥ 3 ≥ 1 2 ≤ and c π/4 if c = 0. 1 ≤ 3

π/2

π/2 O π/2 π/2 c2

Figure 5.14: Illustration of one of the 24 degenerate Weyl chambers located within each unit cell representing the geometry of SU(4). In each Weyl chamber, every individual point corresponds to a unique local equivalence class. By utilizing the local operations shown in Eq. 5.91, the triplet [c1,c2,c3], describing a gate, can be swapped between non-degenerate chambers.

148 5.4. CHARACTERIZATION OF TWO QUBIT DYNAMICS

Entangling properties of two-qubit gates

Using this geometric representation of two-qubit gates and the Cartan decomposition, we are now in a position to classify the entangling properties of two-qubit interactions. First we require a quantifiable measure of entanglement and for two- qubits this quantity is well defined. For a general two-qubit state, φ =(α, β)T , | i entanglement is quantified via the squared concurrence [Woo98], defined as,

2 2 C = φ∗ YY φ (5.92) |h | | i|

2 where φ∗ =(α, β) is the transpose of C varies between 0 (for an un-entangled h | product states) and 1 (for a maximally entangled Bell state). Note that C2 does not represent a natural observable of the system, however it can still be measured by using multiple single qubit basis measurements. This is explained in section 5.4.2. Notice, that the concurrence is deﬁned with regards to a two-qubit state, φ . | i To quantify the entangling properties of a gate, we instead use the deﬁnition,

2 T 2 C (G) = max ψ ψ∗ G YYG ψ , (5.93) | i|h | | i| where ψ is an arbitrary un-entangled state of two-qubits and the concurrence | i is maximized over all possible product states. Using the decomposition of G, we can rewrite C2(G) as,

2 2 T T T 2 C (G)= C (K1GdK2) = max ψ ψ∗ K2 Gd K1 YYK2GdK1 ψ . (5.94) | i|h | | i|

Once again we make the basis transformation

2 T T T 2 C (G) = max ψ ψ∗ Q O1 FO2 Q†YYQO2FO1Q† ψ , (5.95) | i|h | | i| and using the relation that Q†Y Y Q = II, this reduces to, −

2 T T 2 2 C (G) = max ψ ψ∗ Q O1 F O1Q† ψ . (5.96) | i|h | | i|

Note, that since ψ is a product state and K SU(2) SU(2), and O = | i 1 ∈ ⊗ 1 2 Q†K Q, the state ψ = O Q† ψ is also a product state. As C (G) is evaluated 1 | 2i 1 | i

149 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

by maximizing C2 over all possible input states, we can rewrite Eq. 5.96 as,

4 2 2 2 2 2 C (G) = max ψ2 ψ2∗ F ψ2 = max ψ2 γkλk , (5.97) | i|h | | i| | i k=1 X

2 where λk are the eigenvalues of F and,

+ + ψ = γ Φ + γ Φ− + γ Ψ + γ Ψ− , | 2i 1| i 2| i 3| i 4| i + Q† 00 = Φ , | i | i Q† 01 = Φ− , (5.98) | i | i + Q† 10 = Ψ , | i | i Q† 11 = Ψ− . | i | i Using this deﬁnition, we can examine the entangling power of some well known two-qubit gates. First, let us consider the only two gates which are known to generate no entanglement between qubits, namely the Identity operation and the two-qubit SWAP gate. The Cartan decomposition for the Identity operation is trivial and corresponds to the triplet [c1,c2,c3] = 0, while the decomposition for the SWAP operation is,

i π SWAP = K exp (XX + YY + ZZ)K . (5.99) 1 2 4 2

π π π Hence the SWAP operation is deﬁned by the triplet, [c1,c2,c3] = [ 4 , 4 , 4 ]. Fig. 5.15 illustrates the geometric structure of non-entangling two-qubit operations using the degenerate cubical geometry. First notice the elegant Body-Centered-Cubic (BCC) lattice structure of this unit cell. Also notice that while the Identity operation has 8 degenerate points in the unit cell, the SWAP operation only occupies the center and is hence a common point with all, 24 non-degenerate Weyl chambers. Next we can examine the geometric structure of two-qubit gates which can generate a maximally entangled Bell state. The two most common gates which are maximally entangling are the CNOT gate, and the √SWAP gate which is deﬁned as, 10 00 1 iπ/4 1 iπ/4 0 √ e √ e− 0 √SWAP = 2 2 . (5.100) 0 1 e iπ/4 1 eiπ/4 0  √2 − √2    00 01     where (√SWAP).(√SWAP) = SWAP. Performing the Cartan decomposition for

150 5.4. CHARACTERIZATION OF TWO QUBIT DYNAMICS

π 2 Identity Swap

π O 2 π c2 2

Figure 5.15: The location of the two, non-entangling gates within the geometry of SU(4). These gates deﬁned a Body Centered Cubic (BCC) unit cell with eight degenerate points representing the Identity operation. Note that within each cell there is only one point corresponding to the local equivalence class [SWAP] at the center.

both of these gates gives the triplets,

π π π π CNOT[c ,c ,c ]= , 0, 0 , √SWAP[c ,c ,c ]= , , . (5.101) 1 2 3 4 1 2 3 8 8 8 Table 5.1 summarizes the entangling properties of these four gates, as well as the values of the triplet, [c1,c2,c3]. Fig. 5.16 illustrates the geometric location of these gates in the unit cell, where we have indicated all degenerate points which can be reached by applying the local permutations shown in Eq. 5.91. Once again, the geometric structure of these gates are quite elegant. The CNOT gate represents a Face-Centered-Cubit (FCC) lattice, while the √SWAP gate exhibits a smaller cubic structure contained within the larger unit cell. Also notice, that the √SWAP gate lies along the diagonal, half way between the two non-entangling gates, Identity and SWAP, while the CNOT gate lies at the midpoint between two Identity gates.

151 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

2 Gate c1 c2 c3 C CNOT π/4 0 0 1 √SWAP π/8 π/8 π/8 1 Identity 0 0 0 0 SWAP π/4 π/4 π/4 0

2 Table 5.1: Triplet values, [c1,c2,c3], and concurrence values, C , for some well known quantum gates. The SWAP and Identity gates represent non-entangling operations, while the √SWAP and CNOT gates are able to prepare Bell states from un-entangled product states.

c3 π 2 SWAP CNOT

π O 2 π c2 2

Figure 5.16: Locations of two common, maximally entangling, gates in the degenerate cubical geometry of SU(4). The [CNOT] equivalence class deﬁnes a Face Centered Cubit (FCC) unit cell geometry while the [√SWAP] equivalence class deﬁnes a small cubic geometry within the larger structure.

5.4.2 Periodic Entanglement and Two-qubit Hamiltonian Characterization.

The previous analysis examined the geometric structure of certain two-qubit gates, next we can link this geometry to the unitary evolution generated by a

152 5.4. CHARACTERIZATION OF TWO QUBIT DYNAMICS

general two-qubit interaction Hamiltonian. Initially I will focus on a certain subset of two-qubit interactions which conceptually simpliﬁes how the periodic structure of the SU(4) geometry leads to correctly identifying the coupling Hamil- tonian. The protocol developed in this section was in collaboration with Jared Cole and the discussion related to the characterization of Heisenberg and Ising Hamiltonians ﬁrst appeared in his Ph.D thesis [Col07]. Later in section 5.4.3 I will introduce the general protocol which covers a much larger class of interaction dynamics. Consider a general two qubit Hamiltonian of the form,

J J J H = 1 XX + 2 YY + 3 ZZ, (5.102) 2 2 2 representing a general Heisenberg coupling with constants [J ,J ,J ]/2 R3. 1 2 3 ∈ This form of Hamiltonian is initially considered for two reasons. First, many solid-state quantum computing proposals rely on this type of interaction as this general form covers a large class of systems including spin systems [Kan98, LD98] and pseudo spin systems such as charge qubits [MSS01, HDW+04]. Secondly, any unitary evolution generated by this form of Hamiltonian has a simple Cartan decomposition, where the more general local terms (K1 and K2) are simply the identity operations. Therefore, the geometric location of all unitary gates generated by this Hamiltonian is speciﬁed directly by the coupling triplet, [J1,J2,J3]/2. How does the evolution generated by H relate to the gate location? Since H is piecewise constant in time, the unitary gate generated by H is simply,

i U(t) = exp iHt = exp (J tXX + J tY Y + J tZZ) . (5.103) {− } {−2 1 2 3 }

Therefore, the location of any gate, generated by the Hamiltonian evolution in

Eq. 5.102 is given by [c1,c2,c3] = [J1t, J2t, J3t]. The coupling Hamiltonian deﬁnes a velocity vector through the local-equivalence class geometry. Fig. 5.17 examines the entangling power of any gate, within one of the non- degenerate Weyl chambers. As you can see, the entangling power of the gates is periodic along any vector, [J1t, J2t, J3t]. For example, assuming the Hamiltonian J is of the Ising form, H = 2 XX, the trajectory generated by this Hamiltonian progresses along the c1 axis. The resulting gate starts at the Identity, reaches a π maximally entangling class of gates when t = 2J (corresponding to the [CNOT] equivalence class) and comes back to an identity operation at t = π/J.

153 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

Figure 5.17: Density plot of a single Weyl chamber within the larger geometry of all two-qubit interactions. The color coding represents the maximum concurrence for a specific unitary gate parameterized by the coupling triplet, [c1,c2,c3]. Note 2 that all possible trajectories defined by [c1,c2,c3] are periodic in C , for example traversing the trajectory generated by [c1,c2,c3] = [c1, 0, 0] starts at the Identity, passed through the [CNOT] equivalence class and returns to the Identity at c1 = π/2. In contrast, the trajectory generated by [c1,c2,c3] = [c,c,c] passes through the maximal entangling class [√SWAP], finally arriving at the zero-entanglement SWAP gate at c = π/4.

The periodicity of the local-equivalence geometry is how such Hamiltonians can be characterized. Unlike the characterization of single qubit dynamics, where we map out the evolution of the system as a function of time, we instead map out the squared concurrence. The oscillation frequency of this evolution will then be related to the Hamiltonian parameters. As noted in section 5.4.2, the squared concurrence does not represent a quantum observable. However, C2 can be reconstructed from single qubit measurements in two separate basis. We deﬁne the probability of measuring the ith qubit in the η eigenstate of the basis operator α as P η1η2 , where η = 1 and i i α1α2 i ± α = X,Z . For example, in conventional notation, for an arbitrary two qubit i { }

154 5.4. CHARACTERIZATION OF TWO QUBIT DYNAMICS

state ψ , these quantities give, | i

+ 2 P − = 01 ψ (5.104) ZZ |h | i| and, 1 P ++ = 00 ψ + 10 ψ 2 (5.105) XZ 2|h | i h | i| In terms of these probabilities, C2 is given by [SH00],

2 + + ++ ij C =4 P − P − + P −−P 2 P cos(A + B) , (5.106) ZZ ZZ ZZ ZZ − ZZ ij sY where, ++ + 2P P −− P − cos(A)= XZ ZZ ZZ , (5.107) − −+ 2 PZZ−−PZZ− and, p + + 2P − P −− P − 1 cos(B)= XZ ZZ ZZ . (5.108) − + − ++ − 2 PZZ−PZZ Keeping ourselves restricted to Hamiltoniansp of the form shown in Eq. 5.102, we can analytically evaluate C2 for a variety of two-qubit input states, the results are summarised in Table 5.2.

Input state, ψ C2(t) | i 2 00 sin [(J1 J2)t] | i 2 − 01 sin [(J1 + J2)t] 1 ( 0 + 1 |) i( 0 + 1 ) sin2[(J J )t] 2 | i | i ⊗ | i | i 2 − 3 1 ( 0 + 1 ) ( 0 1 ) sin2[(J + J )t] 2 | i | i ⊗ | i−| i 2 2 2 Table 5.2: Functional form of C (t) for the four separable input states ψ i, i [1, .., 4]. In each case the concurrence varies as a sinusoidal signal with frequency| i ∈ given by the Hamiltonian parameters, [J1,J2,J3].

As you can see, the variation in C2 as a function of time is a simple sinusoidal signal with a frequency related to each of the coupling terms in the Hamiltonian. Note that all four input states are simple product states of the two individual qubits. We assume that the multiple basis measurements and the initial state preparation is possible since single qubit conﬁnement and Hamiltonian characterization has already been performed to suﬃcient accuracy. This continues the concept of “bootstrapping” each characterization procedure using the results of

155 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

the previous protocols. This analysis illustrates the basic principle of two qubit Hamiltonian characterization. The two qubit state is initialized in one of the four basis states, ψ , | ii after which an analogous Rabi experiment is performed. However in this case the

η1η2 2 probabilities Pα1α2 are determined and used to evaluate C for each successive value of t. Once the concurrence has been mapped out for a number of oscillation cycles, the peak location in the Fourier spectrum will be directly related to the coupling co-eﬃcients [J1,J2,J3]. Once the oscillation frequency has been determined for all four input states, J1, J2 and J3 can be uniquely deﬁned.

5.4.3 Characterization of a fully non-local interaction.

While in theory, many solid state systems rely on a purely Heisenberg type Hamiltonian and characterization of these systems is straight-forward when concurrence is examined, anisotropic terms can be present when taking into account higher order eﬀects. For example, spin-orbit coupling in quantum dots [SBD+03, BSD01, Kav01] where the Hamiltonian takes the form,

H = J[S S + β~ (S S )+ S ΓS ]. (5.109) 1 · 2 · 1 × 2 1 2

First order corrections are represented by β~, known as the Dzyaloshinski-Moriya (DM) vector [Dzy58, Mor60], Γ is a rank-3 symmetric tensor representing second order corrections and Si = (Xi,Yi,Zi) are the single qubit Pauli matrices. Al- though theoretical estimates on these corrective terms have been made [Kav01], and measurements of the DM corrections made in spin glass systems [PFC02], there has been no experimental measurement of either β~ or Γ for isolated coupled dots. How do we utilize the same methods of periodic concurrence to characterize these much more complicated coupling dynamics? The protocol utilized for full two-qubit characterization will now only require coupling dynamics of the form,

3 H = d σ σ , (5.110) ij i ⊗ j i,j=1 X where σ , σ , σ = X,Y,Z and d R. Since any Hamiltonian only contains { 1 2 3} { } ij ∈ terms within the subgroup p deﬁned in Eq. 5.74 we denote these Hamiltonians

156 5.4. CHARACTERIZATION OF TWO QUBIT DYNAMICS

to be Fully Non-Local (FNL). The Cartan decomposition showed that any two-qubit gate generated by Hamiltonian, H can be expressed in the form,

i G = K exp (c XX + c YY + c ZZ) K , (5.111) 1 {2 1 2 3 } 2 where (K ,K ) SU(2) SU(2). If the coupling Hamiltonian is FNL, it can be 1 2 ∈ ⊗ diagonalized directly,

1 H = K†(c XX + c YY + c ZZ)K = K†H K. (5.112) 2 1 2 3 a

Hence the unitary evolution is given by,

U(t) = exp iHt = exp iK†H tK {− } {− a } i (5.113) = K† exp (c XX + c YY + c ZZ)t K, {−2 1 2 3 } where K SU(2) SU(2). This decomposed form for the unitary gate is very ∈ ⊗ similar to the simple Heisenberg interaction already considered, however a com- plication now arises in the initial state used when mapping out C2. When considering Heisenberg type Hamiltonians the initial state is well defined and fixed during the characterization protocol. This allowed us to fix the initial state to observe a single oscillating frequency which is well related to the coupling parameters [c1,c2,c3]. In the case of a FNL Hamiltonian the initial product state is rotated by the matrix K SU(2) SU(2) before it is acted apon by ∈ ⊗ the interaction term, exp iH t . Since H is unknown, we are unable to deter- {− a } mine what this initial local rotation is. Consequently we are not guaranteed of being able to isolate specific combinations of the coupling terms. In order to fully reconstruct the Hamiltonian, we require enough information to not only determine the coupling parameters, Ha, but also determine the local rotation matrix K.

As with the case of Heisenberg coupling, determining Ha and K can be done by examining the entanglement produced by the operator U(t) on several known, separable, input states. Using the decomposition shown in Eq. 5.113 and assuming we have initialized in a known state, ψ , the analytical form of C2(t) | i0

157 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

can be expressed as,

2 T 2 2 C (t)= (Q†K ψ ) F (t)(Q†K ψ , (5.114) | | 0i | 0i| where, i F (t) = exp (c XX + c YY + c ZZ)t . (5.115) {−2 1 2 3 } Now consider a general 4 4 matrix K SU(2) SU(2). Recall that SU(2) × ∈ ⊗ ⊗ SU(2) is isomorphic to SO(4) where SO(4) Q†SU(2) SU(2)Q and Q is given ≡ ⊗ by Eq. 5.79. Therefore, K = Q†KQ SO(4) and all elements in K are real. Q ∈ Q Let,

+ + Q† φ = K Q† ψ = γ Φ + γ Φ− + γ Ψ + γ Ψ− , (5.116) | 0i Q | 0i 1| i 2| i 3| i 4| i be the rotated input state in the Bell basis. In this case, the concurrence can be rewritten as,

4 2 C2(t)= γ2λ2 = eiǫ1tγ2 + eiǫ2tγ2 + eiǫ3tγ2 + eiǫ4tγ2 2. (5.117) k k | 1 2 3 4 | k=1 X

As with the case of a Heisenberg interaction, two qubits are initialised in a known product state ψ and C2 mapped out as a function of time. Once C2(t) | i has been mapped, the data will have the following form,

4 C2(t)= γ2 2 + γ2 γ2 ei(ωij t+Γij ), (5.118) | i | | i || j | i=1 i=j X X6 where ω =(ǫ ǫ ) and Γ = 2(arg(γ ) arg(γ )). Next, the Fourier spectrum ij i − j ij i − j of the time series, C2(t), is taken and the power density spectrum g(ω) 2 = | | FT[C2(t)] 2 is plotted as a function of ω. | | To detail the reconstruction, consider the case where H decomposes to a form where c = c = c = 0. This represents the most straightforward case for 1 6 2 6 3 6 characterisation and arguably the most likely given a random Hamiltonian. The power density spectrum, g(ω) 2, has the form, | |

4 2 (2π) g(ω) 2 = γ 4 δ(ω)+ γ 4 γ 4δ , (5.119) | | | i| | i| | j| ωij i=1 i,j i X X≥

158 5.4. CHARACTERIZATION OF TWO QUBIT DYNAMICS

with δ = [δ(ω + ω )+ δ(ω ω )]. This functional form for g(ω) 2 illustrates ωij ij − ij | | that the information related to the rotated input state, K Q† ψ , and hence the Q | 0i matrix K is contained within the height of each peak, while all the information relating to the factors [c1,c2,c3] are related to the location of each peak in the frequency spectrum. When c = c = c = 0, the power density contains six 1 6 2 6 3 6 distinct peaks, located at,

ω = 2(ǫ ǫ )=2(c c ), 12 1 − 2 2 − 3 ω = 2(ǫ ǫ )=2(c + c ), 13 1 − 3 1 3 ω14 = 2(ǫ1 ǫ4)=2(c1 c2), − − (5.120) ω = 2(ǫ ǫ )=2(c + c ), 23 2 − 3 1 2 ω = 2(ǫ ǫ )=2(c c ), 24 2 − 4 1 − 3 ω = 2(ǫ ǫ )=2(c + c ), 34 3 − 4 3 2 where overall signs have been omitted as δ is symmetric in ω ω . If we ij ij → − ij now utilize the local symmetry of the SU(4) geometry, we can use one of the non- degenerate Weyl chambers to identify each peak in the spectrum. Restricting ourselves to the chamber where c c c 0 it is clear that the peaks, 1 ≥ 2 ≥ 3 ≥ ω >ω occur at the highest frequency which are separated by ω ω = ω . 23 13 13 − 23 12 Note also that the peaks ω > ω are also separated by ω ω = ω . The 24 14 14 − 24 12 four peaks, ω23, ω14 and ω13, ω24 are centered around the value 2c1 with the ﬁnal peak ω34, which with ω12 is centered around 2c3. Fig. 5.18 in the next section illustrates these symmetries for [c1,c2,c3]=[2.0012, 2.006, 1.998].

Utilizing these peak symmetries and Eq. 5.120, the triplet [c1,c2,c3] can be identiﬁed and hence the decomposed interaction Hamiltonian, Ha = (c1XX + c2YY + c3ZZ)/2 speciﬁed. In order to fully reconstruct the interaction Hamilto- nian, the local rotation K SU(2) SU(2) is still required. ∈ ⊗

Recall that the expansion co-efficients, γk, k = [1, .., 4] are defined for the state K Q† ψ . Hence these co-efficients are related to the matrix K = Q†KQ Q | i Q ∈ SO(4). As K SO(4), each element is real and six parameters are required to Q ∈ specify the matrix. When c = c = c , parameterization of K is not actually 1 6 2 6 3 Q

159 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

required and all 16 elements of KQ can be found directly. If we specify KQ as,

a1 a2 a3 a4

 a5 a6 a7 a8  KQ = , (5.121)  a9 a10 a11 a12   a13 a14 a15 a16     then each parameter can be found if we perform three experiments where in each case we use the following three known input states,

1 ψ = (i 0 + 1 ) ( 0 + i 1 ), | i1 2 | i | i ⊗ | i | i 1 ψ 2 = ( 0 + 1 ) 1 , (5.122) | i √2 | i | i ⊗| i 1 ψ 3 = 1 ( 0 + 1 ). | i √2| i ⊗ | i | i

Using these three input states, the amplitudes for the expansion co-eﬃcients, γ (j), j [1, .., 3], k [1, ..4], are given by, k ∈ ∈ 1 γ 2(1) = (a2 + a2), | 1| 2 1 3 1 1 γ 2(2) = (a + a )2 + (a + a )2, | 1| 4 1 3 4 2 4 1 1 γ 2(3) = (a + a )2 + (a a )2, | 1| 4 1 3 4 2 − 4 1 γ 2(1) = (a2 + a2), | 2| 2 5 7 1 1 γ 2(2) = (a + a )2 + (a + a )2, | 2| 4 5 7 4 6 8 2 1 2 1 2 γ2 (3) = (a5 + a7) + (a6 a8) , | | 4 4 − (5.123) 1 γ 2(1) = (a2 + a2 ), | 3| 2 9 11 1 1 γ 2(2) = (a + a )2 + (a + a )2, | 3| 4 9 11 4 10 12 1 1 γ 2(3) = (a + a )2 + (a a )2, | 3| 4 9 11 4 10 − 12 1 γ 2(1) = (a2 + a2 ), | 4| 2 13 15 1 1 γ 2(2) = (a + a )2 + (a + a )2, | 4| 4 13 15 4 14 16 1 1 γ 2(3) = (a + a )2 + (a a )2. | 4| 4 13 15 4 14 − 16

160 5.4. CHARACTERIZATION OF TWO QUBIT DYNAMICS

The peak heights in g(ω) 2 can also be related to γ as, | | k p p (1/8) p p (1/8) p p (1/8) γ = 12 13 = 12 14 = 14 13 , | 1| 2πp 2πp 4 2πp 4 23 2 3 p p (1/8) p p (1/8) p p (1/8) γ = 12 23 = 12 24 = 24 23 , | 2| 2πp 2πp 2πp 13 14 34 (5.124) p p (1/8) p p (1/8) p p (1/8) γ = 13 23 = 13 34 = 23 34 , | 3| 2πp 2πp 2πp 12 14 24 p p (1/8) p p (1/8) p p (1/8) γ = 14 24 = 24 34 = 14 34 , | 4| 2πp 2πp 2πp 12 23 13 where 2πp = γ 4 γ 4 for each of the three input states. We next utilize the ij | i| | j| group conditions of SO(4), speciﬁcally that the rows of KQ must form an or- thonormal set. Using this restriction we can derive the following relationships,

1 µ (1) = γ 2(1) γ 2(2) + γ 2(3) = (a a )2, 1 | 1| −| 1| | 1| 2 3 − 1 2 2 2 1 2 µ2(1) = γ1 (1) + γ1 (2) γ1 (3) = (a3 + a1) , | | | | −| | 2 (5.125) 1 µ (1) = γ 2(2) γ 2(1) + γ 2(3) = (a a )2, 3 | 1| −| 1| | 1| 2 2 − 4 1 µ (1) = 1 γ 2(1) γ 2(2) γ 2(3) = (a + a )2. 4 −| 1| −| 1| −| 1| 2 2 4

Similar equations can be derived for the other four rows of KQ using the amplitudes, γ , γ and γ for each of the three input states. Since µ represents the | 2| | 3| | 4| square of the matrix elements, simply solving these equations will not result in a unique matrix for KQ. In fact Eq. 5.125 give rise to solutions in the pairs, (a1, a3) and (a2, a4) which implies that multiple matrices can be formed which vary via permutations and/or sign flips of these pairs. This ambiguity in the solution leads to 216 separate matrices which satisfy Eq. 5.125. The final steps require first enforcing the group conditions of SO(4) to these T solutions, specifically that KQKQ = II and det(KQ) = 1. However this still results in a number of valid matrices for KQ. Each of these valid matrices will actually result in a set of locally equivalent Hamiltonians once KQ and the previously determined matrix Ha are combined to form the interaction Hamiltonian, i.e. H = K†HaK. In order to determine which Hamiltonian is correct, the concurrence function C2(t) is analytically constructed for each possible Hamiltonian and compared to the experimental data. This will narrow down the possible Hamilto-

161 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

nians further, however it will not always lead to a unique interaction. There is the possibility that two interaction Hamiltonians have precisely the same analytical form for C2(t) for each input state used in experiment. In this unlikely event, a initial separable state needs to be found which generates unique forms of C2(t) for the remaining possible Hamiltonians. Once this initial state is analytically calculated, one final concurrence experiment is performed. Unlike the original experiment, C2(t) does not need to be mapped out as a function of time, instead a specific value of t is chosen that will discriminate between the possible interactions. Once this final measurement is taken, the analytical and experimental values are compared to finally determine the correct interaction Hamiltonian. The next section illustrates the steps in this protocol using the spin-orbit Hamiltonian shown in Eq. 5.126.

5.4.4 Example: Spin-Orbit corrections in GaAs quantum dots.

Although the characterization process is essential to the eﬃcient construction of arbitrary two qubit gates, it can also be used to experimentally determine other theoretical parameters governing a system. The following is a speciﬁc example to highlight the methodology. Consider the exchange interaction between spin systems in quantum dots. Spin-obit coupling introduces anisotropic corrections to the otherwise ideal isotropic exchange Hamiltonian, H = JS1.S2. When these correction are included, the time independent Hamiltonian describing the two qubit coupling takes the form,

H = J[S S + β~ (S S )+ S ΓS ]. (5.126) 1 · 2 · 1 × 2 1 2

Although gate design schemes have been developed to overcome these anisotropic terms [BSD01], characterisation of this coupling is possible and allows us to determine both the physically relevant parameters Jβ~ and JΓ.

162 5.4. CHARACTERIZATION OF TWO QUBIT DYNAMICS

We consider the trial Hamiltonian where,

J =1, β~ = 0.01, 0.005, 0.02 , { } 0.003 0.005 0.001 (5.127) Γ= 0.005 0.0015 0.0024 . 0.001 0.0024 0.0009     β~ is chosen randomly but is consistent with the analysis of Kavokin [Kav01] for quantum dots in GaAs and Γ is roughly an order of magnitude smaller since it is a second order correction to H. Fig. 5.18 shows the power density spectrum obtained using the input state,

1 ψ 2 = ( 0 + 1 ) 1 . (5.128) | i √2 | i | i ⊗| i which has the following functional form [Eq. 5.119],

4 2 (2π) g(ω) 2 = γ 4 δ(ω)+ γ 4 γ 4δ . (5.129) | | | i| | i| | j| ωij i=1 i,j i X X≥ Each peak is located at,

ω23 =8.039, ω13 =8.025, ω12 =0.012, (5.130) ω24 =0.028, ω14 =0.016, ω34 =8.011, allowing for the determination of [c1,c2,c3] = [2.0012, 2.006, 1.998]. The respective peak heights, p = γ 4 γ 4/2π, and values for (µ ,µ ,µ ,µ ) are given in ij | i| | j| 1 2 3 4 Table 5.3. After constructing all possible combinations of the above values for a , ..., a , { 1 16} 64 matrices are found to satisfy the group conditions of SO(4). These matrices and Ha = (c1XX + c2YY + c3ZZ)/2 are used to ﬁnd 16 distinct Hamiltoni- ans H = K†HaK. Eq. 5.131 shows 4 Hamiltonians in terms of the spin-orbit

163 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

2 Input p p p p p p 10− 12 13 14 23 24 34 × ψ 0 6.81 2.32 0 0 13.35 | i1 ψ .0467 5.68 11.01 .0197 .0383 4.66 | i2 ψ 3 .0473 5.91 10.84 .0201 .0367 4.59 | i 2 µ (i) µ (i) µ (i) µ (i) 10− 1 1 1 1 × i =1 21.21 20.46 56.01 2.33 i =2 0041 .021 4.49 95.48 i =3 50.24 49.74 .0110 .0012 i =4 28.55 29.78 39.49 2.19 Table 5.3: Relevant data from the three power density spectra, one of which is shown in Fig. 5.18. Values for all the respective peak heights and µ (i),µ (i),µ (i),µ (i) can be used to ﬁnd K . { 1 2 3 4 } Q

corrections β~ and Γ.

H : β~ = 0.0015, 0.005, 0.009 , 1 {− − } 2 0.005 2 − − Γ= 0.005 0.01 0.002 ,  −  2 0.002 0.002  − −    H : β~ = 0.01, 0.005, 0.02 , 2 { } 0.003 0.005 0.001 Γ= 0.005 0.0015 0.0024 , 0.001 0.0024 0.0009     (5.131) H : β~ = 0.009, 0.005, 0.0015 , 3 { } 2 0.005 2 − − Γ= 0.005 0.02 0.002 , 2 0.002 0.01  −    H : β~ = 0.02, 0.005, 0.01 , 4 { } 0.001 0.005 0.003 Γ= 0.005 0.0009 0.002  . 0.003 0.002 0.0015     When parameterising Eq. 5.131 with respect to β~ and Γ, we have explicitly used J = 1, this is NOT a requirement to characterise the Hamiltonian using this method. However, since the general form of the Hamiltonian has J as a constant multiplicative factor, extracting the exact values for β~ and Γ requires J to be

164 5.4. CHARACTERIZATION OF TWO QUBIT DYNAMICS

8.01 8.03 8.05

Figure 5.18: Power Density spectrum for the ψ 2 input state. Note that the respective symmetries discussed in section 5.4.3.| Thei values for each peak height are shown in Table 5.3.

known or assumed.

From the 16 possible Hamiltonians generated, clearly the KQ matrices generating H2 are the target matrices, however we need to filter out the other 15 possible Hamiltonians. To do this, analytically construct the unitary operators U (t) = exp( iH t), for i [1, 16] and calculate C2[U (t)] for all i [1, 16] i − i ∈ i,k ∈ and input states, k [1, 3]. The calculated values are then compared with the ∈ 2 values of C [Uk(1)] measured at t = 1 for all three input states. Table 5.4 shows 2 the results for the 4 Hamiltonians in Eq. 5.131. Simulated values of C [Ui,k(1)] immediately eliminate H1 and H3 as possible Hamiltonians. Repeating for the 2 other 12 Hamiltonians shows that the analytical values for C [Ui,k(1)] match the simulated values only for H2 and H4. Here, the analytical forms for concurrence are identical for the three input states ψ . | ik 2 To isolate H2, we analytically construct C and find that the input state ψ = 1 ( 0 i 1 )/√2 generates different functions for H and H . One more | i4 | i⊗ | i− | i 2 4 2 2 measurement of C is taken for this input state at t = 0.5, giving C [U4(0.5)] =

165 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

2 2 2 2 2 Input C [U1(1)] C [U2(1)] C [U3(1)] C [U4(1)] C [U(1)] ψ .581 .581 .581 .581 .581 | i1 ψ .147 .147 .142 .147 .147 | i2 ψ .144 .150 .150 .150 .150 | i3 Table 5.4: Calculations of C2 for all three input states using the 4 unitary operators experimentally obtained for t = 1. The last column represents the data measured for the three input states and unknown unitary operator at t = 1, U(1).

2 2 0.200. Analytically, C [U2,4(0.5)] = 0.200 and C [U4,4(0.5)] = 0.196. This ﬁnal measurement has discriminated between H2 and H4, leaving H2 to satisfy all the simulated measures of C2. As with any characterization protocol, experimental accuracy will need to be extremely high in order for the system to be utilized for information processing.

In this example, the ﬁnal measurement, discriminating between H2 and H4 will be possible as all peak heights and spectrum frequencies will need to be resolved to similar accuracy in order to ensure gates constructed from these Hamiltonians satisfy the error requirements of Fault-tolerant QEC. Signiﬁcant analysis of this protocol will need to be performed in order to determine how experimental errors in two-qubit Hamiltonian characterization propagate to systematic gate errors in QEC.

5.4.5 Conclusions

In conclusion, we have presented a systematic method to experimentally determine the interaction Hamiltonian of a two-qubit system which requires initialisation in a minimum of three product states and measurement in two separate bases. We have illustrated how the geometric structure of two-qubit interactions allows for a natural way to characterize certain quantum gates and ultimately experimentally estimate the coupling Hamiltonian for a large class of systemss. We have demonstrated how this method not only allows for accurate characterisation of system Hamiltonians required for constructing two-qubit quantum gates, but also how characterisation can lead to the determination of other parameters of interacting systems. Specifically, we showed how the first and second order anisotropic corrections in quantum dots can be experimentally determined. There is still a significant amount of work required before a complete experi-

166 5.4. CHARACTERIZATION OF TWO QUBIT DYNAMICS

mental protocol for all two-qubit Hamiltonians is available. Specifically, we need to develop modifications such that characterization in the presence of local terms is still possible and we need to examine how experimental errors filter through to systematic gate errors and consequently, failure probabilies for quantum error correction. These issues are extreamly important and are reserved for future investigation.

167 CHAPTER 5. INTRINSIC CHARACTERIZATION OF QUBIT SYSTEMS

168 Chapter 6

Distributed Quantum Information Processing via Parity Measurements.

Contents 6.1 Introduction ...... 171

6.2 Information free quantum bus ...... 176

6.2.1 Preparation of stabilizer states ...... 182

6.2.2 Universality using the Quantum bus ...... 188

6.2.3 Atom/cavity interaction with photonic qudit . . . . . 189

6.2.4 Implementation of the bus protocol in Solid State Sys- tems...... 191

6.2.5 Conclusion ...... 196

6.3 Scalable quantum error correction in distributed ion trapcomputers...... 199

6.3.1 Architecture ...... 201

6.3.2 Operation...... 201

6.3.3 NodeDesign ...... 213

6.3.4 Conclusion ...... 216

6.4 Photonic Modules ...... 218

6.4.1 Bell State Factories ...... 222

169 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

6.4.2 Arbitrary entangled state preparation...... 224 6.4.3 OperationalTime ...... 227 6.5 Conclusions ...... 229

170 6.1. INTRODUCTION

6.1 Introduction

Since the introduction of quantum information science there has been extensive work on designing effective quantum processors that satisfy the abstract assumptions of large scale algorithms, error correction and Fault-tolerance. The initial proposals for systems such as ion traps [CZ95], Solid state computers [Kan98, LD98], linear optical systems [KLM01] and super-conducting qubits [NPT99, MOL+99] focused heavily on achieving the required single and two-qubit interactions and demonstrating that systematic and decoherence induced errors can be suppressed to a sufficient degree. However, very few of these initial proposals contained significant detail on how they can be scaled up to a large scale computer containing potentially thousands of qubits. As shown in chapters 3 and 4, large scale quantum algorithms and Fault- tolerant Quantum Error Correction have quite stringent assumptions regarding the underlying physical architecture. The original quantum circuit proposals for Shor’s algorithm [Section 3.2] and the original threshold calculations for Fault- tolerant quantum error correction [Kit97, ABO97, Got97], all assumed an optimal physical system where arbitrary pairs of qubits can be coupled (regardless of their physical location in the computer), all isolated two-qubit interactions can be performed simultaneously and no qubit transport is required. Even for the most experimentally successful qubit systems (such as linear optics and ion traps) these assumptions are grossly inadequate. It is commonly accepted that all current proposals for quantum computation do not satisfy the original, ideal assumptions, for large scale information processing. Solid state proposals such as phosphorus in silicon, quantum dots or superconducting qubits are generally linear nearest neighbor systems with no obvious transport mechanism for large scale processors. Linear optical computers suffer from non-determanistic two-qubit interactions and the inevitable problem of qubit storage and ion trap systems are limited to a small finite number of ions within a linear trap. As the field has matured it has become more and more important for re- searchers to concentrate on redesigning quantum architectures in order to slowly satisfy a greater number of the assumptions of large scale quantum computing, while maintaining a certain level of experimental realism in regards to what can actually be built. The general consensus is that all systems will have to be de-

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signed in a distributed manner, where multiple physical systems are incorporated for quantum storage, gate operations and transport. This chapter presents one such method to achieve distributed computation and highlights several systems for which it can be applied. As many theorists and experimentalists recognize the need for highly distributed quantum processors, the subsequent body of work is extensive, hence I will only mention a few of the more prominent distributed schemes. DiVincenzo [DiV00] was one of the first to point out the need for a distributed network of qubits and quantum transport. His original five conditions for a viable computer were augmented to seven with the introduction of flying qubits. This scheme for distribution is quite simple. A small number of qubits which can be directly coupled (for example, in the solid state) are linked together via highly mobile flying qubits (for example, single photons). When interactions are required between these small nodes, information is locally swapped onto this flying qubit, transported to another node, locally coupled to a second stationary qubit and then returned to the original site. Variations of this idea has been proposed for ion- traps [KMW02], phosphorus in silicon [HGFW06], GaAs quantum dots [TED+05] and superconducting systems [BHW+04, FTY+07]. As this method was one of the first proposed, there exists several analytical and numerical studies of flying qubit architectures within the context of Fault-tolerant error correction [SFH07, FTY+07, MCT+04, BKSO05] and even a successful experimental implementation [WSB+04] where coupling between a microwave cavity photon and a single superconducting cooper-pair box was achieved. A second method which is somewhat analogous to flying qubits is the idea or teleportation hubs [OCC02]. This scheme again assumes small nodes of qubits which are distributed within a larger array, but instead of utilizing a flying qubit to directly transport data between nodes, quantum teleportation [BBC+93] is used. This involves preparing multiple two-qubit Bell states locally (commonly referred to as Bell state factories), transporting each half of the Bell state to separate nodes using traditional methods such as the SWAP channel and then utilizing these Bell pairs to teleport quantum data for inter-node interactions. Teleportation hubs do allow for a certain level of distributed computing, but it is limited. Preparing the Bell state locally and transporting each half to the local nodes will generally require purification protocols [BDCZ98, DBCZ99] in order to preserve coherence and hubs do not remove the need for local transport within

172 6.1. INTRODUCTION

each node in order to place relevant data qubits adjacent to the hubs. More recently the ideas for large scale distributed system have been derived from the ideas born out of cluster state and teleportation based quantum computing [RB01, BR01, GC99, Leu04, AL04] [Chapter 2]. Cluster state computing was an extremely important development since it showed that arbitrary quantum algorithms could be implemented using a completely generic resource (namely the initial cluster). It also solved the problem of non-deterministic interactions, since the initial cluster can be prepared off-line using probabilistic gates and the algorithm implemented only when the cluster is ready. Using these theoretical techniques, there have been numerous proposals for distributed architectures. Most of these schemes utilize non-determanistic preparation of distributed entanglement and either cluster state or teleportation based methods. Cluster based distribution is exemplified by proposals such as Bar- rett and Kok [BK05] and brokered graph state computing [BBFM06] where matter qubits are probabilistically entangled into Bell states via single photon emission and path erasure. As the fundamental interaction of these schemes is non-deterministic, cluster state methods are employed to achieve scale up while allowing a massive level of distribution between individual qubits. Sim- ilarly, there have been several proposals where teleported quantum gates are utilized [Got98b, VC04, DBMM04, DMM+06]. In this case, each data qubit is locally paired with an ancilla and long range, probabilistic techniques are used to prepare a non-local Bell state between ancilla qubits [CCGFZ99, PHBK99, BKPV99, SI03, DK03, BPH03]; finally measurement, local operations and classical communication are used to enact a non-local gate operation via teleportation. Each of these ideas has certainly helped to bridge the gap between the assumptions of large scale quantum computation and experimental systems, however, even these schemes suffer from drawbacks. Each of the methods using probabilistic preparation of entanglement either requires the concepts of cluster state computing or teleported gates. The major problem with cluster state computation is the additional qubit overhead associated with the model and the ability to maintain a long lived coherent cluster during the computation. This overhead can be alleviated by preparing the cluster dynamically [Nie04], so that as measurement destroys the cluster during computation, probabilistic gates are continually regenerating it. However, the success of this method is dependent on the fundamental success rate of the probabilistic interaction and how quickly links can be established [BK05, DR05, GKE06, RB07]. Schemes using teleported gates are

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also problematic from the standpoint of error correction. Isolated data/ancilla qubits are ideally linked using probabilistic methods in order to perform gate operations. The probabilistic nature of the entanglement links implies that data qubits will be susceptible to memory errors while waiting for links to be established. Once again, if the linkage operation is slow and/or has a low probability of success then error correction may not be successful. In this chapter I will present an alternative scheme for distributed quantum computation using the concept of Parity measurements. Section 6.2 introduces a protocol which allows for deterministic interactions between isolated data qubits via a photonic bus which mediates the interaction. The benefit of this protocol is two-fold. First, the photonic bus system never carries any computational information of the qubits. Hence, the traditional loss mechanisms introduced when using flying qubits do not apply. Secondly, the interaction is completely deterministic. This alleviates many of the problems associated with probabilistic schemes for distributed computing. In addition to introducing the photonic bus, I will also present an alternative architecture where the same bus protocol can be utilized in solid state computers such as the Kane architecture or quantum dots. The protocol that I will introduce is quite similar to two other ideas, the first put forward by Duan, Wang and Kimble [DWK05], where controlled interactions between spatially isolated atomic qubits is achieved through successive cavity reflection of a bus photon, and a second method proposed by by Spiller et. al. [SNB+06] where a coherent bus probe is used to mediate interactions between isolated data qubits. The major difference in this scheme is that the qubit/bus interaction links elegantly with stabilizer states and hence quantum error correction. It should be noted that there has been several proposed schemes for quantum computation utilizing parity gates in the solid state regime [BDEK04, EL05, TJBB06] which are similar to the protocol presented in the following chapter. However the detailed incorporation of a parity based protocol for computation with an efficient transport mechanism for distributed computation was generally not considered. The benefit of the following schemes (in both the optical and solid state contexts) is that the transport mechanism is inbuilt to the parity gate protocol, allowing for gates to be performed between qubits that are spatially distributed. Section 6.3 presents a detailed proposal for a distributed computer using ion-

174 6.1. INTRODUCTION

traps, which is speciﬁcally designed to exploit the high level of quantum control for small groups of ions already seen in experiments. Finally in section 6.4 I present a design for a “photonic module”, which allows for the deterministic preparation of a large class of multi-photon entangled states. This device will be extremely useful for not only photonic based quantum computation but also in many quantum communication protocols.

Original Material

Sections 6.2 represents original work in collaboration with Andrew Green- • tree, introducing a quantum bus network that can achieve distributed quantum computation. The original motivation for this work came from the adiabatic transport protocol in solid state qubits (MRAP) detailed in section 6.2.4, ﬁrst developed by Greentree and myself. This work ﬁrst appeared in Refs. [GDH06b, DGH07]

Section 6.3 which presents a distributed architecture in Ion-Trap computers • was carried out in collaboration with Daniel Oi who ﬁrst conceived the idea. The actual distributed protocol represents original work that was adapted from the ideas developed for the information free quantum bus. This proposal ﬁrst appeared in Ref. [ODH06]

Section 6.4 which introduces the photonic module represents original work • that ﬁrst appeared in Ref. [DGI+07]

175 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

6.2 Information free quantum bus

The development of viable quantum computers and preparation of effective multi- qubit entangled states depends crucially on transport protocols that can be used to shuttle quantum information and to allow for interactions between isolated qubits. Effective transport of quantum information is essential to the scaling of small, functional elements, and will enable an interpolation between small scale devices and full-blown, massively entangled quantum computers. In this section I will introduce the basic principals of the quantum bus, how it links to the concepts of stabilizers and entanglement preparation and how universal quantum computation can be achieved. The generic properties of the quantum bus will be presented with respect to a specific system, namely as a single photon optical network with atom/cavity systems utilized as computational qubits. Later, in section 6.2.4 I will present an alternative system in the solid state where each individual qubit is a buried donor (or quantum dot) with a bus system consisting of single electrons. The fundamental unit of the bus is a single 4-level quantum system (qudit). This system differs significantly from other systems in that it is defined through spatial eigenstates. The quintessential example of such a system is a single photon and a balanced 50:50 beamsplitter [Fig. 6.1a], where spatial eigenstates for a single photon is defined via the four input/output modes of the beamsplitter. We now assume that the qubit register for the computer is a series of single atoms contained within resonant cavities. Such systems have been investigated in great length due to their potential use as quantum registers, including Nitrogen-Vacancy centers, other rare earth ions [LH00, GOD+06, WKN01] and even super-conducting cooper-pair boxes contained within microwave stripline cavities [BHW+04]. The fundamental difference with our scheme is that each individual qubit is placed within its own isolated cavity system. Therefore, each qubit has no direct coupling to any other atom/cavity qubit and the entire qubit register can be highly distributed, with each atom/cavity system spatially isolated. We also assume that each atom/cavity system has appropriate local control such that single qubit gates, initialization and measurement can be performed on every qubit. First, consider the simple case where we have a single photonic qudit and two, spatially isolated, atom/cavity systems [Fig. 6.1b]. The single photonic qudit

176 6.2. INFORMATION FREE QUANTUM BUS

Laser control for cavity qubit a) b) Single Photon Detector

50:50 Beamsplitter

Single Photon 50:50 Qubit 1 Source & Detector. Beamsplitter

Qubit 2

Figure 6.1: a) Standard four port 50:50 beamsplitter. A single photon in any one of the input modes will be placed into a superposition state via the operator U in Eq. 6.2. The reflective surface for the beamsplitter is on the front surface such that reflection from A to B picks up a eiπ = 1 phase shift. b) The 1 2 − two atom/cavity qubits are placed such that each cavity mode couples to the B1 and B2 optical modes. Once the single photon pulse induces a bit flip on each atom it leaks back out into the respective optical mode and is re-mixed on the beamsplitter. Each atom/cavity system has local optical control, used for single qubit rotations, initialization and readout, while the modes A1 and A2 are contain single photon detectors and the initial single photon source.

is deﬁned via its spatial mode, with the four mode eigenstates given by ψ , | iA1 ψ , ψ and ψ , where A , B are the four optical modes of a four port | iA2 | iB1 | iB2 1,2 1,2 beamsplitter and the internal state of the photon (such as polarization) given by ψ. When a single photon passes through a balanced beamsplitter, a unitary operation is performed over the spatial mode degree of freedom. Deﬁning the state of this spatial qudit as,

α β Ψ = α ψ + β ψ + γ ψ + δ ψ =   , (6.1) | i | iA1 | iA2 | iB1 | iB2 γ   δ     177 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

the operation induced by the beamsplitter is given by,

0 01 1 − 1  0 01 1  U = . (6.2) √ 2  1 10 0     110 0  −    If we have access to an appropriate, on demand, single photon source, the bus qudit is introduced into mode A1, after which U transforms the state to,

1 U ψ A1 = ( ψ B1 ψ B2 ). (6.3) | i √2 | i −| i i.e. the single photon is now in a coherent superposition of two spatial modes. The total system is designed such that the cavity mode for each atom/cavity system is coupled to the spatial modes of the optical qudit (B1 and B2). When a single photon is present in the optical mode it induces a bit-ﬂip (X) operation on the atomic qubit without destroying the photon. If we deﬁne the two-qubit state of the matter qubits as φ , then the total state of the photonic qudit + | i atom/cavity qubits evolves as,

1 1 ( ψ B1 ψ B2 ) φ ( ψ B1 X1I2 ψ B2 I1X2) φ . (6.4) √2 | i −| i ⊗| i→ √2 | i −| i | i where X1 and X2 represent bit-ﬂip operations on the respective qubits coupled to the optical modes B1 and B2 and (I1,I2) are the identity operators. As the optical photon has not been destroyed in this process, the single photon can be reﬂected back to the beamsplitter and the spatial modes B1 and

B2 re-mixed. Using the transformation U, the total state of the system becomes,

1 U( ψ X I ψ I X ) φ √ | iB1 1 2 −| iB2 1 2 | i 2 (6.5) 1 = ( ψ X I + ψ X I + ψ I X ψ I X ) φ , 2 | iA1 1 2 | iA2 1 2 | iA1 1 2 −| iA2 1 2 | i which can be re-written as,

1 1 ψ (X I + I X ) φ + ψ (X I I X ) φ . (6.6) 2| iA1 1 2 1 2 | i 2| iA2 1 2 − 1 2 | i

Two single photon detectors are now placed into the A1 and A2 modes of the

178 6.2. INFORMATION FREE QUANTUM BUS

beamsplitter. After the single photon has been remixed it will be detected at either A1 or A2. If it is localized to A1, the state of the atom/cavity qubits collapses to, 1 (X1I2 + I1X2) φ . (6.7) √2 | i

If the photon is detected in mode A2, then the atom/cavity qubits collapse to the state, 1 (X1I2 I1X2) φ . (6.8) √2 − | i Note in both cases, the single photonic qudit is destroyed due to photodetection. Examinination of Eqs. 6.7 and 6.8 shows that the localization of the photonic qudit collapses the atom/cavity qubits to an entangled state. To see this clearly, assume that both qubits have been initialized in the 0 0 state, then, | i| i 1 1 (X1I2 + I1X2) 0 0 = ( 1 0 + 0 1 ) √2 | i| i √2 | i| i | i| i (6.9) 1 1 (X1I2 I1X2) 0 0 = ( 1 0 0 1 ) √2 − | i| i √2 | i| i−| i| i which represent even and odd parity Bell states. Therefore the interaction of each qubit with the photonic bus has eﬀectively produced entanglement once the photonic qudit is measured. Recall that we have seen Eqs. 6.7 and 6.8 before: In Chapter 4 we showed that to prepare stabilizer states appropriate for Quantum Error Correction, each stabilizer operator was measured over a collection of qubits. This projected arbitrary states into 1 eigenstates of the measured operator. The interaction in- ± duced between the two atom/cavity qubits has the same properties as stabiliser measurements. If the photon is measured in the A1 mode, then the cavity qubits are projected into a +1 eigenstate of the operator X1X2, which can be shown by applying X1X2 to the resulting state,

1 1 X1X2(X1I2 + I1X2) φ = (X1I2 + I1X2) φ . (6.10) √2 | i √2 | i

Similarly, if the photon is measured in the mode A2, then the cavity qubits are projected into a 1 eigenstate of X X , − 1 2 1 1 X1X2(X1I2 I1X2) φ = (X1I2 I1X2) φ . (6.11) √2 − | i −√2 − | i

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Therefore, the fundamental interaction exhibited by the system is a parity check of XX. This method for coupling two isolated qubits is extremely beneficial compared with traditional schemes for distributed interactions, such as flying qubits and photonic path erasure. When utilizing flying qubits, quantum data is stored on immobile and potentially long lived data qubits. When two data qubits need to be coupled, the quantum data is swapped onto an extremely mobile flying qubit, transported to the physical site of the second data qubit, coupled via short-range interactions, transported back and then swapped back to the original data qubit. One problem with this method is that the qubit can be lost during transport. For example, if the flying qubit is a single photon, it will be extremely difficult to guarantee that the photon will not be lost on-route. This is potentially fatal to the implementation of large scale computing, since data loss is extremely difficult to handle with standard quantum error correction. Ideally, if data loss is significant, non-demolition detection needs to be employed which can determine if the flying qubit has been lost, without discriminating between the computational states. In the photonic regime, this non-demolition detection [POW+04] will be a significant bottleneck for any large scale error correction protocol. The benefit of our scheme is that the single photonic qudit, which mediates the interaction between data qubits, never carries any information regarding the state of the qubits. Instead the photon only carries parity information. Hence, loss of the photon does not lead to loss of information. In fact, the nature of the interaction is such that photon loss will only induce (at most), a single coherent error on one of the qubits. For example, consider a loss event where the photon is effectively localized by the environment between the steps detailed in Eq. 6.5. In this case, the photon is localized by the environment and then lost. This will collapse the qubits to a classical mixture of the states,

X I φ , I X φ . (6.12) { 1 2| i 1 2| i}

Therefore, the loss event has induced a single bit ﬂip error on one of the qubits. As this error is restricted to the computational state of the qubits, standard error correction protocols will detect and correct this failure without any further modiﬁcation.

180 6.2. INFORMATION FREE QUANTUM BUS

Additionally, since a loss event results in no-detection at either A1 or A2 this error can be detected and a new photon can be injected into the bus and the interaction repeated. This will guarantee that the intended coupling between the two data qubits is achieved, even though previous loss event may have introduced a coherent X error. The result that photonic loss does not lead to loss of qubit information is the reason why we have dubbed this bus “Information free”. Although the photon does carry parity information, it does not carry computational information. Consequently loss of the bus photon will never lead to a traditional loss event. A second method to achieve distributed interaction between qubits is photonic path erasure, exempliﬁed by the Barrett and Kok [BK05] protocol. This method involves classically exciting two qubits and waiting for them both to re- lax, emitting two photons. These two photons are then mixed together on a beamsplitter and detected. The major problem with these types of entangling schemes is that they are non-deterministic. For example, in the Barrett and Kok scheme, there is a 50% chance that the photon detection collapses the qubits into an un-entangled state at the end of the protocol. Therefore, even theoretically, any entangling gate utilizing this method will never have a failure rate less than pf =0.5. The extremely high failure rate of emission schemes implies that standard gate based quantum computation can never be used (as error correction can never handle error rates of 50%). Instead, cluster state computation [BR01, RB01] is employed. This solves the problem of non-deterministic interactions, but it requires many more qubits to implement algorithms and the eﬃcient preparation of an appropriate, highly entangled, cluster prior to computation. As with loss errors, our scheme solves this problem, in that the interaction is completely deterministic. The photon measurement is random, but regardless of the result the photons are always projected into an entangled state. The measurement result only determines the parity of the resultant projection. Since the parity of the projected entangled states are interchangeable through local gates, and we have assumed that each atom/cavity qubit has appropriate local control, classically controlled local gates can be applied to convert any 1 eigenstate into − +1.

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6.2.1 Preparation of stabilizer states

As detailed in chapter 4, parity gates are closely linked to the concepts of stabilizer states. Stabilizer states were introduced in reference to quantum error correction, however there are a many more useful entangled states which can be expressed in terms of stabilizers. Using the bus protocol we are able to prepare any multiparticle stabilizer state. First I detail how the basic protocol (utilizing two data qubits per interaction) allows for the preparation of N qubit GHZ states and Linear Cluster states, after- which I will demonstrate how a small modiﬁcation to the protocol can be used to prepare any N qubit stabilizer state. As the fundamental interaction of the quantum bus takes an arbitrary qubit state φ to (X I I X ) φ , I will assume that after the completion of the | i 1 2 ± 1 2 | i protocol, the local operation X1 is always applied. This will simply ensure that the projected state is (I I X X ) φ . This is only relevant for this section, 1 2 ± 1 2 | i since applying the X1 correction preserves the parity of the conjugate operator

(Z1Z2).

GHZ and Linear Cluster State Preparation

Using the stabiliser formalism, highly entangled multi-qubit states, speciﬁcally GHZ and linear cluster states can be prepared when only two-qubit parity measurements and single qubit gates are available. The method requires the ability to initialise qubits in the 0 state, apply single qubit Hadamard, X and Z gates | i and the ability to perform two qubit XX parity measurements. Combining the quantum bus interaction with the ability to do local operations directly on data qubits satisﬁes these conditions. The preparation of multi-qubit GHZ states using the quantum bus is straightforward. The stabilizer structure for an N qubit GHZ state is completely described by the operators,

N K = X⊗ , K = Z Z , j [1, ..., N 1]. (6.13) 1 j j j+1 ∈ − where the signs are omitted for notational convenience and identity operators ⊗ are implied. We ﬁrst consider an array of N atom/cavity qubits, each initialized

182 6.2. INFORMATION FREE QUANTUM BUS

in the state, 0 . The stabilizer structure for this initial state is given by, | i

K = Z , j [1, ..., N]. (6.14) j j ∈

For N cavity qubits, we introduce N/2 photonic qudits, the optical modes for each photon are arranged so that any two qubits can be coupled and the maximum number of interactions can be performed simultaneously. We use each single photon to perform a parity check of the operator X2j 1X2j, for j [1, ..., N/2]. − ∈ Assuming that each interaction projects the qubits into a +1 eigenstate of XX, the stabilizer set will now be described by the operators,

Kj = X2j 1X2j, Ki = Z2j 1Z2j, j [1, .., N/2], (6.15) − − ∈ which is the stabilizer structure for N/2 Bell states. Next, the photonic routing is changed such that each photonic qudit links each Bell pair. Local Hadamard gates are applied to every qubit in the register, after which another pairwise interaction in performed. After the interactions, local Hadamard gates are again applied to every qubit. This application of local Hadamard gates is needed since we wish to perform a parity check of the operator Z Z , j [1, ..., N 2]. 2j 2j+1 ∈ − The photonic interaction naturally measures the operator XX, hence performing local Hadamard gates before and after the parity check will convert X to Z and visa-versa (as HXH = Z). After the application of this second parity check, the N qubits are projected to an eigenstate of Z Z for j [1, ..., N 2]. Since the stabilizer structure for 2j 2j+1 ∈ − any given state must be formed from an Abelian group, the previous stabilizers which anti-commute with Z Z for j [1, ..., N 2] will no longer be members 2j 2j+1 ∈ − of the stabilizer group, while all commuting combinations will remain. As the initial state consisted of N/2 Bell pairs, the only X stabilizers which commutes N with Z Z for j [1, ..., N 2] is K = X⊗ , in contrast all Z stabilizers in 2j 2j+1 ∈ − 1 Eq. 6.15 commute and hence remain in the group. Therefore after this operation the state is described by the operators,

N K = X⊗ , K = Z Z , j [1, ..., N 1]. (6.16) 1 j j j+1 ∈ − which is an N qubit GHZ state. Using local operations and pairwise applications of the bus protocol an N qubit GHZ state can be prepared in two steps. This assumes that all pairwise interactions of isolated qubits can be performed

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simultaneously by utilizing multiple photonic qudits. Note, that since the interactions are mediated by single photons, the topological links (i.e. which two qubits we decide to link via an XX parity check) do not necessarily correspond to physical links of qubits. Physically, each atom/cavity qubit can be anywhere within the qubit register for a certain topological geometry. For each single photon source and beamsplitter required by the bus protocol, the output modes can be physically connected to any two cavity systems, provided that no two modes from separate photonic qudits overlap. An N qubit GHZ state is not the only multi-qubit entangled state that can be eﬃciently prepared using the two-qubit version of the bus protocol. Raussendorf and Briegel [BR01, RB01] demonstrated that any i-qubit cluster state, CS , is | ii deﬁned by the eigenvalue equation, K(a) CS = CS where, | ii | ii

K(a) = X Z a =1...i, (6.17) a b ∀ b ngbh(a) ∈ O and ngbh(a) represents qubits linked to site a in the cluster (neighbours), in arbitrary dimensions. Linear cluster states for 2 and 3 qubits are equivalent to Bell and three qubit GHZ states respectively (up to local operations). For N > 3 the number of basis terms for cluster states grow quickly, hence it is more convenient to express large cluster states via the eigenvalue equations. For example, a 4 qubit linear cluster state can be generated by the operators,

K(1) = XZII, K(2) = ZXZI, (6.18) K(3) = IZXZ, K(4) = IIZX.

Since a four-qubit cluster state satisﬁes, K(a) CS = CS , the operators K(a) | i4 | i4 form a basis set of the stabiliser group for a 4 qubit linear cluster state. The stabiliser group can be used to specify the topology of a given cluster state, without having to write out the state directly. Linear cluster state preparation using the bus protocol can be achieved by examining the stabiliser structure. The stabilisers for N qubits are generated by

184 6.2. INFORMATION FREE QUANTUM BUS

(neglecting identity operators),

(1) K = X1Z2, (N) K = ZN 1XN , (6.19) − (j) K = Zj 1XjZj+1, j = [2, 3, ..., N 1]. − − Preparing a state that satisﬁes this stabiliser structure using only XX and ZZ parity measurements (where as noted before, ZZ parity measurements can be achieved by applying local Hadamard gates before and after the interaction), combined with direct single qubit operations, requires linking the cluster together sequentially. To show the method explicitly, we detail the required steps needed to prepare a 4 qubit linear cluster state, after which adding links and expanding the cluster is straight forward. The analysis to follow assumes that we always measure the +1 eigenstate of any given operator (i.e. the photon is measured at A ), if the 1 eigenstate is obtained, simply apply local X and Z gates to 1 − convert the state to a +1 eigenstate. However, since X and Z gates are part of the Cliﬀord group, all these corrections can be applied at the end of the state preparation. Begin by initialising four qubits in the state φ = 0 0 0 0 , where | i | iQ1 | iQ2 | iQ3 | iQ4 [Q1, Q2, Q3, Q4] are the four atom/cavity qubits. The stabiliser group can be generated by the 4 operators, Zj, j = [1, 2, 3, 4]. Measuring the operator IXXI, via coupling a single photon to cavities two and three, will project the state into a stabilised eigenstate of IXXI and remove all existing stabilisers that anti-commute with IXXI. In this case, the stabilisers IZII and IIZI are removed, while IZZI commutes with IXXI and hence remains in the group. After measurement, the state of the computer will be stabilised by the basis operators, K(1) = IXXI, K(2) = IZZI, K(3) = IIIZ and K(4) = ZIII. Qubits two and three are now in an entangled Bell state described by the basis stabilisers IXXI and IZZI, while qubits one and four remain un-entangled. We now perform single qubit Hadamard rotation on Q1 which transforms the above basis stabilisers to, K(1) = IXXI, K(2) = IZZI, K(3) = IIIZ and K(4) = XIII. Combining the stabilisers (IXXI,XIII) and (IZZI,IIIZ) shows that the qubits are also stabilised by the operators (XXXI,IZZZ). To produce the four qubit linear cluster state now requires ZZII and IIXX parity measurements. As these operator measurements act on independent qubits they can be performed in parallel given independent photonic qudits. This will

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project the qubits into an eigenstate of these stabilisers and remove all previous non-commuting operators. From the above stabilisers, K(1) = IXXI, K(2) = IZZI, K(3) = IIIZ and K(4) = XIII anti-commute with either ZZII or IIXX, while the stabilisers XXXI and IZZZ commute and hence remain in the set. The qubit register will now be in the stabilised state generated by the operators, K(1) = ZZII, K(2) = XXXI, K(3) = IZZZ and K(4) = IIXX. If a Hadamard rotation is performed on qubits one and three the stabiliser group is rotated to,

K(1) = XZII, K(2) = ZXZI, (6.20) K(3) = IZXZ, K(4) = IIZX, which is the basis set of stabiliser operators describing a four qubit linear cluster state. Extending this scheme to an N qubit linear cluster state is similar. Initially prepare N qubits (for simplicity assume N even) in the state φ = 00...00 . | iN | iN Hadamard rotations are performed on all odd numbered qubits except for the two at the centre of the chain and the following operators are measured,

Step 1 XN/2XN/2+1,

Step 2 ZN/2 1ZN/2 and XN/2+1XN/2+2, − (6.21) Step 3 XN/2 2XN/2 1 and ZN/2+2ZN/2+3, − − Step N/2 Z1Z2 and XN 1XN . − Hadamard gates are again applied to all odd numbered qubits, after which the generators of the stabiliser set are identical to a linear cluster state. Although additional links in the cluster are created sequentially, the first link is made from the centre of the chain and subsequent links formed from this point, increasing the number of possible operations that can be done in parallel. Using this method, an N qubit linear cluster state can be prepared using N/2 time steps (for N even) or N/2+1 time steps (for N odd). The preparation of linear cluster states via the photonic bus is useful in the preparation of multi-qubit entangled systems, however Nielsen [Nie06] has shown that linear cluster states are insufficient for universal quantum computation. The original proposal of Raussendorf and Briegel showed that a 2-D tiled cluster state must be used, however this version of the bus protocol, combined with single qubit gates is insufficient to create such a state directly.

186 6.2. INFORMATION FREE QUANTUM BUS

Preparation of Higher order stabilizer states

The previous analysis demonstrates how a certain subset of stabilizer states can be prepared using only pairwise interactions enabled by the photonic bus. GHZ states and Linear cluster states are unique in this regard as they can be sequentially constructed from only 2-dimensional parity checks (exhibited by the bus interaction). However, we are able to slightly modify the physical architecture such that the parity checks can be of arbitrary dimension. In the original version of the protocol, a single atom/cavity qubit is coupled to the output modes of the beamsplitter. Now consider the situation where we instead couple N/2 atom/cavity systems to each output mode in the beamsplitter. As the atom-photon interaction does not destroy the bus photon it is able to induce a bit-ﬂip in one cavity, leak out, enter the next cavity and induce a bit-ﬂip etc.. Hence for an arbitrary N qubit state, φ the interaction with the photon | iN leads to,

1 U ψ A1 φ N = ( ψ B1 ψ B2 ) φ N | i ⊗| i √2 | i −| i ⊗| i 1 ( ψ B1 X1X2...XN/2 ψ B2 XN/2+1XN/2+2...XN ) φ N → √2 | i −| i | i (6.22) Hence, once the photon has been re-mixed on the beamsplitter the total state of the system is given by,

1 ψ A1 (X1X2...XN/2 + XN/2+1XN/2+2...XN ) φ N 2| i | i (6.23) 1 + ψ (X X ...X X X ...X ) φ 2| iA2 1 2 N/2 − N/2+1 N/2+2 N | iN

Measuring the location of the photon will collapse the qubit state to a 1 eigen- ± N state of the operator X⊗ . Therefore, any higher dimensional operator can be measured by simply coupling the photonic qudit to more and more atomic qubits before it is re-mixed on the beamsplitter prior to measurement. This implies that any state which can be described in terms or stabilizers can be prepared via local operations and the transport bus. Instead of slowly constructing the stabilizer structure of the state with 2-dimensional parity checks, each of the N stabilizers describing the state can be directly measured by classical routing of the photonic qudit. Once each of the N stabililzers have been measured, the ﬁnal state of the system will be (up to local operations) equal to the desired

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stabililzer state. This ability to prepare any stabilized state can therefore be utilized to quickly prepare codeword states for quantum error correction or the 2-D cluster geometry required for universal quantum computation. However, as I mentioned before, immediately moving to cluster state quantum computation has associated penal- ties, namely many more qubits are required to run a quantum algorithm and even eﬃcient preparation of a large cluster is useless unless physical qubits can maintain coherence long enough to perform large algorithms. As the fundamental interaction exhibited by the photonic bus is deterministic, can we still use a quantum gate based method for computation?

6.2.2 Universality using the Quantum bus

Quantum gate universality can be achieved if we employ the results of Aliferis and Leung [AL04] and their work into teleportation based quantum computing. The quantum bus protocol already assumes that single qubit gates can be performed directly on data qubits, hence the ability to simulate any entangling gate is suﬃcient for universality [DNBT02, BDD+02]. We demonstrate explicitly how a controlled phase rotation (CZ gate) can be enacted over two qubits, using the quantum bus combined with local operations. Consider an arbitrary two qubit state φ = α 00 + β 01 + γ 10 + δ 11 | i12 | i | i | i | i and a third data qubit prepared in a + = ( 0 + 1 )/√2 state that acts as | i3 | i3 | i3 an ancilla. Using the bus interaction, a Z1Z3 operator measurement is performed on the control and ancilla qubit (again we assume that the photon is localized at

A1 and +1 eigenstates are projected, if not the classical measurement record can be used to correct the state using local X and/or Z gates). This operation takes the combined qubit/ancilla state to,

′ φ = φ + α 000 + β 010 γ 101 δ 111 . (6.24) | i | i⊗| i→ | i | i − | i − | i

A Hadamard gate is applied to the target qubit taking the combined qubit/ancilla state to, ′ φ = α 0+0 + β 0 0 γ 1+1 δ 1 1 . (6.25) | i | i | − i − | i − | − i

An X2X3 operator measurement is now performed on the ancilla and target qubits

188 6.2. INFORMATION FREE QUANTUM BUS

taking the state to,

′ φ = α 0++ β 0 γ 1++ δ 1 . (6.26) | i | i − | − −i − | i − | − −i

Performing a second Hadamard rotation on the target qubit leads to

′ φ =(α 00 β 01 γ 10 δ 11 )12 0 3+ | i | i − | i − | i − | i ⊗| i (6.27) (α 00 + β 01 γ 10 + δ 11 ) 1 . | i | i − | i | i 12 ⊗| i3 The ancilla qubit is now locally measured in the computational basis. If it is measured in the 0 state, local Z gates are applied to both the control and target | i qubit. If the ancilla is measured in the 1 state, a local phase gate is applied to the | i control qubit. After these corrections the state has been transformed from φ to | i CZ φ . Therefore, using specific parity measurements and local gates, a CZ gate | i can be effectively simulated across two spatially isolated qubits by introducing a third ancilla. Since a CZ gate can be directly implemented in this scheme, and we have assumed that single qubit operations can be implemented directly on data qubits, universal computation is possible using the transport system. Using this information free quantum bus, we are not only able to prepare an extreamly large class of useful, multi-qubit entangled states but we are also able to achieve universal quantum computation with gate based computation. This gives this system a huge amount of flexibility when optimizing any potential large scale, distributed quantum computer.

6.2.3 Atom/cavity interaction with photonic qudit

The required atom/photon interaction, crucial to the implementation of the quantum bus, has already been demonstrated at microwave frequencies. Schuster et. al. [SHS+07] has demonstrated the non-destructive interaction we require where a single microwave cavity photon produces an eﬀective stark shift on a classically driven transition on a Cooper-pair box. These results were presented from the perspective of individual photon number detection, but the scheme can be inverted and used as the primary resource for a microwave version of the photonic bus. In the optical regime, we can consider two separate schemes. The proposal of

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Figure 6.2: Four level atomic system required for the quantum bus. The atomic qubit is defined as the 1 and 3 states, with classical pumping fields defined | i | i by their Rabi frequencies, (Ω1, Ω2), used for single qubit operations. A single photon pulse is introduced to the cavity in a controlled manner using Q- switching [GSPH06]. The single photon pulse induces a light shift on the state 3 , with strength β2/∆. Provided the photon remains in the cavity long enough,| i a π phase shift can− be induced on the atom without destroying the photon. After the atom/photon interaction is complete, the photon is switched out of the cavity and back to the free space optical modes.

Duan, Wang and Kimble [DK03, DWK05] considers a single atom/cavity system, where a single photon reflecting from the cavity will produce a π phase shift if the atomic qubit is in the state 0 . Hence any arbitrary state of the atomic | i qubit will transform as, α 0 +β 1 β 1 α 0 without destroying the photon. | i | i→ | i− | i Therefore, to achieve the required X operation on the atomic qubit, Hadamard and X gates are locally applied before and after the interaction with the photonic qudit. A second method employs a four-level atom in the N configuration, shown in Fig. 6.2. The general principle is to induce a phase shift Z σ on the atom, ≡ z conditional on the presence or absence of a photon in the cavity mode. Here, the atomic qubit is defined via the levels 1 and 3 and classical fields (defined | i | i via the Rabi frequencies, Ω1, Ω2) are used to perform single qubit rotations on each atomic system. Readout for each qubit can be achieved through classical pumping of the 3 4 cycling transition and observing photoluminescence. | i↔| i To achieve the required atom/photon interaction, we employ the idea of cavity Q-switching [GSPH06] in order to control the input/output pulse into the

190 6.2. INFORMATION FREE QUANTUM BUS

cavity system. A single photon is adiabatically switched into the cavity, where it is off-resonant with the 3 4 transition. Therefore, a light-shift will be | i→| i induced on the state 3 . The magnitude of the shift is well known and is given by, 2 | i δ = β , where the detuning of the photon frequency, ∆, satisfies ∆ β, where − ∆ ≫ β is the atom/photon coupling strength. Therefore, to induce a phase shift of γ, γ∆ the photon must be present in the cavity for a time given by, t = β2 . Consistent with the analysis in [GSPH06], this implies that the photon storage time, κ =1/t β2 must be κ = γ∆ . Taking γ = π ensures a full phase flip is applied to the atomic system. The photon is then Q-switched out of the cavity back into the optical mode using shaping techniques [GSPH06, FRDM07] to ensure correct mixing on when the photonic qudit is re-mixed on the beamsplitter. As the interaction on the atomic qubit is actually a phase flip, Z, Hadamard gates are applied locally to the qubits before and after the atom/photon interactions to ensure that the operator XX is measured over the two qubit system. Later, in section 6.4.3, we revisit this interaction from the perspective of entangled photon preparation and detail some of the experimental cavities that have been reported [Tab. 6.1] and how close these systems are to achieving the required transformation.

6.2.4 Implementation of the bus protocol in Solid State Systems

Although I have introduced the quantum bus protocol from the standpoint of an optical system coupled to atom/cavity qubits, the generalized protocol can be applied to any system where an effective single particle spatial superposition can be established. In particular we can consider a second implementation for a solid state quantum computer. It is well known that the original Kane quantum computer suffers from numerous problems related to large scale applications. Two specific examples is the high qubit density and the linear nature of the architecture design. In the original proposal of Kane, donor spin qubits are separated by an approximate distance of 20-25nm, and each pair of qubits are controlled by three surface gates, an A gate, utilized for single qubit rotations through the donor electron-nuclear hyperfine coupling and a J gate used for controlled interactions between qubits [Kan98]. This requirement of three classical gates approximately every 50nm presents an

191 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

enormous problem. The classical wiring of these surface gates [COI+03], possible cross-talk problems [KWH06] (where one gate influences not only the target qubit but neighboring spins as well) and the inevitable quantum behavior in otherwise classical structures implies that for the Kane system to be feasible, a high level of distribution is needed for the qubit array. Additionally, the Kane computer is an inherently linear design, with qubit- qubit coupling restricted to nearest neighbors. As seen in chapters 3 and 4, this not only makes algorithm design more difficult, but also leads to problems with Fault-tolerant computation as all transport is achieved via the SWAP channel. It has also been shown [KHS01] that donor alignment is crucial for the coupling between spins. If donor spins are improperly aligned in the substrate, the qubit- qubit coupling can drop to zero, effectively breaking a linear register into two smaller computers. These (and other) problems is why transport and a high level of distribution is critical for solid state architectures such as Kane. There is in fact a mechanism for realizing our quantum bus protocol in the Kane system where each data qubit is the nuclear spin of a phosphorus atom and the bus particle is a single electron which can be placed into spatial super-positions using adiabatic tunneling. It was first proposed by Greentree et. al. [GCHH04] that an adiabatic transport protocol can be implemented in solid state quantum dot systems, allowing for the movement of a single electron between two spatial locations. Further work by Greentree and myself [GDH06b] demonstrated that a slight modification to the protocol can be used to prepare a coherent spatial superposition of the wavefunction for a single electron. This protocol has been named Multiple Receiver via Adiabatic Passage (MRAP). To illustrate the protocol, consider the system shown in Fig. 6.3 that contains a single electron. Here, each node represents an ionized phosphorus donor (one site will be neutral due to the presence of the electron), while each connecting line represents a non-zero, controllable, tunneling rate between donor sites. Un- like the photonic system presented earlier, the electron does not form a spatial qudit (4-level system), but instead forms a qutrit (3-level system), where the positional eigenstates are denoted A , B and B . Like the photonic system, the | i | 1i | 2i transport protocol induces a unitary operation over the spatial degree of freedom, hence any internal state of the electron (for example, spin) is carried along as a spectator and will therefore be ignored.

192 6.2. INFORMATION FREE QUANTUM BUS

Bob 1

Alice

Bob 2

Figure 6.3: (From Ref. [DGH07]) Schematic of the conﬁguration required to demonstrate the qutrit transport protocol using a spatially deﬁned bus qutrit. By utilizing the MRAP protocol, the central site, C is never occupied. Here the single lines correspond to controlled tunnelling matri| i x elements for the single control particle Hamiltonian, and the double lines correspond to the controlled interactions between the Bob sites and the donor qubits. Once the protocol is complete, the spatial location of the electron is measured, using a single electron transistor. Depending on if the electron is localized at A , the data qubits Q1 and Q are projected into 1 eigenstates of XX. | i 2 ±

If we deﬁne the coupling between each site and the central dot, C ,asΩ for | i α α = A, B1, B2 (assumed to be real and positive), the Hamiltonian for the system can be written as,

H = Ω (t) C A + Ω (t) B C + Ω (t) C B + h.c. (6.28) A | ih | B1 | 1ih | B2 | ih 2|

Here we have assumed that Alice, A , and the two Bob sites, ( B , B ) can | i | 1i | 2i time-vary the tunneling matrix element (through local control of surface gates) and the on-site energies for each site remains constant and equal. Using this Hamiltonian we are able to determine which sites are adiabadically connected at zero energy by examining the null-space of the Hamiltonian. These

193 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

are,

ΩB1 ΩA D1 = A B1 , | i 2 2 | i − 2 2 | i ΩA + ΩB1 ΩA + ΩB1 (6.29) q ΩB2 q ΩA D2 = A B2 . | i 2 2 | i − 2 2 | i ΩA + ΩB2 ΩA + ΩB2 q q Where the time dependance of the Ω’s have been omitted. The null-Space of the Hamiltonian does not contain the state C and provided we remain | i in the null-Space, the only states which can be populated are A , B and | i | 1i B . Hence this single electron deﬁnes a spatial qutrit over the positional state | 2i ψ =( A , B , B )T . | i | i | 1i | 2i

Assuming that local barrier control at each site can be tuned, ΩA, ΩB1 and

ΩB2 can be chosen to transport the electron from Alice to either of the Bob sites. If we consider the case where either Ω (t)=0 t or Ω (t)=0 t, then the B1 ∀ B2 ∀ null states D and D adiabadically connect A to B or B . This is the | 1i | 2i | i | 1i | 2i standard coherent transport protocol ﬁrst introduced in [GCHH04].

We instead choose the following functional forms for ΩB1 and ΩB2 ,

t max [1 erf( σ )] ΩB1 = ΩB2 =ΩB = ΩB − , 2 (6.30) [1 + erf( t )] Ω = Ωmax σ . A A 2

max max where σ is the roll-oﬀ of the error function (erf) and ΩB and ΩA are the maximum values of the tunneling matrix elements. These pulse functions are known as the counter-intuitive pulse sequence since ΩB is pulsed before ΩA to transport the electron from Alice to Bob. The state which connects A to B | i | 1i and B2 is, | i 2Ω A Ω ( B + B ) D = B| i − A | 1i | 2i , (6.31) 3 2 2 | i 4ΩB + 2ΩA where D = D + D up to re-normalizationp and is also in the null-Space of | 3i | 1i | 2i H. Therefore, assuming that the electron is initially localized in the state A | i (when Ω 0), each tunneling term is adiabadically swept in the counter- A ≈ intuitive direction such that Ω Ω . In response, the electron will coherently B ≫ A

194 6.2. INFORMATION FREE QUANTUM BUS

tunnel to a spatial superposition of the sites B and B , i.e., | 1i | 2i 1 A ( B1 + B2 ). (6.32) | i→ √2 | i | i

As these sites are adiabadically connected, it can be shown that the reverse application of the tunnelling pulses (i.e. applying ΩA before ΩB ) will take,

1 1 B1 A + ( B1 B2 ), | i→ √2| i 2 | i−| i (6.33) 1 1 B2 A ( B1 B2 ). | i→ √2| i − 2 | i−| i

Therefore we can summarize theses sets of transformations as an eﬀective unitary operation over the spatial degrees of freedom which describes the transport process of the electronic qutrit as,

0 1 1 √2 √2 U =  1 1 1  . (6.34) √2 2 − 2 1 1 1  √ 2 2   2 − −    The quantum bus protocol is now very similar to the photonic case already introduced. We assume that a donor qubit is placed in spatial proximity to the qutrit sites B and B , and that a standard CNOT gate can be applied between | 1i | 2i the electron located at each Bob site and the data qubit, Qi, i = [1, 2]. The bus electron is initialized at the site A and the MRAP protocol per- | i formed. Assuming the two data qubits, Q and Q are in the state φ we have, 1 2 | i 1 Φ = A φ ( B1 + B2 ) φ , (6.35) | i | i⊗| i→ √2 | i | i ⊗| i

A standard electron spin CNOT gate [LD98, HHF+05] is performed between the qubit donors and the local Bob sites, where we have assumed that the internal spin of the transport electron is polarized in the 1 state throughout the transport | i protocol, giving,

1 1 Φ = ( B1 + B2 ) φ ( B1 X1I2 + B2 I1X2) φ , (6.36) | i √2 | i | i ⊗| i→ √2 | i | i | i where X1 and X2 are bit-ﬂips on the donor qubits and (I1, I2) are identity oper-

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ations. The MRAP protocol is then reversed and the electron transported back. This gives,

1 Φ = U( B1 X1I2 + B2 I1X2) φ | i √2 | i | i | i (6.37) 1 1 = A (X1I2 + I1X2) φ + ( B1 B2 )(X1I2 I1X2) φ . 2| i | i √2 | i−| i − | i

As with the photonic case, the electron is measured at A , using a single electron | i transistor (SET). If the electron is found at A , the donor qubits are projected | i into a +1 eigenstate of XX. If the electron is not found at A then the donor | i qubits are projected to a 1 eigenstate of XX. If the charge measurement results − in no detection at A , then a charge phase ﬂip needs to be performed on the state | i ( B B )/√2, after which the MRAP protocol will take the electron back to | 1i−| 2i A . The other option is to simply “ﬂush” the electron from the bus system and | i re-inject a new electron to A for further reuse. | i The MRAP transport method for single electrons exhibits exactly the same interaction between spatially isolated data qubits as the photonic bus qudit. Fig 6.4 illustrates the transformations which entangle two qubits as well as the classical pulse sequence applied to the surface gates.

6.2.5 Conclusion

Here we have presented a quantum bus protocol which can be used both for gate based universal computation and as a method for efficient preparation of large multi-qubit entangled states. We have introduced the concept of the quantum bus using a specific system, where atom/cavity qubits are coupled via a photonic spatial qudit which never carries computational information. The extremely mobile nature of the bus photons immediately allows for a highly distributed scheme for quantum information processing that is not susceptible to traditional error mechanisms associated with schemes such as flying qubits or non-determanistic photon emission protocols. We have shown how the natural protocol exhibited by the bus interaction is equivalent to parity measurements, leading to an immediate scheme for the preparation of large multi-qubit states that can be expressed in terms of stabilizers. Through specific examples we have demonstrated how the two-qubit protocol + appropriate local control can be used to prepare N qubit GHZ states and linear

196 6.2. INFORMATION FREE QUANTUM BUS

Cluster states and with a small modification to the photonic routing how any multi-qubit stabilizer state can be deterministically prepared. We have detailed the basic operating principles of the atom-photon interaction, which lies at the heart of the protocol, from the standpoint of several atom/photon schemes already proposed [DK03, SHS+07] and also a new method based on a 4-level atomic system. Finally, we have presented an alternate scheme for the quantum bus in the solid state regime, where the interaction is mediated by a spatial qutrit, utilizing the methods of adiabatic transport (MRAP). This approach to direct synthesis of parity measurements may have significant application to improving the efficiency of quantum operations, and con- stitutes a different approach to the generation of remote entanglement and/or universal quantum computation from more traditional methods. Although this section has been focused heavily on the obvious benefits of the quantum bus, parity measurements can have significant application on their own for non-determanistic, distributed quantum computation. The next section introduces a distributed model for ion-trap quantum computers using this fundamental interaction. Unlike solid state computers and atom/cavity systems, we have yet to find a sufficient physical system to directly implement the full quantum bus in ion trap computers. However, the high degree of experimental control which has already been demonstrated in ion traps does lead to an elegant distributed system, using parity measurements, which may be implemented in the near future.

197 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

(I) (II) (III) (IV) tne

]stin 1.0 m erusae

0.8 2 u .bra[ u 0.6 , X 1 X M

0.4 ecil Ω 0.2

0 A time [arb. units] (I) Initialisation

(II) Qutrit Transport

(III) Qubit coupling

(IV) Measurement

Figure 6.4: (From Ref. [DGH07]) Pulsing sequence for tunneling matrix elements (top) and schematics showing system evolution through the MRAP protocol to realise two-qubit parity measurements. (I) The qutrit is initialised at A with the two qubits in some arbitrary state φ . (II) MRAP takes the qutrit| toi the state | i ( B1 + B2 )/√2, (III) the controlled unitaries (X1,X2) are performed between each| i Bob| sitei and the qubits, conditional on the presence of the qutrit at the appropriate site, and the system is transformed to ( B X + B X ) φ /√2. | 1i 1 | 2i 2 | i (IV) the transport is reversed and a projective measurement of the spatial qutrit at A performed (using, for example a SET charge detection). The results of this| measurementi projects the qubits into an eigenstate of the operator XX, and depending on the measurement result, a phase ﬂip at either B1 or B2 and further MRAP protocol can be used to return the qutrit to Alic|e fori reuse.| i

198 6.3. SCALABLE QUANTUM ERROR CORRECTION IN DISTRIBUTED ION TRAP COMPUTERS.

6.3 Scalable quantum error correction in distributed ion trap computers.

A promising candidate for quantum information processing is the ion trap, with superb coherence and few qubit control having been already demonstrated [LKS+05, HHR+05]. However, it is difficult to effectively control more than a few tens of ions in a single trap, hence several ideas have been proposed to overcome this lim- itation. Multiple microtraps can be constructed in the same structure with ions shuttled between them (CCD architecture) [KMW02]. The main disadvantage of a CCD trap is the difficulty in designing a micro-trap structure that allows for maximum parallelizability for both inter- and intra-logical operations. Shuttling heats ions up, requiring additional cooling in the interaction regions and slowing operation. Additionally, large numbers of electrodes and lasers would be required in a single device [Ste07]. Alternatively, ions in separate trap structures may be made to interact via a flying qubit photonic bus [CZKM97, Pel97, SL00, KLH+03, DBMM04, DMM+06, vEKCZ99]. If used directly to implement two-qubit gates, photon loss from the bus is a major problem and requires additional QEC overhead [RHG05, RRM07]. Alternatively, the photonic bus can mediate the generation of entanglement between traps which can then be used to perform gates [DB03]. The use of entanglement for intra-computer communication is not a new idea. For example, this has been proposed in CCD ion trap designs where EPR pairs are created locally and then the halves sent to entanglement stations distributed among a sea of qubits [MTC+05]. These entangled pairs would then be used to teleport qubits between memory, storage and processing regions, circumventing the problem of directly transporting data across the whole computer. However, since the entangled pairs themselves are created locally and then the separate halves physically moved to where they are required, this neccessitates the use of quantum repeaters and extensive purification. Furthermore in [MTC+05], all 49 qubits of a second level encoded logical qubit are teleported requiring many EPR pairs for transport in both directions. This section outlines a proof-of-concept architecture based around an ion trap processing node containing a relatively small number of ions, representing an encoded first level error corrected qubit, ancillas for Fault-tolerant operation, and an interface ion which can be entangled with its counterpart in another node

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SegmentedElectrodes

Data Ancilla Ions Ions ControlLasers

Optical OpticalLink Interface toOther Traps Ion

RFElectrodes

Figure 6.5: An ion trap node. A single ion trap contains enough ions for a single encoded qubit, ancillas and an interface ion. Conventional single and two-qubit operations are performed via axial phonon modes of the trap. The interface ion may be entangled with its counterpart in another identical trap via photon interference and path erasure. The resultant Bell link is used to perform inter- trap operations. The simplest node consists of a single optically coupled ion, ﬁve ions encoding a single logical qubit, and several ancilla for Fault-tolerant operations and singlet puriﬁcation.

200 6.3. SCALABLE QUANTUM ERROR CORRECTION IN DISTRIBUTED ION TRAP COMPUTERS.

(Fig. 6.5). An abstract basis of the scheme was suggested in [DB03], but here we analyze a concrete realization, taking particular attention to the requirements of error correction and Fault-tolerant operation. In particular, we show how local and non-local logical operations can be reliably performed directly between two nodes via Parity measurements, introduced in the previous section within the context of an information free quantum bus, from which scaling to an arbitrary sized quantum computer follows. A small prototype is presented which is within reach of current experiment.

6.3.1 Architecture

The basic architecture is illustrated in Fig. 6.6. A network of local processing nodes are connected by optical ﬁbres and a multiplexing switch. In each trap node is a small array of ions upon which conventional single and intra-trap two- qubit operations can be performed. Pairs of nodes can be optically linked to beamsplitters and single photon detectors which entangle the interface qubits when subjected to appropriate laser excitation and conditioned upon a correct sequence of detector clicks. The resulting Bell pair is then used to perform inter- node operations.

6.3.2 Operation

We start off with all qubits initialized. Intra-trap operations are used to prepare encoded qubits. We assume that each trap can hold a sufficient number of ions to encode a logical qubit plus an appropriate number of ancilla ions for error correction in at least the first level of concatenation. Single qubit, non-trivial, logical operations (for example the T gate [Fow04]) are performed with the assistance of ancilla qubits in the local trap. For inter-node two-qubit logical operations, instead of directly interacting data qubits, we create Bell pairs spanning the nodes. By local operations and classical communication (LOCC), two-qubit gates can be performed without risking data loss between nodes.

Inter-node operations and encoded Bell state preparation

As an example of inter-node operations, consider the preparation of a logically encoded Bell state between two separate nodes. Each node houses between seven

201 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

PhotonDetectors

Multiplexer Beamsplitters

Node1 Node2 Node3 Node4

Figure 6.6: Distributed Ion Trap Architecture. The whole computer consists of a set of identical nodes, each holding a few physical qubits encoding a logical qubit, and associated ancillas. The nodes are connected by optical ﬁbre linking the interface ions in each node. A heralded probablistic procedure entangles pairs of ions in separate nodes via interference and path erasure. An optical multiplexer allows arbitrary pairs of nodes to be entangled, and parallel operation is achieved using multiple beam-splitters and detectors.

202 6.3. SCALABLE QUANTUM ERROR CORRECTION IN DISTRIBUTED ION TRAP COMPUTERS.

1 M M1 a) 0 H H M X Non-FT: Measure K Z ϕ L Z

X 1

M1 b) 0 X X M 0 H H M

0 X X FT: Measure K 0 X X 0 X X

Repeat if M = 1. X Z

ϕ 1 L Z X

Repeat at most three times, take majority.

Figure 6.7: Quantum circuit measuring the stabiliser K1 for the [[5,1,3]] quantum code. a) non-Fault-tolerant circuit b) basic Fault-tolerant circuit. The Fault- tolerant circuit first requires the preparation and verification of a four qubit GHZ state. If the verification measurement = 1, then the ancilla block is reset and prepared again. To protect against Z errors in the ancilla block, the circuit is repeated up to three times and a majority vote of the syndrome results is taken.

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and fourteen ions depending on whether Fault-tolerant error correction and gate operations are employed. The data ions in each trap will be encoded using the [[5,1,3]] code [LMPZ96, DS96], which is the smallest full quantum code, requiring ﬁve ions for a single logically encoded qubit protected from at most one error. The stabiliser structure [Got97] for the [[5,1,3]] code, Ki, i 1, 2, 3, 4 , and the ∈{ } logical bit (X¯) and phase (Z¯) operations are speciﬁed by,

K1 = XZZXI, K2 = IXZZX, K3 = XIXZZ, K4 = ZXIXZ, X¯ = XXXXX, Z¯ = ZZZZZ, (6.38) where X and Z are the Pauli σ and σ operators, I is the 2 2 identity ma- x z × trix, and the tensor product is implied. Error correction using stabiliser codes was detailed in chapter 4 where each of the four generators Ki are measured [Fig. 6.7(a)] either sequentially using a single ancilla, or simultaneously using four ancillas. Each of the sixteen possible four-bit results represent one of the correctable single qubit errors, as well as the case where no error occured. At a minimum, Fault-tolerant measurement of the stabilisers requires a four qubit GHZ state as an ancilla block [Fig. 6.7(b)]. Additionally, a ﬁfth qubit is used to verify the GHZ state against possible X errors which can subsequently propagate to the data block. Therefore the minimum number of ions in a single trap needed for logical encoding and correction is six, while a total of ten ions are needed to employ full Fault-tolerant correction sequentially. The interaction between logical qubits in separate nodes is mediated by interface ions entangled into Bell pairs by any one of a number of methods [CCGFZ99, PHBK99, BKPV99, SI03, DK03, BPH03]. Two-qubit gates can be performed directly using Bell pairs via LOCC [Got98b, VC04, DBMM04, DMM+06] with all

Calderbank-Shor-Steane (CSS) codes allowing logical controlled-σx (CNOT) and controlled-σz (CZ) gates to be applied block-wise between two data blocks, which are also inherently Fault-tolerant. However, the [[5,1,3]] quantum code is not a CSS code and block-wise CNOT or CZ gates are not possible. This also means that the more rapid method of error correction introduced by Steane [Ste97, Ste02], requiring a larger ancilla block, will not work with the [[5,1,3]] code. However, logical CNOT or CZ gates can be performed for any code that allows for block-wise single-qubit bit and/or phase operations, and using fewer interface qubits than the standard block-wise approach. This method utilizes the Parity

204 6.3. SCALABLE QUANTUM ERROR CORRECTION IN DISTRIBUTED ION TRAP COMPUTERS.

measurement gates introduced in the previous section. We will use the [[5,1,3]] code as an example to demonstrate the method, and to minimize the number of qubits required in each node. A CZ gate between two qubits can be written in terms of operators on an arbitrary two-qubit state ψ as, | i 1 CZ ψ = (II + ZI + IZ ZZ) ψ . (6.39) | i 2 − | i

To achieve this transformation, we append an ancilla qubit prepared in the state + = ( 0 + 1 )/√2, and measure the operators Z I Z and I Z X over the | i | i | i 1 2 3 1 2 3 three qubit system. After these measurements, and assuming that the qubits are always projected to a +1 eigenstate of these operators (otherwise local corrections can be applied), the ﬁnal state is given by

1 (I I I + Z I Z + I Z X + Z Z (X.Z) ) ψ + . (6.40) 2 1 2 3 1 2 3 1 2 3 1 2 3 | i| i

Since the Pauli operators X and Z anti-commute and that X + = + and | i | i Z + = , the state can be re-written as, | i |−i 1 ((II + IZ) ψ + +(ZI ZZ) ψ ), (6.41) 2 | i| i3 − | i|−i3 after which the ancilla is then measured in the computational basis. If the measurement result is 0 , ψ is projected to CZ ψ , otherwise it is projected to | i | i | i (IZ).CZ ψ upon which a local IZ correction is then applied. | i We use the above method to perform a logical CZ across two nodes. A single physical Bell state is prepared between two nodes each containing a logical qubit. Each half of the Bell state is used as a control qubit on the respective data block of an encoded qubit. For example, to measure the logical Z¯Z¯ operator across two logical blocks, local CZ gates are applied between each Bell pair qubit and the ﬁve ions representing the single logical qubit in each node. For a general state of two logical qubits ψ , the transformation is | iL 1 1 ( 00 + 11 ) ψ L ( 00 II + 11 Z¯Z¯) ψ L, (6.42) √2 | i | i | i → √2 | i | i | i where Z¯ is as in Eq. 6.38, a logical phase gate for the [[5,1,3]] code. A local

205 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

Hadamard gate is applied to both interface qubits, leading to the state,

1 1 j+k jk (II +( 1) Z¯Z¯) ψ L. (6.43) 2√2 | i − | i j,kX=0 Measuring the parity of interface qubits projects the data qubits into a 1 eigen- ± state of Z¯Z¯ for an even/odd parity result, hence performing the required parity check. Measuring an appropriate sequence of operators will enact a logical controlled phase rotation across two nodes. To perform the full CZ gate, an ancilla is needed which is finally measured in the computational basis. This ancilla need not be a fully encoded logical qubit in its own trap, it can just as easily be a single ion contained in either the control or target trap. However, to maintain Fault- tolerance this ancilla qubit should be encoded. By using operator measurements between traps, inter-logical operations can be performed directly on the [[5,1,3]] encoded data using only one interface qubit per trap. Localizing a single logical qubit plus appropriate ancilla ions for local error correction has several advantages. Intra-trap operations have been demonstrated on up to eight ions [HHR+05] and local operations and error correction should in- cur minimal overhead. Probabilistic entanglement generation [CCGFZ99, PHBK99, BKPV99, BPH03] does not pose a problem for inter-node operations as local error correction can preserve coherence between the generation of entangled links. This is substantially more advantageous than other highly distributed schemes [DBMM04] where all ions interact via these non-local linkages as qubits would be highly susceptible to memory errors while waiting for non-local links to be established. Local control over a few qubits also enables the use of entanglement purification of the interface ion Bell pairs in order to increase inter-trap gate fidelity [DB03]. Also, by structuring all inter-logical operations such that they are mediated by entangled links, larger trap structures, for example the CCD design of Kielpinski et. al. [KMW02], need only be designed and optimized for local error correction and dispensing with long range ion transport and routing. From an experimental standpoint, preparing a logically encoded Bell pair does not require the operator CZ gate in full. If the initial state is 0 0 , the | iL| iL measurement of the operator X¯X¯ is sufficient to produce the state ( 0 0 + | iL| iL 1 1 )/√2. Hence experimental demonstration of encoded Bell state prepara- | iL| iL tion does not need the third ancilla qubit required by the operator CZ gate.

206 6.3. SCALABLE QUANTUM ERROR CORRECTION IN DISTRIBUTED ION TRAP COMPUTERS.

Repeat N Times

0 0 0 Non-FT: Measure K Non-FT: Measure K Non-FT: Measure K Non-FT: Measure K 0 X Non-FT: Encoded Bell State Preparation Correction noeEncode 0 Encode X 5 5 0 X 0

0 1 2 3 4 X ION TRAP #1 0 X 0 H M 0 0 Prepare Bell Link 0 H M 0 0 X Z

0 Correction Non-FT: Measure K Non-FT: Measure K Non-FT: Measure K Non-FT: Measure K 0 X Z 5 5 0 X Z 0 0 X Z

0 1 2 3 4 X Z ION TRAP #2 0 0 0

Repeat N Times

Figure 6.8: Full quantum circuit for non-Fault-tolerant, non-local preparation of an encoded Bell state across two nodes. Five ions in each trap are ﬁrst encoded into the 0 L state after which local error correction is repeated continuously (say N times)| toi protect against memory errors while the Bell link is created. Once the Bell link is created, each interface ion is used as a control qubit for a blockwise X gate on each trap. The interface qubits are then measured locally and a classical Z¯ gate is applied to the second trap if the measurement result has odd parity. The ﬁnal state of the two traps is the encoded Bell state, ( 0 0 + 1 1 )/√2. | iL| iL | iL| iL

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Without maintaining Fault-tolerance, we only require ﬁve (data block) + one (ancilla) + one (interface) ions per trap to implement the protocol. Fig. 6.8 shows the complete quantum circuit required to implement the state preparation non- Fault-tolerantly, assuming that N full local error correction cycles are performed in the time required to prepare the inter-trap Bell link.

Fault-tolerant encoded state preparation.

This general method for preparing a distributed, encoded Bell state is not Fault- tolerant as utilizing a single qubit for correction allows errors to cascade into the data block. Also, the Bell pair interface can induce logical errors if it is not prepared correctly. Maintaining Fault-tolerance for local error correction is fairly straightforward [Chapter 4]. The stabilizers for the [[5,1,3]] code have a maximum weight of four, hence the ancilla ion used for correction is replaced with five ions, four of which are prepared in the entangled state ( 0000 + 1111 )/√2, after which the | i | i fifth is used to verify the ancilla state against X errors that can propagate to the data block. If verification fails, the ancilla block is reset and re-prepared. Once the state is verified, each of the four ancilla ions are coupled to the data block, with local CNOT and Hadamard gates (depending on the stabiliser structure) and measured to determine the syndrome. To protect against Z errors, occurring or propagating to the ancilla block, the syndrome is measured multiple times. At least two syndrome measurements are made, if they disagree a third syndrome is measured and a majority vote taken [Fig. 6.7(b)]. We adapt this general method to operator measurement gates between traps. Errors in the Bell link between nodes, either during preparation or during operation, can lead to multiple errors propagating to the data blocks. For Fault- tolerance we again use several ancillas, thereby ensuring that only one ancilla qubit interacts with one qubit within the data block. Measuring the operator U¯ U¯ , where U¯ X,¯ Z¯ , requires a Bell link between the two nodes and the 1 ⊗ 2 j ∈{ } ability to perform CNOT or CZ gates between each qubit in the Bell pair and their respective data block. If an X error occurs on the Bell pair then this can propagate to possibly all of the qubits within one of the nodes. If this occurs, the single qubit error will induce a logical error. To counter this, we introduce several more ancilla qubits into each node and verify the interface qubit state before coupling ions to the data block.

208 6.3. SCALABLE QUANTUM ERROR CORRECTION IN DISTRIBUTED ION TRAP COMPUTERS.

Repeat for odd parity measurement. 0 X 0 X Bell State 0 Prepare ION TRAP #1 0 Prepare X M 0 Bell State X M 0 0 X 0 X ION TRAP #2

Figure 6.9: Circuit to prepare and verify the interface ancilla blocks for Fault- tolerant operator measurement on the [[5,1,3]] code. The ancilla state requires the preparation of two Bell links between the separate data traps. After the local CNOT gates the second Bell link is measured. If the measurement result has odd parity, the interface block is reset and re-prepared. Local error correction can be performed on each data block while waiting for a veriﬁed interface ancilla block.

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The required circuit needed to prepare a sufficient interface ancilla for the [[5,1,3]] code is shown in Fig 6.9. We use two Bell pairs and two additional ancilla qubits in each node that are coupled to the original Bell pair through CNOT gates. After preparation, the ancilla blocks of the two nodes are in the state, 1 = ( 000 000 + 111 111 )( 00 + 11 ). (6.44) |Ai 2 | i1| i2 | i1| i2 | iv | iv The subscripts 1 and 2 represent the three ancilla qubits within each node, while the subscript v represents a second Bell link between the two nodes used for ancilla verification. Now a CNOT gate is performed between the last qubit in each node and the verification Bell state. If no errors have occurred, then the CNOT operations will leave the verification Bell pair invariant. Considering all the possible single X error locations during the preparation of the ancilla state we find the following unique states are possible.

1 = ( 000 000 + 111 111 )( 01 + 10 ), |Ai1 2 | i1| i2 | i1| i2 | iv | iv 1 = ( 000 111 + 111 000 )( 01 + 10 ), |Ai2 2 | i1| i2 | i1| i2 | iv | iv 1 = ( 000 011 + 111 100 )( 01 + 10 ), |Ai3 2 | ii| ij | ii| ij | iv | iv 1 = ( 000 001 + 111 110 )( 01 + 10 ), |Ai4 2 | ii| ij | ii| ij | iv | iv (6.45) where [i, j] [1, 2], [2, 1] . The verification qubits are measured in the com- ∈ { } putational basis and the ancilla is verified if an even parity result is obtained. Otherwise, either the ancilla or verification qubits have experienced a single X error and we repeat the preparation. The inclusion of the second Bell link between two nodes and the additional ancillas allows the verification of the ancilla state prior to coupling it to the data qubits, protecting the ancilla state from a single X error. Phase errors in the ancilla block result in an incorrect determination regarding which eigenstate the data qubits are projected to. To protect against this, the operator is measured two to three times and a majority vote taken. At each stage, error correction can be continuously performed on each data block while the interface ancilla block is prepared and verified. In Section 6.3.2 we showed how the Bell link enables the measurement of a

210 6.3. SCALABLE QUANTUM ERROR CORRECTION IN DISTRIBUTED ION TRAP COMPUTERS.

given Hermitian operator by performing a controlled operation from the Bell link qubits to each of the data qubits. For the [[5,1,3]] code, this allows logical X¯ and/or Z¯ operator measurements since these can be performed block-wise. To maintain Fault-tolerance, this would require six interface ions in each node, five of which are used to produce the non-local GHZ state across the two traps and the sixth used to verify the GHZ state. However we can reduce this to four in each node by exploiting the stabiliser structure of the [[5,1,3]] code. Any given logical state ψ encoded with the [[5,1,3]] code is stabilised by | iL the operators K1 to K4. Therefore ψ = Ki ψ , i 1, 2, 3, 4 . If a logical Z¯ | iL | iL ∈{ } (X¯) operation is performed on the state, it is not necessary to apply five single qubit Z (X) gates, but we can redefine the logical operators in terms of the stabilisers. Consider the first stabilizer K1 = XZZXI. Then,

Z¯ ψ = ZK¯ 1 ψ =(X.Z)II(X.Z)Z ψ . (6.46) | iL | iL | iL

Therefore only three non-trivial operators are needed to enact a Z¯ gate, hence within the interface block only four interface ions are needed, three of which are used to create the ancilla state (between two traps),

1 = ( 000 1 000 2 + 111 1 111 2), (6.47) |Ai √2 | i | i | i | i and the forth to verify the interface block. The total number of ions in each trap for full Fault-tolerant local correction and coupling between the traps is fourteen. Five ancilla ions are needed for local Fault-tolerant error correction of the five ion logical qubits, while four ions are needed as the interface ancilla block including two non-local Bell links, one to actually link the traps and one to verify the ancilla block. It should be noted that if continuous QEC is performed while the interface block is being prepared and verified that the ancilla ions used in the interface block are separate from the ancilla ions required for standard QEC. The total number of qubits needed for this scheme and the number of non-local Bell links for a general quantum code is significantly less depending on the size of the code used. For a general CSS n-qubit code correcting a single error, and assuming that no purification protocols are used, n Bell links are required to perform a block-wise CZ gate in one time step. In contrast, this scheme only requires two Bell links (regardless of the code size) to perform a CZ gate in several time steps. In between each step,

211 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

local error correction can be performed to protect against memory errors.

Extending the scheme to larger architectures

The above scheme of preparing a non-local encoded Bell state between two separate nodes can easily be extended to a much larger distributed system. Each node would be designed to house a single logical qubit, or several logical qubits for a general [[n,k,d]] code with k > 1. If multiple concatenation levels are war- ranted, then the node system would also have the requisite number of qubits and routing system to allow full Fault-tolerant error correction at all levels. The inter-logical operations, at the highest level of encoding are then performed using the non-local Bell links and the operator measurement protocol. For a multi- ply concatenated (mth) level qubit, the operator measurement formalism can be extended in a straightforward manner. For all CSS codes, block-wise Z and X operations are possible. Hence to measure these operators across two nodes at the mth level of encoding, a CZ or CNOT gate is performed between each half of a Bell link pair of qubits and all of the physical qubits in the two separately encoded nodes. Fault-tolerance would require a similar ancilla system as that used in the [[5,1,3]] example. Instead of using two Bell links and four ancilla qubits per trap, the total number of ancillas will be equal to Number of Ancilla = Wt(U¯)m +1, (6.48) where m is the concatenation level and Wt(U¯) is the minimum weight of the n qubit operation that invokes a blockwise logical U operation on the kth qubit (if multiple qubits are encoded within a single node). The number of Bell links required between nodes remains constant at two, unless a quantum code is employed that encodes multiple logical qubits. In this case inter-logical operations between states located in diﬀerent nodes will require two Bell links for each pair of qubit interactions between nodes. Figure 6.10 shows an example structure for a distributed computer using the CCD trap design. Each CCD chip is designed exclusively for a single logical qubit encoded with the [[7,1,3]] Steane code. Each chip houses seven data qubits, an additional 28 ancilla qubits which would allow for the simultaneous preparation and veriﬁcation of two separate ancilla blocks using the rapid method of Steane [Ste97, Ste02], and the four interface qubits which are required for Fault-tolerant operator measurements using the Steane

212 6.3. SCALABLE QUANTUM ERROR CORRECTION IN DISTRIBUTED ION TRAP COMPUTERS.

code. Each chip is manufactured and characterized separately and would be plugged in to the optical multiplexer, linking it to the rest of the computer. Within a larger architecture, the logical qubits needed for a given quantum algorithm are interspersed with logically encoded ancilla traps that are then used to perform logical CZ gates using the methods described in Section 6.3.2.

6.3.3 Node Design

To summarize the architecture, each node should satisfy the key requirements:

Contain a suﬃcient number of long lived physical qubits for an error cor- • rected logical qubit.

Contain an additional number of ancilla qubits for error correction and • operator measurements. There should be a fast and reliable measurement mechanism. The absolute coherence time of these ancilla may be traded against fast operations.

Possess a qubit which can be entangled with its counterpart in another • node. This process can be probabilistic but heralded.

Fast and reliable single and two-qubit operations within the node for sin- • gle logical qubit operations, error correction, operator measurements, and entanglement puriﬁcation.

A crucial element of the architecture is the ability to create entanglement between separate nodes. There are many proposed ways of achieving this for ions and atomic systems, for instance Refs. [DBMM04, DMM+06, CCGFZ99, PHBK99, BKPV99, SI03, DK03, BPH03]. The basic technique (originally proposed in [CCGFZ99], adapted to nitrogen vacancy centres in diamond in [BK05, BBFM06]) is to use a multilevel system consisting of the qubit levels, 0 , 1 {| i | i} and a third excited state 2 and the ability to coherently drive the 1 2 | i | i↔| i transition. Both qubits to be entangled are placed in the superposition state ( 0 + 1 )/√2 (e.g. by conventional qubit manipulations) and then subjected to | i | i a π-pulse on the 1 2 transition. The state 2 is only allowed to decay back | i↔| i | i to the state 1 , e.g. by selection rules. Either collection optics or cavities and op- | i tical ﬁbres are used to direct any spontaneously emitted photons to diﬀerent input ports of a 50 : 50-beamsplitter whose output ports are monitored by single photon

213 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

QEC DataIons Ancilla Ions

Ancilla Verification Ions

InterfaceIons LogicalQubitNode

SingleLevel[[7,1,3]] SingleLevel[[7,1,3]] SingleLevel[[7,1,3]] SingleLevel[[7,1,3]] QCCDLogicalQubit QCCDLogicalQubit QCCDLogicalQubit QCCDLogicalQubit 2 AncillaBlocks 2 AncillaBlocks 2 AncillaBlocks 2 AncillaBlocks

OpticalMultiplexer

Figure 6.10: CCD micro-trap structure [KMW02] for a single logical qubit using the [[7,1,3]] Steane code. Each chip houses 39 ions: 7 data ions, 28 ancilla ions (allowing for simultaneous preparation and veriﬁcation of two ancilla blocks using Steane’s rapid correction method [Ste97, Ste02]) and 4 interface ions for coupling to other logical qubits. The interface state required for the [[7,1,3]] code is identical to the [[5,1,3]] code since each of the seven dimensional stabilizers for the Steane code has weight four, hence Wt(Z¯) = Wt(X¯) = 3. Each of these chip nodes can then be connected to the optical multiplexer, increasing the total size of the quantum computer as needed.

214 6.3. SCALABLE QUANTUM ERROR CORRECTION IN DISTRIBUTED ION TRAP COMPUTERS.

detectors (Fig. 6.6). If, after a suitably long time, only one photon is detected, assuming perfect detection efficiency and no losses, the final state of the qubits is projected onto a maximally entangled state (depending on which detector fired) ( 0 1 1 0 )/√2 due to path erasure by the beamsplitter. Modifications to | i| i±| i| i this basic scheme have relaxed the need for perfect photon detectors, high cavity coupling and low losses [BPH03, BK05]. Schemes based on two-photon entanglement [SI03, DK03, DBMM04, DMM+06] utilize entanglement between the qubit levels and photonic properties such as polarization or frequency. These have the advantage of not requiring cavities for practical implementation. Experimental progress on the required optical excitation of ions [MMM+06] and two photon interference from separately trapped atoms [BJD+06] is encouraging. If multiple Bell links are required with a single node, the state of the interface qubit can be swapped to an ancilla for storage and the interface qubit re- entangled. Of course, multiple interface qubits would allow for parallel entangling operations but are not strictly necessary. By creating and storing many entangled pairs, standard entanglement purification techniques [BBP+96] could be used to increase the fidelity of the link between nodes. However, to achieve a sufficiently high purity, the initial number of entangled pairs may be too many to contain in a single trap. Techniques, such as nested entanglement pumping [DB03], may be able to greatly reduce the number of simultaneously stored Bell pairs required for a final high fidelity link, hence reduce the number of ancilla in each node. The basic technique uses one source of pairs (interface ions) to iteratively purify a fixed pair (ancilla) [BDCZ98, DBCZ99]. This latter pair can then be used to purify a third pair etc. Numerical studies in Ref. [DB03] indicate that only three nesting levels (the interface ion and three auxiliaries in each node) are required to achieve extremely high purity Bell links. Compared to standard recurrence methods, entanglement pumping carries only a polynomial overhead. Segmentation of a linear trap could be used to isolate the interface ion from the rest of the trap until required. By suitable geometry, the interface region would not impinge on intra-trap operation, either by phonon coupling or photon scattering. When entanglement is needed, the trap potentials are rearranged so that an ancilla ion could be placed into a common mode with the interface ion and quantum state transfer performed, afterwards which the ancilla would be brought back to the rest of the ions for further processing.

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Though we have primarily considered linear ion trap nodes, any other small qubit systems could be substituted as long as the above requirements are met, e.g. a CCD trap with an optical interface region (Section 6.3.2). The Penning trap [CPOC+06] has also been suggested as a candidate for quantum computation, with hundreds or thousands of ions in a single two-dimensional Coulomb crystal [DBr+98, PC06] and two-qubit gates via transverse phonon modes [ZMD06]. The large number of physical qubits would allow larger code words protecting against multiple errors and/or optimized for different error models. However, the rotation of the crystal would complicate ion addressing and optical coupling to the interface ion. Within each node, the physical qubits play different roles opening up the possibility of optimizating their separate properties. Data qubits require long coherence times, ancilla qubits could be optimized for fast operations and measurements. The interface qubit should have suitable optical properties the entanglement generation procedure. In a Paul ion trap, different ionic species may be utilized and loaded in order by frequency selective ionization [BBH+06]. Using heavier ions (such as Cd+) for the data qubits may reduce gate errors due to spontaneous decay from intermediate metastable states [WBB+03], or else direct microwave driving of hyperfine transitions could eliminate this entirely [WB03]. Using lighter ions as ancillas would lower the effective mass of the ion string and hence raise the axial phonon frequencies which would enhance cooling and two-qubit gate times. To reduce the required number of ions in each node, the use of multiple levels in the ground hyperfine manifold to encode multiple qubits could be considered [ARD04]. Since the measurement process is likely to distinguish the joint state of all the encoded qubits of an individual ion, this encoding method may not be suitable for data qubits, but is not necessarily a drawback for use for ancilla qubits which en bloc are measured and initialized repeatedly.

6.3.4 Conclusion

We have proposed the use of entanglement to directly implement non-local operations between separately housed logical qubits. These ideas may also be applicable to other physical quantum computing implementations which satisify the requirements in Section 6.3.3. The entanglement is created by a point-to-

216 6.3. SCALABLE QUANTUM ERROR CORRECTION IN DISTRIBUTED ION TRAP COMPUTERS.

point process which reduces routing diﬃculties and enables parallel operation. Logical operations via parity measurements require minimal entangled resources compared to a directly teleported sequence of block-wise gates but still retains Fault-tolerance. By keeping data local to a single node, the node can be of comparatively simple design and size, optimized for local high ﬁdelity operations. The technique should be able to be generalized to multi-qubit operations utilizing multi-partite entangled states and may serve as the basis for a full scalable quantum computing architecture. A proof of principle demonstration with two traps containing seven or eight ions and an optical interface each is within the reach of current experiment [KLH+03, HHR+05]. Even simpler to demonstrate are parity gates, the optical interface could be omitted and a gate performed between two three-qubit encoded (single X or Z-error) logical qubits coupled via a single Bell pair. Such a distributed architecture is a strong alternative to monolithic designs.

217 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

6.4 Photonic Modules

Multi-partite entanglement is arguably the most important resource needed when attempting to perform quantum processing. It is well known that entanglement forms the basis of the most well known quantum algorithms [Sho97, Gro97], secure cryptographic protocols [Eke91], increased resolution for optical lithography [BKA+00] and even a generic resource for a universal quantum computer [RB01, BR01]. However, it has proven to be a difficult challenge to efficiently generate multi-qubit entangled states that can be used as a resource for quantum information and/or communications. The problem of generating large multi-qubit entanglement, which can later be utilized, falls into two broad categories. Systems such as trapped ions [CZ95], solid-state qubits (including donor [Kan98] and superconducting systems [NPT99, MOL+99]) and color centers [LH00, GOD+06, WKN01] may allow for deterministic and accurate multi-qubit gates which can then be used to prepare entanglement. However, these systems have a high sensitivity to decoherence and information transport is difficult. For large scale computation, transport issues can be overcome through various quantum bus protocols [SNB+06, GCHH04, GDH06b, DGH07] [Section 6.2] and non-deterministic atomic fusion methods [BK05, BES05, LBK05, BBFM06] but qubit immobility generally prohibits these systems from being utilized in quantum communications and cryptography. In contrast, photonic qubits are extremely easy to move and are quite robust against decoherence. However, performing appropriate gate operations to prepare photonic entanglement is extremely difficult. It is generally acknowledged that the deterministic preparation of multi- qubit entanglement using photons requires some additional non-linear component. That can be achieved through a physical optical non-linearity or through offline ancilla preparation, photon measurement and post-selection. Photonic entanglement was first considered by utilizing Kerr [Mil89] non- linearities, however, using this resource to directly synthesize quantum gates does not appear to be practical due to the noise induced when preparing entangled photonic states. It was shown by Knill, Laflamme and Milburn [KLM01] that a system of linear optical elements, combined with measurement and post-selection could be used to perform conditional operations between two photonic qubits and in recent years this idea has been experimentally demonstrated [OPW+03]

218 6.4. PHOTONIC MODULES

and refined such that two-photon gates require far fewer optical elements [KYI01, YR03]. Currently the use of optical fusion gates [ZZH97, PJF01, LKCD02, BR05] are proposed to prepare photonic entanglement, probabilistically, which can be used to create small cluster states, dynamically, for universal computation [Nie04]. Here we illustrate the construction of the photonic module [Fig. 6.11]. A single atom/cavity system which leads to an extremely versatile device that can be used as a static resource for preparing entangled photonic states for computation and/or communication quickly, and with complete determinism. Although utilizing a single atomic qubit to mediate the preparation of photonic entanglement is not a new concept [SHW+07] the nature of the interaction exhibited by this scheme leads to a simple and versatile plug and play device. A single module, or multiple connected modules, can be constructed and with classical routing prepares entanglement without the downsides of probabilistic interactions or single photon detection. Atom/cavity mediated entanglement is well understood and there exists several schemes which can be adapted for use in the modules. These include the cavity-assisted interaction proposed by Duan, Wang and Kimble [DK03, DWK05] who utilized a similar network to achieve gate based photonic quantum computation and recent experimental schemes from Schuster et al. [SHS+07] which demonstrated a photon non-demolition interaction using a Cooper-pair box qubit and microwave cavity photons. We describe photonic modules for which the natural operation allows for the preparation of any N photon entangled state which can be described via the stabilizer formalism of Gottesman [Got97]. As such, the internal construction of each module is independent of the state being prepared, no single (or multi) photon detection is needed and the coherence time required for the atomic qubit is limited only by the time required to measure specific stabilizers describing the state (which can be small, even for large, highly entangled, multi-photon states). Therefore, the module is a completely generic resource, which can be applied to a vast variety of quantum applications. The possible uses for these modules are extensive. The ability to prepare any stabilizer state allows for the deterministic preparation of any geometric graph state, including states appropriate for optical cluster state computation [RB01, BR01, Nie04]. Bell state analyzers and factories are useful resources for quantum cryptographic protocols [Eke91], quantum dense coding [BW92, SBL+04], purification protocols and quantum repeaters [BDCZ98, DBCZ99]. Additionally,

219 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

quick and deterministic preparation of N photon GHZ states can be utilized in loss protection schemes for optical quantum computing [RHG05] and secret sharing protocols [HBB99]. The version of the module illustrated in Fig. 6.11a represents a free space version of the module, consisting of two Half Wave-Plates (HWP), two Polariz- ing Beam Splitters (PBS) and the single atom/cavity system with a polarization independent interaction. The module illustrated in Fig. 6.11b represents a photonic crystal implementation of the module where the atom/photon interaction is polarization dependent and hence the single photon interferometer is no longer required. The atom/cavity system contains a single multi-level atom which is the primary resource needed to prepare photonic entanglement. Additionally, the atomic system must be controllable such that it can be initialized in a known state and read out. To detail the operation of the modules, the following example uses the freespace version illustrated in Fig. 6.11a.

220 6.4. PHOTONIC MODULES g e. ines ingle h the witched Photonic module a. -1 +1 . The second Q-switch cavity is then i ± σ tom/photon version of the photonic module, |±i → | the photon back into the waveguide mode once the ich rotates . In both schematics, the atom/cavity qubit has appropriate which contains the atomic system with a differential couplin HZ states. The initial two cavities represent a Q-switched s ic module in freespace and photonic crystals. e measurement result of the atom qubit (green and red) determ ly the vertical component of the single photon interacts wit tion, requiring two single photon HWP and two PBS in free spac witched from the source to the first Q-switch cavity and then s i → |±i ± Optical Output σ | ical Output ical PBS Opt -1 +1 s -1 aser +1 -1 Control L Control +1 Delay Atom/Cavity qubit PBS the exiting photon train is projected to. Input N A photonic bandgap structure for a polarization dependent a ⊗ Optical X b. photons. The final Q-switch cavity is then used to out-couple Optical Input i ± σ | Figure 6.11: Schematics showing the basic design of a photon which eigenstate of used to adiabaticallybetween load the photon into the module cavity interaction is complete, where a second QWP rotates design for a polarizationThe independent polarization atom/photon dependent interac atom/cavity interferometer qubit. ensures thatrequired on to prepare twophoton photon source Bell [GSPH06]. states Single and photonsinto are higher the adiabatically order s waveguide G containing a Quarter Wave Plates (QWP) wh laser control such that it can be initialized and read out. Th

221 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

6.4.1 Bell State Factories

The basic operation of the module is best understood if we choose to use it as a factory for two photon Bell states, deﬁned through polarization as,

H H + V V Φ+ = | i| i | i| i. (6.49) | i √2

Given an appropriate single photon source, which can produce a train of single photon pulses of known polarization, separated by an interval ∆t, a two photon train is prepared in the product state H I H I, and sequentially sent through | i2| i1 the module. The indices, 1, 2 = 0, ∆t , represent the temporal mode of each { } { } single photon pulse, I the spatial mode (in this case the optical input) and ∆t is predefined and must be greater than the total time a single photon is present within the network. For a single photon passing through the module, the natural operation of the module, M, is given by, M + I φ + O φ , | i | i→| i | i (6.50) I O M φ φ′ . |−i | i → |−i | i Where =( H V )/√2, φ = α 0 + β 1 is the state of the atomic qubit, |±i | i±| i | i | i | i φ′ = X φ = α 1 + β 0 and the indices I,O represent the input and output | i | i | i | i { } optical modes. The atom/cavity system is positioned such that the cavity mode is coupled to the spatial mode B2, where denotes the photon polarization and cavity |◦i ◦ Q-switching [GSPH06] (which allows for the adiabatic loading of a single photon into a cavity) is employed before and after the atom/photon interaction to ensure appropriate in- and out-coupling to and from the cavity. The mode B1 contains |◦i an optical delay equal to the time required for the photon/atom interaction which is specific. A single photon passing through the atom/cavity system must induce a photonic non-demolition bit-flip on the two-level atom, releasing the photon again into B2 once the interaction is complete. |◦i If the photonic state is + , the initial Half Wave Plate (HWP) will rotate the | i state to H after which it will continue into the mode B1 and not interact with | i |◦i the atom. The second Polarizing Beam Splitter (PBS) and HWP will then couple B1 to the output mode and rotate H back to + . If the initial photonic state |◦i | i | i is , the HWP will rotate the state to V and the PBS will reflect the photon |−i | i

222 6.4. PHOTONIC MODULES

into the B2 mode, where it ﬂips the state of the atomic qubit. The photon is |◦i then released back into B2 where the second PBS and HWP will reﬂect the |◦i photon into the output mode and rotate it from V to . Therefore, the two | i |−i basis states, , of a single photon passing through the module will enact the |±i transformation M shown in Eq. 6.50. For a two photon train, polarized in the state H I H I =( + I + I)( + I + | i2| i1 | i2 |−i2 | i1 I )/2, we are able to enact the same transformations on the photon/atom |−i1 interaction, giving,

I I 1 I O I O M2,1 H H φ = M2 H + φ + H φ′ | i2| i1| i √2 | i2| i1 | i | i2|−i1 | i 1 O O O O = + + + φ′ (6.51) 2 |−i2 | i1 | i2 |−i1 | i 1 O O 1 O O + + + φ′′ + φ , 2| i2 | i1 | i 2|−i2 |−i1 | i

2 2 where φ′′ = X φ′ = X φ . As X is Hermitian, X = I and we can expand out | i | i | i the states to give, |±i

I I 1 O O O O M2,1 H 2 H 1 φ = H 2 H 1 + V 2 V 1 φ | i | i | i 2 | i | i | i | i | i (6.52) 1 O O O O + H H V V φ′ . 2 | i2 | i1 −| i2 | i1 | i After both photons have passed through the module the ﬁnal step is to measure the state of the atom/cavity qubit. If prior to the interactions, the atomic qubit is initialized in the state φ = 0 and the subsequent measurement | i | i is also 0 , the photons are projected to the state, | i 1 ( H O H O + V O V O), (6.53) √2 | i2 | i1 | i2 | i1 which represents an even parity Bell state. If the atom is measured in the state 1 , the photons are projected to, | i 1 ( H O H O V O V O), (6.54) √2 | i2 | i1 −| i2 | i1 which is an odd parity Bell state. The output pulse consists of the original two photon train which is now polarization entangled into a two photon Bell state. Unlike other schemes, the measurement result of the atom/cavity system never

223 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

collapses the photons to un-entangled states. In fact, since the odd and even parity Bell states differ through a local phase flip on any photon, either result is acceptable and a positive parity state can be prepared by applying a local, classically controlled phase flip on any photon once the atom/cavity qubit is measured. The preparation of the Bell state is therefore completely deterministic, with the classical result only giving parity information of the photon state. Addition- ally, since positive and negative parity states are interchangeable through local Clifford gates, correction can be fed forward to the end of subsequent operations on the photonic state. Although we explicitly considered the case when the induced operation was a bit-flip, any Hermitian operation is acceptable provided it transforms the state of the atomic qubit between two orthogonal states. The transformation, M, shown in Eq. 6.50 is also exhibited by the module illustrated in Fig. 6.11b for a polarization dependent interaction. For appropriate atomic systems it is well known that there exists states with a differential dipole coupling between σ = ( H i V )/√2 polarized photons, e.g. NV− | ±i | i ± | i diamond [ASMRB07] or Rubidium. If the atomic coupling to the cavity mode is chosen such that only σ polarized photons interacts with the atomic qubit, we − are able to eliminate the interferometer shown in Fig. 6.11a. Instead, single photon wave plates are used to rotate σ before and after the atom/cavity |±i ↔ | ±i interaction. As the atom/photon interaction is polarization dependent, the transformation, M, for this modified version of the module still hold. The engineering of the module when a polarization dependent interaction is available is beneficial. The lack of the interferometer implies that this structure can be directly fabricated in systems such as photonic bandgap crystals, with cavity Q-switching protocols [GSPH06] fabricated on-chip to control the in- and out-coupling of the single photon pulses [Fig. 6.11b].

6.4.2 Arbitrary entangled state preparation.

The potential of these modules goes far beyond the preparation of Bell states. In fact, the unit can be augmented with appropriate single photon routing and local operations to prepare any entangled photon state that can be expressed in terms of stabilizers [Got97]. These include, codeword states for Quantum Error Correction, Bell states, GHZ states and arbitrary graph states (of which cluster

224 6.4. PHOTONIC MODULES

states are a speciﬁc topological subset). A remarkable property of the module is that the number of entangled photons that are prepared depends only on the number actually sent through the module, no internal structure of the module needs to be altered to entangle more photons. The natural operation of the module is to project the train of photons into N a 1 eigenstate of the X⊗ operator, i.e. any arbitrary N photon state will be ± transformed to,

1 N 1 N M Ψ 0 = Ψ + X⊗ Ψ 0 + Ψ X⊗ Ψ 1 . N,..,1| iN | i 2 | iN | iN | i 2 | iN − | iN | i (6.55) Where 0 , 1 are the states of the atom/cavity qubit and all N photons have {| i | i} been passed through the module. To show this, consider an N photon train, with each single photon pulse separated by ∆t. Each basis element, ψ , of the state, Ψ , can be written in the | i | i form, N 1 N 1 − − ψ = c = + +( 1)ca , (6.56) | i | aia | ia − |−ia a=0 a=0 O O where c = H 0 , V 1 and = ( H V )/√2, with each single | ai {| ≡ i | ≡ i} |±i | i±| i photon pulse centered at time t = a∆t. The transformation of the basis |±i states are given by M in Eq. 6.50,

I O M + φ q = + φ q, | i | i | i | i (6.57) I O M φ = φ′ , |−i | iq |−i | iq where, φ is the state of the atomic qubit, φ′ = X φ , I and O are the | iq | iq | iq input/output modes of the module and, for clarity, we have omitted the time index for the pulse. Assuming that the atomic system is initialized in the state φ = 0 , an arbitrary basis state of Ψ transforms as, | iq | i | iN

N 1 − M ψ 0 = M + +( 1)ca 0 N,..,1| i| iq N,..,1 | i − |−i | iq a=0 N 1O N 1 − − ca ca = 0 q + +( 1) + 1 q + +( 1) , | i | i − |−i Ev | i | i − |−i Od a=0 |−i a=0 |−i O O (6.58) where the ﬁrst term represents all states of the tensor product formed with an even number of states and the second term represents all tensor products |−i

225 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

formed with an odd number of states. The even and odd components of the |−i basis terms, ψ , can be written in the following way, | i

N 1 N 1 N 1 − 1 − 1 − + +( 1)ca = + +( 1)ca + + +( 1)ca+1 , | i − |−i Ev 2 | i − |−i 2 | i − |−i a=0 |−i a=0 a=0 NO1 NO1 NO1 − 1 − 1 − + +( 1)ca = + +( 1)ca + +( 1)ca+1 . | i − |−i Od 2 | i − |−i − 2 | i − |−i a=0 |−i a=0 a=0 O O O (6.59) N Noting that the second term in each equation is simply the state X⊗ ψ , each | i basis term, ψ , transforms as, | i

1 N 1 N M ψ 0 = ( ψ + X⊗ ψ ) 0 + ( ψ X⊗ ψ ) 1 , (6.60) N,..,1| i| iq 2 | i | i | iq 2 | i − | i | iq and consequently, the total state, Ψ = β ψ , transforms as, | i j j| ij P 1 N 1 N M Ψ 0 = Ψ + X⊗ Ψ 0 + Ψ X⊗ Ψ 1 , (6.61) N,..,1| i| iq 2 | i | i | iq 2 | i − | i | iq as required. The measurement outcome of the atomic system will determine which eigenstate is projected, with local operations applied to switch between eigenstates. To prepare any N photon stabilized state, a parity check is performed on the N stabilizers which describe the state. As each of the stabilizers for an arbitrary N photon state are described via an N-fold tensor product of the op- N ′ erators I,X,Y,Z , the ability to perform a parity check of the operator X⊗ , { } for N ′ N, and apply local operations is suﬃcient to stabilize an arbitrary state ≤ with respect to any operator of this form. Therefore, if we assume that we can selectively route photons within the train (which is possible, as each pulse is tem- porally tagged) and apply local operations to any photon, the parity measurement performed by the module is suﬃcient to prepare any stabilizer state. For a general N photon state, N parity checks are required. This can either be done by constructing and utilizing N separate modules, or it can be done by sequentially utilizing only one. If multiple modules are available, many parity checks can be done in parallel without waiting for atomic readout, potentially speeding up state preparation. As the stabiliser structure of the desired state dictates the number of pho-

226 6.4. PHOTONIC MODULES

tons passed through the module for each parity check, the coherence time of the atom/cavity system does not depend on the total number of photons in the entangled state. Instead, the atomic system only has to maintain coherence until the parity of a specific stabiliser operator is measured, this is extremely beneficial. The number of non-Identity operators in any given stabilizer operator (which we denote the “Parity-weight”), dictates the number of photons passed through the module in any one step and therefore the coherence time required for the atom/cavity system. For example, an N photon cluster state appropriate for quantum computation, has a well known stabilizer structure [RB01], with a maximum Parity-weight of five. Hence, regardless of the total size of the cluster, the atomic system only needs to maintain coherence long enough for five photons to pass through the module between initialization and measurement. Conversely, if the coherence time of the atomic system is short compared to Pm∆t, where Pm is the maximum Parity-weight of the state and ∆t is the time required for a single photon to pass through the module, then fusion methods [LKCD02, BR05, Nie04] can be employed to prepare states with large Pm.

For example, N photon GHZ states have Pm = N, corresponding to the sta- N bilizer, K = X⊗ . If the coherence time of the atomic system only allows for

N ′ < N-dimensional parity checks to be performed at any one time, then multiple N ′-GHZ states can be prepared and fused together via two-photon ZZ parity measurements.

6.4.3 Operational Time

The interaction required for the photonic module is identical to the photonic version of the quantum bus protocol introduced in section 6.2.3. In this case we consider a train of single photon pulses which will sequentially ﬂip the state of the atomic qubit between 0 and 1 . The freespace and photonic crystal | i | i versions of the module shown in Fig. 6.11 can utilize the same atom/photon interaction, however the photonic crystal module [Fig. 6.11b] requires the single photon coupling in Fig. 6.2 to be polarization dependent. As mentioned in section 6.2.3, there are several methods to achieve the required interaction at both microwave [SHS+07] and optical [DK03, DWK05] frequencies. In order to estimate a rough operational time for the module, we will focus on the atom/photon

227 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

interaction utilizing a 4-level atom, as illustrated in Fig. 6.2 in section 6.2.3. For this speciﬁc method, we examine the transition time for a single photon in the module and compare our cavity requirements with systems currently in existence. In general, we wish to maintain single-photon absorption probabilities, ζ, on the 3 4 transition of less that ζ 1, hence ∆ β/√ζ, where ∆ is | i↔| i ≪ ≥ the detuning between the single photon frequency and the 3 4 resonance | i↔| i transition which couples to the photon with strength β. For a π phase shift, 1 π and choosing the equality, this corresponds to, t = (κ)− = β√ζ . For a proof of concept device, we assume ζ =0.1, which corresponds to an average of one in ten photons being absorbed. Consequently, we can examine t as a function of some of the current experimental values for β and κ [Tab. 6.1]. The last column in Tab. 6.1

1 1 Cavity β (MHz) t at ζ = 10− Exp. t = κ− Γ (MHz) Cs [BBM+06] 34 0.29µs 0.24µs 2.6 Rb [THE+05] 366 27ns 1.7ns 6.3 4 NV− [SNAA05] 10 1ns 3.4ns 83 ≈ ≈ ≈ Table 6.1: Estimates on Cavity requirements for various systems. The Cavity from Boozer et. al. [BBM+06] has been experimentally demonstrated, while the atom chip cavity of Trupke et. al. [THE+05] and the photonic bandgap cavity of Song et. al. [SNAA05] in silicon have yet to couple the atomic qubit. Hence we use theoretical estimates for the coupling, β, and atomic decay rates, Γ, for Rubidium + and NV− [GSPH06, GOD 06] qubits. The first column quotes the atom/cavity coupling while the second column estimates the required photon storage time in the cavity to invoke a π phase shift in the atom with a single photon absorption probability of 10%. The final column quotes the current photon storage time which has been experimentally demonstrated (estimated) for each cavity system. For both the Boozer and Trupke cavities, approximately an order of magnitude improvement in either the coupling constant or cavity lifetime is required. Current estimates suggest that the photonic bandgap cavity will be able to exhibit the interaction with the fastest operational time of all the systems and is also more amenable to current cavity Q-switching protocols [GSPH06]. The last column details estimates on atomic decay rates for the systems considered for each cavity. The ratio of the required photon storage time to the coherence time of the atomic system dictates the maximum Parity-weight a single module can measure in any one step. specifies the atomic decay rates, Γ, for the atomic systems used, or estimated, for each cavity system (Cesium, Rubidium and NV− diamond). The coherence time of the atomic system will dictate the maximum Parity-weight which can be measured in any one step using the module. Hence taking the ratio of the

228 6.5. CONCLUSIONS

required photon storage time to Γ, the Cs cavity of Boozer et. al. [BBM+06] falls slightly short of being able to perform two-photon parity measurements (which are sufficient to prepare Bell states, GHZ states and linear cluster states [DGH07]). The micro-cavity of Trupke et. al. [THE+05], using Rb, theoretically has sufficient coherence to allow a parity check over six photons (sufficient for universal cluster states), while NV− in photonic bandgap cavities could allow for Parity-weights up to twelve [SNAA05, GOD+06], allowing for a huge amount of flexibility in preparing highly entangled graph states very quickly.

6.5 Conclusions

We have detailed the construction of a photonic module which, given a steady source of single photons, can deterministically prepare a large class of useful photonic entangled states. The construction of each module is generic and independent of which entangled state is being prepared, and the stabiliser nature of the entangled states implies that the coherence time of the atomic system only needs to long compared with the maximum Parity-weight (P ) pulse separa- m × tion (∆t) of the desired state (which can be small, even for large multi-photon entangled states). The practical uses of these modules is quite extensive. Multi-photon entangled states can be utilized for quantum computation, quantum cryptography, quantum dense coding, and quantum repeaters. As the internal design of the photonic module is completely independent of the state being prepared, multiple modules, combined with an appropriate single photon source, optical wave plates and classical routing can be used to construct a static, on chip system, tailored for fast preparation of speciﬁc entangled states. For example, pumping out large cluster states for computation, or multiple Bell pairs, in succession, for communications and cryptography. The engineering of an appropriate atom/photon interaction is still some- thing that needs to be experimentally investigated. Cavity experiments reported in [BBM+06, THE+05, SNAA05, WSB+04, SHS+07] show exceptional promise at both optical and microwave frequencies. Ideally, once the required interaction has been experimentally demonstrated, additional engineering to realize the photonic module should be comparatively straight-forward.

229 CHAPTER 6. DISTRIBUTED QUANTUM INFORMATION PROCESSING VIA PARITY MEASUREMENTS.

230 Chapter 7

Concluding Remarks.

Neglecting Chapter 2, which contained a brief review of the major achievements in quantum information processing, this thesis has presented several new results, experimental protocols and device inventions which attempts to link experimental restrictions in nano-engineering and quantum device construction to the more abstract concepts of large scale quantum computing and control. In Chapter 3, I presented a general introduction to Shor’s factoring algorithm and a quantum circuit appropriate for implementing the Quantum Period Finding (QPF) subroutine on a physical system restricted to a linear array of qubits with nearest neighbor coupling (LNN). This circuit was developed in collaboration with Austin Fowler and demonstrated that the LNN restriction of many solid state computing architectures is not a prohibitive factor in efficient circuit design. Section 3.3 then numerically examined the stability of this circuit under a discrete error model, commonly utilized in error correction analysis. I showed how both the LNN circuit introduced in Section 3.2 and the non-LNN circuit first proposed by Beauregard [Bea03] is extremely sensitive to quantum errors and how the success probability of the QPF subroutine drops significantly with only a small number of errors in the entire circuit. These results demonstrated that within any realistic implementation of large scale factoring, significant error correction will need to be employed. In Chapter 4, I presented a detailed overview of the more important concepts in Quantum Error Correction (QEC). I began with first illustrating the 3-qubit code, which provides a good example of how encoded states and projective measurement can be used to detect and correct for errors in the quantum

231 CHAPTER 7. CONCLUDING REMARKS.

regime. Next I reviewed the basic principles of the Stabilizer formalism of Gottes- man [Got97] and how it can be used to directly synthesize quantum circuits for state preparation and correction and in section 4.3 I briefly reviewed how the discrete error model, used in QEC analysis, is linked with more physically relevant error processes, namely environmental decoherence and systematic errors. Section 4.4 reviewed the concepts of Fault-tolerant quantum computation and how this leads to the threshold theorem for concatenated Quantum Error Correction, with section 4.6 illustrating Fault-tolerant circuit design for encoded state preparation. Finally in Section 4.7 I presented new results, illustrating the effectiveness of Fault-tolerant circuits compared to simpler non-Fault-tolerant versions. I showed, numerically, how Fault-tolerant state preparation only becomes more efficient at extremely low error rates, implying that fast preparation of codeword states for error correction protocols introduced by Steane [Ste97, Ste02] should definitely be done in a non-Fault-tolerant manner. Chapter 5 introduced several experiemental protocols designed to identify and characterize single and two-qubit behaviour for computational systems. Sec- tion 5.2 introduced a single qubit confinement protocol designed specifically to test if an “ideal” qubit is a controlled two-level system. This protocol made very few assumptions regarding the underlying physical system and only utilized machinery that is required for standard qubit operations. I also presented several numerical simulations showing the effectiveness of this scheme, accuracy of analytical error bounds and finally, demonstrated that the inclusion of Markovian decoherence does not invalidate the protocol. Section 5.3 reviewed single qubit characterization schemes first developed by Cole et. al. [CSG+05, CGO+06, Col07] and how these schemes are “bootstrapped” from the results of the confinement experiments. Finally, section 5.4 reviewed the geometric properties of two-qubit interactions and how this picture leads to an elegant method for characterizing two-qubit interaction Hamiltonians. I presented an experimental protocol for two-qubit Hamiltonian characterization, which like single qubit characterization, is bootstrapped from previous experiments and can be used to fully reconstruct a large class of interaction dynamics via the examination of two-qubit concurrence. I then presented numerical simulations, demonstrating how this method can be used to determine the anisotropic Hamiltonian corrections in GaAs quantum dots. Chapter 6 was focused on large scale architecture designs for quantum pro-

232 cessors. I introduced a protocol for distributed quantum computation using an information free quantum bus consisting of spatial qudits defined through optical modes. I demonstrated how this protocol can be used to prepare entangled states between spatially isolated qubits constructed from an atomic system placed in a cavity. The benefit of this scheme, in contrast to other distributed protocols, is that the photonic mode only carries parity information related to the qubit/qubit coupling. Therefore, if the bus mode is lost during computation, no computational information is lost. Instead, a loss event will induce, at most, a coherent error on one of the qubits which can be corrected via standard quantum error correcting protocols. I demonstrated how this scheme can prepare large multi-qubit stabilizer states and simulate a controlled interaction between qubits. Combined with local control, this allows for universal computation with the bus. Section 6.2.3 presented several alternative schemes to achieve the required qubit/photon interaction required by our distributed network. Atom/cavity designs which have been experimentally constructed, such as the Cesium cavity of Boozer et. al. [BBM+06], the Rubidium cavity of Trupke et. al [THE+05], the photonic bandgap cavity of Song at. al. [SNAA05] in silicon and the microwave strip-line cavity of Wallraff and Schuster et. al [WSB+04, SHS+07] show enormous promise as candidates for this distributed network. I also presented an alternative physical architecture based on the same idea in the solid state. Utilizing the coherent transport protocol developed by Andrew Greentree and myself [GDH06b] for electronic systems (MRAP), I showed how spatially isolated, solid state qubits can be coupled together. This would con- ceivably allow for a distributed network in GaAs quantum dots or P:Si systems, solving many significant problems which currently exist for large scale quantum computation in the solid state. Section 6.3 then introduced a distributed scheme for ion-trap computation. The basic operational element for the information free bus, namely the parity measurement, was utilized between small ion-trap nodes. In this case, a system was presented where a small collection of ions (representing local code-states in error correction) are controlled via local intra-trap operations. This allows for local error correction to be performed locally and quickly to maintain coherence between logical qubit operations. These logical operations are then performed using probabilistic inter-trap links and the parity gate. The benefit of this scheme is that the number of inter-trap connections is small and operations can be per-

233 CHAPTER 7. CONCLUDING REMARKS.

formed on quantum data encoded with the [[5,1,3]] code. As ion-trap technology is by far the most advanced in quantum control, a small two-trap network can hopefully be built in the near future to prepare a two-qubit, logically encoded Bell state, utilizing full Fault-tolerant error correction. Finally in section 6.4 I introduced the photonic module. This small scale optical device would allow for the deterministic preparation of any large, multi- photon entangled state which is described via stabilizers. The uses for such a device is extensive with clear applications found in quantum cryptography, dense coding, quantum repeaters and large scale optical quantum computation. The device itself is an inversion of the computational system introduced in section 6.2 but has much more flexibility. As the qubit/photon interaction is identical to that required for the quantum bus, current atom/cavity system are very close to being realized for such a device. With all of the topics presented in this thesis, there is clear avenues for future investigation. The stochastic nature of the simulations for Shor’s algorithm meant that quantitative error behavior for the circuit could not be identified. Re-simulating this algorithm using the density matrix formalism would remove this restriction and allow us to find functional forms for each of the probability curves as a function of error rate. Secondly, density matrix simulations will allow us to include error models that accurately simulate Markovian decoherence and include multi-qubit correlated errors, introduced by gate coupling or non- Markovian effects. Characterization protocols for qubit systems also requires further work. At this stage we have yet to develop protocols for two-qubit confinement, single qubit characterization under non-Markovian decoherence, characterization using weak measurement or two-qubit characterization with the inclusion of decoherence. Additionally, we still require a complete protocol for two-qubit characterization that accounts for Hamiltonian systems that are not fully non-local. The completion of a full set of characterization protocols, from confinement up to full Hamiltonian identification, will be a significant area of future investigation. Finally, the architecture proposals introduced in Chapter 6 still require extensive theoretical modeling and detailed protocols for Quantum Error Correction and Fault-tolerance. Being able to construct a spatially distributed network of qubits is a significant first step, however we still need to develop efficient methods to perform Fault-tolerant error correction and also ensure that the basic mod-

234 els for the atom/photon interaction are realistic from the standpoint of cavity input/output coupling and phase noise on the quantum bus. The ﬁeld of quantum information science is progressing extremely fast, both theoretically and experimentally. The basic concepts of quantum engineering will become extremely relevant in the next decade when small, controllable quantum system come out of the physics laboratories and into the commercial world. The ability to piece together the large amount of work in experimental methods, algorithm design, error correction and large scale information processing will be vital to the continuing success of quantum information and computation. As with any other area of science, quantum information and computation may lead to exciting and revolutionary ideas not only in nano-engineering but also in our fundamental conceptions regarding quantum theory and our understanding of the universe.

235 CHAPTER 7. CONCLUDING REMARKS.

236 References

[ABJ+03] J.B. Altepeter, D. Branning, E. Jeﬀrey, T.C. Wei, P.G. Kwiat, R.T. Thew, J.L. O’Brien, M.A. Nielsen, and A.G. White. Ancilla- assisted quantum process tomography. Phys. Rev. Lett., 90:193601, 2003. [ABO97] D. Aharonov and M. Ben-Or. Fault-tolerant Quantum Computa- tion with constant error. Proceedings of 29th Annual ACM Sym- posium on Theory of Computing, page 46, 1997. [AG04] S. Aaronson and D. Gottesman. Improved Simulation of Stabilizer Circuits. Phys. Rev. A., 70:052328, 2004. [Aga91] B.K. Agarwal. X-ray Spectroscopy: An Introduction. Springer - Verlag, Berlin ; New York, 1991. [AJW+01] A. Aassime, G. Johansson, G. Wendin, R.J. Schoelkopf, and P. Delsing. Radio-Frequency Single-Electron Transistor as Readout Device for Qubits: Charge Sensitivity and Backaction. Phys. Rev. Lett., 86:3376, 2001. [AL04] P. Aliferis and D.W. Leung. Computation by measurements: A unifying picture. Phys. Rev. A., 70:062314, 2004. [Ali07] P. Aliferis. PhD Thesis (Caltech), 2007. [ALKH02] D. Ahn, J. Lee, M.S. Kim, and S.W. Hwang. Self-Consistent non-Markovian theory of a quantum-state evolution for quantum- information-processing. Phys. Rev. A., 66:012302, 2002. [APN+05] O. Astaﬁev, Yu. A. Pashkin, Y. Nakamura, T. Yamamoto, and J.S. Tsai. Quantum Noise in the Josephson Charge Qubit. Phys. Rev. Lett., 93:267007, 2005. [ARD04] ARDA. Quantum information science and technology roadmap project, http://qist.lanl.gov, 2004. [ASMRB07] T.P. Mayer Alegre, C. Santori, G. Medeiros-Rubeiro, and R.G. Beausoleil. Polarization-selective excitation of NV centers in diamond. arxiv:0705.2006, 2007.

237 REFERENCES

[AT07] P. Aliferis and B.M. Terhal. Fault-tolerant quantum computation for local leakage faults. Quant. Inf. Comp., 7:139, 2007.

[Avi87] A. Avizienis. The Evolution of Fault-Tolerant Computing. Springer -Verlag, New York, 1987.

[Bac03] D. Bacon. PhD Thesis (Caltech). quant-ph/0305025, 2003.

[Bac06] D. Bacon. Operator Quantum Error-Correcting Subsystems for self-correcting quantum memories. Phys. Rev. A., 73:012340, 2006.

[BBC+93] C.H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. Peres, and W.K. Wooters. Teleporting an Unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett., 70:1895, 1993.

[BBC+95] A. Barenco, C.H. Bennett, R. Cleve, D.P. DiVincenzo, N. Margo- lus, P.W. Shor, T. Sleator, J. Smolin, and H. Weinfurter. Elemen- tary Gates For Quantum Computation. Phys. Rev. A., 52:3457, 1995.

[BBFM06] S.C. Benjamin, D.E. Browne, J. Fitzsimons, and J.J.L. Morton. Brokered Graph state quantum computing. New J. Phys., 8:141, 2006.

[BBH+06] C. Balzer, A. Braun, T. Hannemann, C. Paape, M. Ettler, W. Neuhauser, and C. Wunderlich. Electrodynamically trapped Yb+ ions for quantum information processing. Phys. Rev. A., 73:041407(R), 2006.

[BBM+06] A.D. Boozer, A. Boca, R. Miller, T.E. Northup, and H.J. Kim- ble. Cooling to the Ground State of Axial Motion for One Atom Strongly Coupled to an Optical Cavity. Phys. Rev. Lett., 97:083602, 2006.

[BBP+96] C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, and W.K. Wooters. Puriﬁcation of Noisy Entanglement and Faithful Teleportation via noisy channels. Phys. Rev. Lett., 76:722, 1996.

[BDCZ98] H.J. Briegel, W. D¨ur, J.I. Cirac, and P. Zoller. Quantum Repeaters: The role of imperfect local operations in quantum communication. Phys. Rev. Lett., 81:5932, 1998.

[BDD+02] M.J. Bremner, C.M. Dawson, J.L. Dodd, A. Gilchrist, A.W. Har- row, D. Mortimer, M.A. Nielsen, and T.J. Osborne. Practical Scheme for Quantum Computation with any Two-Qubit Entan- gling Gate. Phys. Rev. Lett., 89:247902, 2002.

238 REFERENCES

[BDEK04] C.W.J. Beenakker, D.P. DiVincenzo, C. Emary, and M. Kinder- mann. Charge Detection Enables Free-Election Quantum Compu- tation. Phys. Rev. Lett., 93:020501, 2004.

[Bea03] S. Beauregard. Circuit for Shor’s algorithm using 2n+3 qubits. Quant. Inf. Comp., 3:175, 2003.

[Ben73] C.H. Bennett. Logical Reversibility of Computation. IBM J. Res. Develop, 17:525, 1973.

[BES05] S.C. Benjamin, J. Eisert, and T.M. Stace. Optical generation of matter qubit graph states. New J. Phys., 7:194, 2005.

[BHPC03] N. Boulant, T.F. Havel, M.A. Pravia, and D.G. Cory. Robust Method for Estimating the Lindblad operators of a dissipative quantum process from measurements of the density operator at multiple time points. Phys. Rev. A., 67:042322, 2003.

[BHW+04] A. Blais, R-S. Huang, A. Wallraﬀ, S.M. Girvin, and R.J. Schoelkopf. Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A., 69:062320, 2004.

[BJD+06] J. Beugnon, M.P.A. Jones, J. Dingjan, B. Darqui´e, G. Messin, A. Browaeys, and P. Grangier. Quantum Interference between two single photons emitted by independently trapped atoms. Nature (London), 440:779, 2006.

[BK05] S.D. Barrett and P. Kok. Eﬃcient high-ﬁdelity quantum computation using matter qubits and linear optics. Phys. Rev. A., 71:060310(R), 2005.

[BKA+00] A.N. Boto, P. Kok, D.S. Abrams, S.L. Braunstein, C.P. Williams, and J.P. Dowling. Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diﬀraction Limit. Phys. Rev. Lett., 85:2733, 2000.

[BKD04] G. Burkard, R.H. Koch, and D.P. DiVincenzo. Multilevel quntum description of decoherence in superconducting qubits. Phys. Rev. B., 69:064503, 2004.

[BKPV99] S. Bose, P.L. Knight, M.B. Plenio, and V. Vedral. Proposal for teleportation of an Atomic State via Cavity Decay. Phys. Rev. Lett., 83:5158, 1999.

[BKSO05] S. Balensiefer, L. Kregor-Stickles, and M. Oskin. An Evaluation Framework and Instruction Set Architecture for Ion-Trap based Quantum Micro-architectures. SIGARCH Comput. Archit. News, 33(2):186, 2005.

239 REFERENCES

[BLWZ05] M.S. Byrd, D.A. Lidar, L.-A. Wu, and P. Zanardi. Universal leakage elimination. Phys. Rev. A., 71:052301, 2005.

[BM03] S.D. Barrett and G.J. Milburn. Measuring the Decoherence rate in a semiconductor charge qubit. Phys. Rev. B., 68:155307, 2003.

[BPH03] D.E. Browne, M.B. Plenio, and S.F. Huelga. Robust Creation of Entanglement between Ions in Spatially Separate Cavities. Phys. Rev. Lett., 91:067901, 2003.

[BR01] H.-J. Briegel and R. Raussendorf. Persistent Entanglement in Ar- rays of Interacting Particles. Phys. Rev. Lett., 86:910, 2001.

[BR05] D.E. Browne and T. Rudolph. Resource-Eﬃcient Linear Optical Quantum Computation. Phys. Rev. Lett., 95:010501, 2005.

[Bra00] R.N. Bracewell. The Fourier Transform and its applications. McGraw-Hill series in electrical and computer engineering. Mc- Graw Hill, Boston, 3rd edition, 2000.

[Bra02] D. Braun. Quantum chaos and quantum algorithms. Phys. Rev. A., 65:042317, 2002.

[BSD01] N.E. Bonesteel, D. Stepanenko, and D.P. DiVincenzo. Anisotropic Spin Exchange in Pulsed Quantum gates. Phys. Rev. Lett., 87:207901, 2001.

[BvDBZ03] A. Blais, A.M. van Der Brink, and A.M. Zagoskin. Tunable Cou- pling of Superconducting Qubits. Phys. Rev. Lett., 90:127901, 2003.

[BW92] C.H. Bennett and S.J. Wiesner. Communication via one- and two- particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett., 69:2881, 1992.

[Car99] H.J. Carmichael. Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations. Springer, Berlin, 1999.

[CCD+88] J. Clarke, A.N. Cleland, M. Devoret, D. Esteve, and J. Martinis. Quantum mechanics of a macroscopic variable: The phase diﬀer- ence of a Josephson junction. Science, 239:992, 1988.

[CCGFZ99] C. Cabrillo, J.I. Cirac, P. Garci´a-Fern´andez, and P. Zoller. Creation of entangled states of distant atoms by interference. Phys. Rev. A., 59:1025, 1999.

[CCL01] A.M. Childs, I.L. Chuang, and D.W. Leung. Realization of quantum process tomography in NMR. Phys. Rev. A., 64:012314, 2001.

240 REFERENCES

[CDH06] J.H. Cole, S.J. Devitt, and L.C.L. Hollenberg. Precision Charac- terization of Two-Qubit Hamiltonians via Entanglement Mapping. J. Phys. A:Math.Gen, 39:14649–14658, 2006. [CFH97] D.G. Cory, A.F. Fahmy, and T.F. Havel. Ensemble Quantum Com- puting by NMR Spectroscopy. Proc. National Academy of Science, 94:1634–1639, 1997. [CG97] R. Cleve and D. Gottesman. Efficient Computations of Encodings for Quantum Error Correction. Phys. Rev. A., 56:76, 1997. [CGJH05] V.I. Conrad, A.D. Greentree, D.N. Jamieson, and L.C.L. Hol- lenberg. Analysis and Geometric Optimization of Single Elec- tron Transistors for Read-Out in Solid-State Quantum Comput- ing. Journal of Computational and Theoretical Nanoscience, 2:214, 2005. [CGK98] I.L. Chuang, N. Gershenfeld, and M. Kubinec. Experimental Im- plementation of fast quantum searching. Phys. Rev. Lett., 80:3408, 1998. [CGO+06] J.H. Cole, A.D. Greentree, D.K.L. Oi, S.G. Schirmer, C.J. Wellard, and L.C.L. Hollenberg. Identifying a two-state Hamiltonian in the presence of decoherence. Phys. Rev. A., 73:062333, 2006. [Cha] RSA Factoring Challenge. http://www.rsa.com/rsalabs/node.asp?id=2092. [Cha70] R. Chang. Basic principles of spectroscopy. McGraw Hill, New York, 1970. [CLS+04] J. Chiaverini, D. Leibfried, T. Schaetz, M.D. Barrett, R.B. Blakestad, J. Britton, W.M. Itano, J.D. Jost, E. Knill, C. Langer, R. Ozeri, and D.J. Wineland. Realization of quantum error correction. Nature (London), 432:602, 2004. [CLSZ95] I.L. Chuang, R. Laflamme, P.W. Shor, and W.H. Zurek. Quantum Computers, factoring, and decoherence. Science, 270:1633, 1995. [CN97] I.L. Chuang and M.A. Nielsen. Prescription for experimental determination of the dynamics of a quantum black box. J. Mod. Opt., 44:2455, 1997. [CNHM03] I. Chiorescu, Y. Nakamura, C.J.P.M Harmans, and J.E. Mooij. Co- herent quantum dynamics of a superconducting flux qubit. Science, 299:1869, 2003. [COI+03] D. Copsey, M. Oskin, F. Impens, T. Metodiev, A. Cross, F.T. Chong, I.L. Chuang, and J. Kubiatowicz. Towards a Scalable, Silicon-Based Quantum Computing Architecture. IEEE J. Sel. Top. Quant. Elec., 9(6):1552, 2003.

241 REFERENCES

[Col07] J.H. Cole. PhD Thesis (Melbourne). 2007.

[CPM+98] D.G. Cory, M.D. Price, W. Maas, E. Knill, R. Laﬂamme, W.H. Zurek, T.F. Havel, and S.S. Somaroo. Experimental quantum error correction. Phys. Rev. Lett., 81:2152, 1998.

[CPOC+06] J.R. Castrej´on-Pita, H. Ohadi, D.R. Crick, D.F.A. Winters, D.M. Segal, and R.C. Thompson. Novel Designs for Penning Ion Traps. quant-ph/0603195, J. Mod. Opt. (to appear), 2006.

[CRSS98] A.R. Calderbank, E.M. Rains, P.W. Shor, and N.J.A. Sloane. Quantum Error Correction via Codes Over GF(4). IEEE Trans. Inform. Theory, 44:1369, 1998.

[CS96] A.R. Calderbank and P.W. Shor. Good Quantum Error-Correcting codes exist. Phys. Rev. A., 54:1098, 1996.

[CSG+05] J.H. Cole, S.G. Schirmer, A.D. Greentree, C.J. Wellard, D.K.L. Oi, and L.C.L. Hollenberg. Identifying an experimental two-state Hamiltonian to arbitrary accuracy. Phys. Rev. A., 71:062312, 2005.

[CW00] R. Cleve and J. Watrous. Fast Parallel circuits for the quantum fourier transform. Proc. 41st Annual IEEE Symposium on Foun- dations of Computer Science (FOCS 2000), pages 526–536, 2000.

[CZ95] J.I. Cirac and P. Zoller. Quantum Computations with Cold Trapped Ions. Phys. Rev. Lett., 74:4091, 1995.

[CZKM97] J.I. Cirac, P. Zoller, H.J. Kimble, and H. Mabuchi. Quantum state transfer and entanglement distribution among distant nodes in a quantum network. Phys. Rev. Lett., 78:3221, 1997.

[DA07] D.P. DiVincenzo and P. Aliferis. Eﬀective Fault-Tolerant Quantum Computation with slow measurement. Phys. Rev. Lett., 98:020501, 2007.

[DB03] W. D¨ur and H.-J. Briegel. Entanglement puriﬁcation for quantum computation. Phys. Rev. Lett., 90:067901, 2003.

[DBCZ99] W. D¨ur, H.-J. Briegel, J.I. Cirac, and P. Zoller. Quantum repeaters based on entanglement puriﬁcation. Phys. Rev. A., 59:169, 1999.

[DBMM04] L.-M. Duan, B.B. Blinov, D.L. Moehring, and C. Monroe. Scalable trapped ion quantum computation with a probabilistic Ion-Photon mapping. Quant. Inf. Comp., 4:165, 2004.

[DBr+98] M. Drewsen, C. Brodersen, L. Hornekaerlig r, J.S. Hangst, and J.P. Schiﬀfer. Large Ion Crystals in a Linear Paul trap. Phys. Rev. Lett., 81:2878, 1998.

242 REFERENCES

[DCH06] S.J. Devitt, J.H. Cole, and L.C.L. Hollenberg. Scheme for Direct Measurement of a General Two-Qubit Hamiltonian. Phys. Rev. A., 73:052317, 2006.

[Deu85] D. Deutsch. Quantum Theory, the Church-Turing Principle and the universal Quantum Computer. Proc. Royal Society of London A., 440:97, 1985.

[Deu89] D. Deutsch. Quantum Computational Networks. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 425:73, 1989.

[DFH05a] S.J. Devitt, A.G. Fowler, and L.C.L. Hollenberg. Investigating the Practical Implimentation of Shor’s Algorithm. Proc. SPIE, 5650:483, 2005.

[DFH05b] S.J. Devitt, A.G. Fowler, and L.C.L. Hollenberg. Practicality of Fault-Tolerant Quantum Computation. Proc. Australian Institute of Physics 16th National Congress, 2005.

[DFH06] S.J. Devitt, A.G. Fowler, and L.C.L. Hollenberg. Robustness of Shor’s Algorithm. Quant. Inf. Comp., 6:616–629, 2006.

[DGH07] S.J. Devitt, A.D. Greentree, and L.C.L. Hollenberg. Informa- tion free quantum bus for generating stabiliser states. quant- ph/0511084, Quant. Inf. Proc. (To Appear), 2007.

[DGI+07] S.J. Devitt, A.D. Greentree, R. Ionicioiu, J.L. O’Brien, W.J. Munro, and L.C.L. Hollenberg. The Photonic Module: an on- demand resource for photonic entanglement. arXiv:0706.2226, 2007.

[Dia] Moore’s Law Diagram. http://en.wikipedia.org/wiki/Moore’slaw.

[DiV00] D.P. DiVincenzo. The Physical Implementation of Quantum Com- putation. Fortschr. Phys., 48:771, 2000.

[DJ92] D. Deutsch and R. Jozsa. Rapid solutions of problems by Quantum Computation. Proc. Royal Society of London Ser. A. Math. Phys. Sci., 439:553, 1992.

[DK03] L.-M. Duan and H.J. Kimble. Eﬃcient Engineering of Multiatom Entanglement through Single-Photon Detections. Phys. Rev. Lett., 90:253601, 2003.

[DMM+06] L.-M. Duan, M.J. Madsen, D.L. Moehring, P. Maunz, R.N. Kohn Jr., and C. Monroe. Probabilistic quantum gates between remote atoms through interference of optical frequency qubits. Phys. Rev. A., 73:062324, 2006.

243 REFERENCES

[DN06] C.M. Dawson and M.A. Nielsen. The Solovay-Kitaev Algorithm. Quant. Inf. Comp., 6(1):81, 2006.

[DNBT02] J.L. Dodd, M.A. Nielsen, M.J. Bremner, and R.T. Thew. Universal quantum computation and simulation using any entangling Hamil- tonian and local unitaries. Phys. Rev. A., 65:040301(R), 2002.

[DR05] L.-M. Duan and R. Raussendorf. Eﬃcient Quantum Computation with Probabilistic Quantum Gates. Phys. Rev. Lett., 95:080503, 2005.

[DS96] D.P. DiVincenzo and P.W. Shor. Fault-Tolerant error correction with eﬃcient quantum codes. Phys. Rev. Lett., 77:3260, 1996.

[DS00] M.H. Devoret and R.J. Schoelkopf. Amplifying Quantum Signals with the Single-Electron Transistor. Nature (London), 406:1039, 2000.

[DSO+06] S.J. Devitt, S.G. Schirmer, D.K.L. Oi, J.H. Cole, and L.C.L. Hol- lenberg. Subspace Conﬁnement of Qubit Systems. Proc. Australian Institute of Physics 17th National Congress, 2006.

[DSO+07] S.J. Devitt, S.G. Schirmer, D.K.L. Oi, J.H. Cole, and L.C.L. Hol- lenberg. Subspace Conﬁnement: How good is your Qubit? quant- ph/0702123, 2007.

[DWK05] L.-M. Duan, B. Wang, and H.J. Kimble. Robust quantum gates on neutral atoms with cavity-assisted photon scattering. Phys. Rev. A., 72:032333, 2005.

[DWM03] S. Daﬀer, K. W´odkiewicz, and J.K. McIver. Quantum Markov Channels for Qubits. Phys. Rev. A., 67:062312, 2003.

[Dzy58] I. Dzyaloshinski. J. Phys. Chem. Solids, 4:241, 1958.

[Eke91] A.K. Ekert. Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett., 67:661, 1991.

[EL05] H.-A. Engel and D. Loss. Fermionic Bell-State Analyzer for Spin Qubits. Science, 309:586, 2005.

[FDH04] A.G. Fowler, S.J. Devitt, and L.C.L. Hollenberg. Implementation of Shor’s algorithm on a Linear Nearest Neighbour Qubit Array. Quant. Inf. Comp., 4:237–251, 2004.

[FDH05] A.G. Fowler, S.J. Devitt, and L.C.L. Hollenberg. Constructing Steane code Fault-Tolerant Gates. Proc. SPIE, 5650:557, 2005.

[Fey82] R.P. Feynman. Simulating Physics with Computers. Int. J. Theor. Phys., 21:467, 1982.

244 REFERENCES

[FH04] A.G. Fowler and L.C.L. Hollenberg. Scalability of Shor’s algorithm with a limited set of rotation gates. Phys. Rev. A., 70:032329, 2004.

[Fow04] A.G. Fowler. Constructing arbitrary single-qubit Fault-Tolerant gates. quant-ph/0411206, 2004.

[Fow05] A.G. Fowler. PhD Thesis (Melbourne). quant-ph/0506126, 2005.

[FPS99] R. Fazio, G. Massimo Palma, and J. Siewert. Fidelity and Leakage of Josephson Qubits. Phys. Rev. Lett., 83:5385, 1999.

[FRDM07] M.J. Fern´ee, H. Rubinsztein-Dunlop, and G.J. Milburn. Improving single-photon sources with Stark tuning. Phys. Rev. A., 75:043815, 2007.

[FTY+07] A.G. Fowler, W.F. Thompson, Z. Yan, A.M. Stephens, B.L.T Plourde, and F. K. Wilhelm. Long-range coupling and scalable architecture for superconducting ﬂux qubits. cond-mat/0702620, 2007.

[Gar91] C.W. Gardiner. Quantum Noise, volume 56 of Springer Series in Synergetics. Springer -Verlag, Berlin ; New York, 1991.

[GBP97] M. Grassl, Th. Beth, and T. Pellizzari. Codes for the quantum erasure channel. Phys. Rev. A., 56:33, 1997.

[GC97] N. Gershenfeld and I.L. Chuang. Bulk Spin Resonance Quantum Computing. Science, 275:350, 1997.

[GC99] D. Gottesman and I.L. Chuang. Demonstrating the viability of Universal Quantum Computation using teleportation and single qubit operations. Nature (London), 402:390, 1999.

[GCHH04] A.D. Greentree, J.H. Cole, A.R. Hamilton, and L.C.L. Hollenberg. Coherent electronic transfer in quantum dot systems using adia- badic passage. Phys. Rev. B., 70:235317, 2004.

[GDH06a] A.D. Greentree, S.J. Devitt, and L.C.L. Hollenberg. Adiabatic Pro- tocols for Operator Measurement based Entanglement and Quan- tum Computing. Proc. Australian Institute of Physics 17th Na- tional Congress, 2006.

[GDH06b] A.D. Greentree, S.J. Devitt, and L.C.L. Hollenberg. Quantum Information Transport to Multiple Receivers. Phys. Rev. A., 73:032319, 2006.

[GH95] G. Gilmore and J.D. Hemingway. Practical Gamma-Ray Spectrom- etry. John Wiley and Sons, Chichester, 1995.

245 REFERENCES

[GHSZ90] D.M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger. Bell’s theorem without inequalities. Am. J. Physics, 58:1131, 1990. [GHW05] J. Gorman, D.G. Hasko, and D.A. Williams. Charge-Qubit operation of an Isolated double quantum dot. Phys. Rev. Lett., 95:090502, 2005. [GHZ89] D.M. Greenberger, M.A. Horne, and A. Zeilinger. Bell’s theorem, Quantum, and Conceptions of the Universe. Kluwer Academic, Dordrecht, 1989. [GKE06] D. Gross, K. Kieling, and J. Eisert. Potential and limits to cluster- state quantum computing using probabilistic gates. Phys. Rev. A., 74:042343, 2006. [GN96] R.B. Griﬃths and C-S. Niu. Semiclassical Fourier Transform for Quantum Computation. Phys. Rev. Lett., 76:3228, 1996. [GOD+06] A.D. Greentree, P. Olivero, M. Draganski, E. Trajkov, J.R. Rabeau, P. Reichart, B.C. Gibson, S. Rubanov, S.T. Hunting- ton, D.N. Jamieson, and S. Prawer. Critical components for diamond-based quantum coherent devices. J.Phys. Condens. Mat- ter, 18(21):S825, 2006. [Got97] D. Gottesman. PhD Thesis (Caltech). quant-ph/9705052, 1997. [Got98a] D. Gottesman. A theory of Fault-Tolerant quantum computation. Phys. Rev. A., 57:127, 1998. [Got98b] D. Gottesman. The Heisenberg Representation of Quantum computers. quant-ph/9807006, 1998. [GPSW06] M. R. Geller, E. J. Pritchett, A. T. Sornborger, and F. K. Wil- helm. Quantum Computing with Superconductors 1: Architec- tures. quant-ph/0603224, 2006. [Gro97] L.K. Grover. Quantum Mechanics Helps in Searching for a Needle in a Haystack. Phys. Rev. Lett., 79:325, 1997. [GSPH06] A.D. Greentree, J. Salzman, S. Prawer, and L.C.L. Hollenberg. Quantum gate for Q switching in monolithic photonic bandgap cavities containing two-level atoms. Phys. Rev. A., 73:013818, 2006. [HBB99] M. Hillery, V. Buzek, and A. Berthiaume. Quantum Secret Sharing. Phys. Rev. A., 59(1829), 1999. [HDW+04] L.C.L. Hollenberg, A.S. Dzurak, C.J. Wellard, A.R. Hamilton, D.J. Reilly, G.J. Milburn, and R.G. Clark. Charge-based quantum computing using single donors in semiconductors. Phys. Rev. B., 69:113301, 2004.

246 REFERENCES

[Hei25] W. Heisenberg. On the quantum interpretation of kinematical and mechanical relationships. Z. Phys., 33(12):879–893, 1925.

[Hel76] C.W. Helstrom. Quantum detection and estimation theory, volume 123 of Mathematics in Science and Engineering. Academic Press, New York, 1976.

[Hel78] S. Helgason. Diﬀerential geometry, Lie groups, and symmetric spaces. Academic Press, New York, 1978.

[HGFW06] L.C.L. Hollenberg, A.D. Greentree, A.G. Fowler, and C.J. Wellard. Two-Dimensional Architectures for Donor-Based Quantum Com- puting. Phys. Rev. B., 74:045311, 2006.

[HHF+05] C.D. Hill, L.C.L. Hollenberg, A.G. Fowler, C.J. Wellard, A.D. Greentree, and H.-S. Goan. Global control and fast solid-state donor electron spin quantum computing. Phys. Rev. B., 72:045350, 2005.

[HHR+05] H. Häffner, W. Hänsel, C.F. Roos, J. Benhelm, D. Chek al kar, M. Chwalla, T. Körber, U.D. Rapol, M. Riebe, P.O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt. Scalable multiparticle entanglement of trapped ions. Nature (London), 438:643, 2005.

[HMCS00] J.J. Hope, G.M. Moy, M.J. Collett, and C.M. Savage. Steady-State quantum statistics of a non-Markovian atom laser. Phys. Rev. A., 61:023603, 2000.

[Hol73] A.S. Holevo. Probl. Inf. Trans., 9:177, 1973.

[Hol82] A.S. Holevo. Probabilistic and statistical aspects of quantum theory, volume 1 of North-Holland series in statistics and probability. North-Holland Pub. Co., Amsterdam; New York, 1982.

[Hsi98] W.Y. Hsiang. Lectures on Lie groups. World Scientiﬁc, 1998.

[HTW+06] M. Howard, J. Twamley, C. Wittmann, T. Gaebel, F. Jelezko, and J. Wrachtrup. Quantum Process tomography and Lindblad estimation of a solid state qubit. New J. Phys., 8:33, 2006.

[JKMW01] D.F.V. James, P.G. Kwiat, W.J. Munro, and A.G. White. Mea- surement of qubits. Phys. Rev. A., 64:052312, 2001.

[JM98] J.A. Jones and M. Mosca. Implementation of a Quantum Algorithm on a nuclear magnetic resonance quantum computer. J. Chem. Phys., 109:1648–1653, 1998.

[Kan98] B.E. Kane. A Silicon-Based nuclear spin Quantum Computer. Na- ture (London), 393:133, 1998.

247 REFERENCES

[Kav01] K.V. Kavokin. Anisotropic exchange interaction of localized conduction-band electrons in semiconductors. Phys. Rev. B., 64:075305, 2001.

[KC01] B. Kraus and J.I. Cirac. Optimal creation of entanglement using a two-qubit gate. Phys. Rev. A., 63:062309, 2001.

[KHS01] B. Koiller, X. Hu, and S. Das Sarma. Exchange in Silicon-Based Quantum Computer Architecture. Phys. Rev. Lett., 88:027903, 2001.

[Kit97] A.Y. Kitaev. Quantum Computations: algorithms and error correction. Russ. Math. Serv., 52(6):1191, 1997.

[KL03] K. Khodjasteh and D.A. Lidar. Quantum computing in the presence of spontaneous emission by a combined dynamical decoupling and quantum-error-correction strategy. Phys. Rev. A., 68:022322, 2003.

[KLH+03] M. Keller, B. Lange, K. Hayasaka, W. Lange, and H. Walther. Deterministic Coupling of single ions to an optical cavity. Appl. Phys. B: Laser Opt., 76:125, 2003.

[KLM01] E. Knill, R. Laﬂamme, and G.J. Milburn. A Scheme for Eﬃ- cient Quantum Computation with linear optics. Nature (London), 409:46, 2001.

[KLP05] D. Kribs, R. Laﬂamme, and D. Poulin. Uniﬁed and Generalized Ap- proach to Quantum Error Correction. Phys. Rev. Lett., 94:180501, 2005.

[KLZ96] E. Knill, R. Laﬂamme, and W.H. Zurek. Accuracy threshold for Quantum Computation. quant-ph/9610011, 1996.

[KMW02] D. Kielpinski, C. Monroe, and D.J. Wineland. Architecture for a large-scale ion-trap quantum computer. Nature (London), 417:709, 2002.

[KWH06] G. Kandasamy, C.J. Wellard, and L.C.L. Hollenberg. cross-talk compensation of hyperﬁne control in donor qubit architectures. Nanotechnology, 17:4572, 2006.

[KYI01] M. Koashi, T. Yamamoto, and N. Imoto. Probabilistic manipulation of entangled photons. Phys. Rev. A., 63:030301(R), 2001.

[Lau99] W.S. Lau. Infrared characterization for microelectronics. World Scientiﬁc, 1999.

248 REFERENCES

[LBK05] Y.-L. Lim, A. Beige, and L.C. Kwek. Repeat-Until-Success Lin- ear Optics Distributed Quantum Computing. Phys. Rev. Lett., 95:030505, 2005.

[LD98] D. Loss and D.P. DiVincenzo. Quantum Computation with quantum dots. Phys. Rev. A., 57:120, 1998.

[Leu04] D.W. Leung. Quantum computation by measurements. Int. J. Quant. Inf., 2:33, 2004.

[LH00] M.D. Lukin and P.R. Hemmer. Quantum Entanglement via Optical Control of Atom-Atom Interactions. Phys. Rev. Lett., 84:2818, 2000.

[LKCD02] H. Lee, P. Kok, N.J. Cerf, and J.P. Dowling. Linear optics and projective measurements alone suﬃce to create large photon-number path entanglement. Phys. Rev. A., 65:030101(R), 2002.

[LKS+05] D. Leibfried, E. Knill, S. Seidelin, J. Britton, R.B. Blakestad, J. Chiaverini, D.B. Hume, W.M. Itano, J.D. Jost, C. Langer, R. Ozeri, R. Reichle, and D.J. Wineland. Creation of a six-atom Schr¨odinger cat state. Nature (London), 438:639, 2005.

[LMP03] C. Lavor, L.R.U. Manssur, and R. Portugal. Shor’s Algorithm for Factoring Large Integers. quant-ph/0303175, 2003.

[LMPZ96] R. Laﬂamme, C. Miquel, J.P. Paz, and W.H. Zurek. Perfect Quan- tum Error Correcting Code. Phys. Rev. Lett., 77:198, 1996.

[Mak02] Y. Makhlin. Non-local properties of two-qubit gates and mixed states and optimization of quantum computations. Quant. Inf. Proc., 1:243, 2002.

[MCM+05] J.M. Martinis, K.B. Cooper, R. McDermott, M. Steﬀen, M. Ans- mann, K.D. Osborn, K. Cicak, S. Oh, D.P. Pappas, R.W. Sim- monds, and C.C. Yu. Decoherence in Josephson Qubits for Dielec- tric Loss. Phys. Rev. Lett., 95:210503, 2005.

[MCT+04] T. Metodiev, A. Cross, D. Thaker, K. Brown, D. Copsey, F.T. Chong, and I.L. Chuang. Preliminary Results on Sim- ulating a Scalable Fault-Tolerant Ion Trap system for quantum computation. In 3rd Workshop on Non-Silicon Computing (NSC-3), online:www.csif.cs.ucdavis.edu/ metodiev/papers/NSC3- setso.pdf, 2004.

[MI05] R. Van Meter and K.M. Itoh. Fast Quatum Modular Exponentia- tion. Phys. Rev. A., 71:052320, 2005.

249 REFERENCES

[Mil89] G.J. Milburn. Quantum optical Fredkin gate. Phys. Rev. Lett., 62:2124, 1989. [ML06] M. Mohseni and D.A. Lidar. Direct Characterization of Quantum Dynamics. Phys. Rev. Lett., 97:170501, 2006. [MMK+95] C. Monroe, D.M. Meekhof, B.E. King, W.M. Itano, and D.J. Wineland. Demonstration of a Fundamental quantum logic gate. Phys. Rev. Lett., 75:4714, 1995. [MMM+06] M.J. Madsen, D.L. Moehring, P. Maunz, R.N. Kohn Jr., L.-M. Duan, and C. Monroe. Ultrafast coherent excitation of a trapped ion qubit for fast gates and photon frequency qubits. Phys. Rev. Lett., 97:040505, 2006. [MNA02] J.M. Martinis, S. Nam, and J. Aumentado. Rabi Oscillations in a Large Josephson-Junction Qubit. Phys. Rev. Lett., 89:117901, 2002. [Moc04] C. Mochon. Anyon computers with smaller groups. Phys. Rev. A., 69:032306, 2004. [MOL+99] J.E. Mooij, T.P. Orlando, L. Levitov, L. Tian, C.H. van der Wal, and S. Lloyd. Josephson Persistent-Current Qubit. Science, 285:1096–1039, 1999. [Moo65] G.E. Moore. Cramming more Components onto Integrated Cir- cuits. Electronics, 38(8), 1965. [Mor60] T. Moriya. Anisotropic superexchange Interaction and Weak Fer- romagnetism. Phys. Rev., 120:91, 1960. [MPP96] C. Miquel, J.P. Paz, and R. Perazzo. Factoring in a dissipative quantum computer. Phys. Rev. A., 54:2605, 1996. [MSS01] Y. Makhlin, G. Sch¨on, and A. Shnirman. Quantum state engineering with Josephson-junction devices. Rev. Mod. Phys., 73:357, 2001. [MTC+05] T.S. Metodi, D. Thaker, A.W. Cross, F.T. Chong, and I.L. Chuang. A general purpose architecture layout for arbitrary quantum computations. Proc. SPIE, 5815:91, 2005. [NC00] M.A. Nielsen and I.L. Chuang. Quantum Computation and Infor- mation. Cambridge University Press, second edition, 2000. [NCT97] Y. Nakamura, C.D. Chen, and J.S. Tsai. Spectroscopy of energy- level splitting between two macroscopic quantum states of charge coherently superposed by josephson coupling. Phys. Rev. Lett., 79:2328, 1997.

250 REFERENCES

[Nie04] M.A. Nielsen. Optical quantum computation using cluster states. Phys. Rev. Lett., 93:040503, 2004.

[Nie06] M.A. Nielsen. Cluster-state quantum computation. Rep. Math. Phys., 57:147, 2006.

[NKL98] M.A. Nielsen, E. Knill, and R. Laﬂamme. Complete quantum teleportation using nuclear magnetic resonance. Nature (London), 396:52, 1998.

[NPT99] Y. Nakamura, Yu. A. Pashkin, and J.S. Tsai. Coherent control of Macroscopic Quantum states in a Cooper-Pair box. Nature (Lon- don), 398:786, 1999.

[OCC02] M.H. Oskin, F.T. Chong, and I.L. Chuang. A practical architecture for reliable quantum computers. IEEE Computer, 35:79, 2002.

[ODH06] D.K.L. Oi, S.J. Devitt, and L.C.L. Hollenberg. Scalable Error Correction in Distributed Ion Trap Computers. Phys. Rev. A., 74:052313, 2006.

[OFvdW+98] T.H. Oosterkamp, T. Fujisawa, W.G. van der Wiel, K. Ishibashi, R.V. Hijman, S. Tarucha, and L.P. Kouwenhoven. Microwave spectroscopy of a quantum-dot molecule. Nature (London), 395:873, 1998.

[OPG+04] J.L. O’Brien, G.J. Pryde, A. Gilchrist, D.F.V. James, N.K. Lang- ford, T.C. Ralph, and A.G. White. Quantum process tomography of a controlled-NOT gate. Phys. Rev. Lett., 93:080502, 2004.

[OPW+03] J.L. O’Brien, G.J. Pryde, A.G. White, T.C. Ralph, and D. Bran- ning. Demonstration of an all optical quantum controlled-NOT gate. Nature (London), 426:264, 2003.

[PC06] D. Porras and J.I. Cirac. Quantum manipulation of trapped ions in two dimensional Coulomb crystals. Phys. Rev. Lett., 96:250501, 2006.

[PCZ97] J.F. Poyatos, J.I. Cirac, and P. Zoller. Complete Characterization of a quantum process; the two-bit quantum gate. Phys. Rev. Lett., 78:390, 1997.

[Pel97] T. Pellizzari. Quantum Networking with optical Fibres. Phys. Rev. Lett., 79:5242, 1997.

[PFC02] D. Petit, L. Fruchter, and I.A. Campbell. Ordering in Heisenberg Spin Glasses. Phys. Rev. Lett., 88:207206, 2002.

251 REFERENCES

[PHBK99] M.B. Plenio, S.F. Huelga, A. Beige, and P.L. Knight. Cavity-loss- induced generation of entangled atoms. Phys. Rev. A., 59:2468, 1999.

[PJF01] T.B. Pittman, B.C. Jacobs, and J.D. Franson. Probabilistic quantum logic operations using polarizing beam splitters. Phys. Rev. A., 64:062311, 2001.

[POW+04] G.J. Pryde, J.L. O’Brien, A.G. White, S.D. Bartlett, and T.C. Ralph. Measuring a photonic qubit without destroying it. Phys. Rev. Lett., 92:190402, 2004.

[PR04] M. Paris and J. Rehacek. Quantum State Estimation, volume 649 of Lecture notes in Physics. Springer -Verlag, Berlin ; New York, 2004.

[Pre98] J. Preskill. Introduction To Quantum Computation. World Scien- tiﬁc, Singapore, 1998.

[RB01] R. Raussendorf and H.-J. Briegel. A One way Quantum Computer. Phys. Rev. Lett., 86:5188, 2001.

[RB07] P.P. Rohde and S.D. Barrett. Strategies for the preparation of large cluster states using non-deterministic gates. quant-ph/0701068, 2007.

[RBdM94] M.G. Raymer, M. Beck, and d. McAlister. Complex wave-ﬁeld reconstruction using phase space tomography. Phys. Rev. Lett., 72:1137, 1994.

[RHG05] T.C. Ralph, A.J.F. Hayes, and A. Gilchrist. Loss Tolerant Optical Qubits. Phys. Rev. Lett., 95:100501, 2005.

[RRM07] P.P. Rohde, T.C. Ralph, and W.J. Munro. Error tolerance and tradeoﬀs in loss- and failure-tolerant quantum computing schemes. Phys. Rev. A., 75:010302(R), 2007.

[RSA78] R. Rivest, A. Shamir, and L. Adleman. A method for obtaining digital signatures and public key cryptosystems. Comm. of the ACM, 21(2):120–126, 1978.

[SAM+06] M. Steﬀen, M. Ansmann, R. McDermott, N. Katz, R.C. Bialczak, E. Lucero, M. Neeley, E.M. Weig, A.N. Cleland, and J.M. Martinis. State Tomography of capacitively shunted phase qubits with high ﬁdelity. Phys. Rev. Lett., 97:050502, 2006.

[SBD+03] D. Stepanenko, N.E. Bonesteel, D.P. DiVincenzo, G. Burkard, and D. Loss. Spin-orbit coupling and time-reversal symmetry in quantum gates. Phys. Rev. B., 68:115306, 2003.

252 REFERENCES

[SBF+06] T. Szkopek, P.O. Boykin, H. Fan, V.P. Roychowdhury, E. Yablonovitch, G. Simms, M. Gyure, and B. Fong. Threshold Error Penalty for Fault-Tolerant Computation with Nearest Neigh- bour Communication. IEEE Trans. Nano., 5(1):42, 2006.

[SBL+04] T. Schaetz, M.D. Barrett, D. Leibfried, J. Chiaverini, J. Britton, W.M. Itano, J.D. Jost, C. Langer, and D.J. Wineland. Quantum Dense Coding with atomic qubits. Phys. Rev. Lett., 93:040505, 2004.

[SBRF93] D.T. Smithey, M. Beck, M.G. Raymer, and A. Faridani. Mea- surement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to Squeezed States and the Vacuum. Phys. Rev. Lett., 70:1244, 1993.

[Sch26] E. Schr¨odinger. Quantisation as a problem of characteristic values. Ann. Phys., 79(4):361–376, 1926.

[SDF+06] A.M. Stephens, S.J. Devitt, A.G. Fowler, J.C. Ang, and L.C.L. Hollenberg. Gate Level Simulation of Logical State Preparation. quant-ph/0608112, 2006.

[SDT07] K.M. Svore, D.P. DiVincenzo, and B.M. Terhal. Noise Threshold for a Fault-Tolerant Two-Dimensional Lattice Architecture. Quant. Inf. Comp., 7:297, 2007.

[SFH07] A. Stephens, A.G. Fowler, and L.C.L. Hollenberg. Universal Fault- Tolerant Computation on bilinear nearest neighbor arrays. quant- ph/0702201, 2007.

[SH00] J.M.G. Sancho and S.F. Huelga. Measuring the entanglement of bipartite pure states. Phys. Rev. A., 61:042303, 2000.

[Sho95] P.W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A., 52:R2493, 1995.

[Sho96] P.W. Shor. Fault-Tolerant quantum computation. Proc. 37th IEEE Symp. on Foundations of Computer Science., pages 56–65, 1996.

[Sho97] P.W. Shor. Polynomial-Time algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM Journal of Sci. Statist. Comput., 26:1484, 1997.

[SHS+07] D.I. Schuster, A.A. Houck, J.A. Schreier, A. Wallraﬀ, J.M. Gam- betta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M.H. Devoret, S.M. Girvin, and R.J. Schoelkopf. Resolving photon number states in a superconducting circuit. Nature (London), 445:515, 2007.

253 REFERENCES

[SHW+07] C. Sch¨on, K. Hammerer, M.M. Wolf, J.I. Cirac, and E. Solano. Se- quential generation of matrix-product states in cavity QED. Phys. Rev. A., 75:032333, 2007. [SI03] C. Simon and W.T.M. Irvine. Robust Long-Distance Entanglement and a Loophole-Free Bell Test with Ions and Photons. Phys. Rev. Lett., 91:110405, 2003. [SI05] A.M. Steane and B. Ibinson. Fault-Tolerant Logical Gate Networks for Calderbank-Shor-Steane codes. Phys. Rev. A., 72:052335, 2005. [Sim94] D.R. Simon. On the Power of Quantum Computing. Proc. 35th IEEE Symp. on Foundations of Computer Science, pages 116–123, 1994. [SKK+00] C.A. Sackett, D. Kielpinski, B.E. King, C. Langer, V. Meyer, C.J. Myatt, M. Rowe, Q.A. Turchette, W.M. Itano, D.J. Wineland, and C. Monroe. Experimental entanglement of four particles. Nature (London), 404:256, 2000. [SKO04] S.G. Schirmer, A. Kolli, and D.K.L. Oi. Experimental Hamilto- nian Identiﬁcation for controlled two-level systems. Phys. Rev. A., 69:050306(R), 2004. [SL00] A.M. Steane and D.M. Lucas. Quantum Computing with trapped ions, atoms and light. Fortschr. Phys., 48:839, 2000. [SNAA05] B.-S. Song, S. Noda, T. Asano, and Y. Akahane. Ultrahigh-Q photonic double-heterostructure nanocavity. Nature Materials (Lon- don), 4:207, 2005. [SNB+06] T.P. Spiller, K. Nemoto, S.L. Braunstein, W.J. Munro, P. van Loock, and G.J. Milburn. Quantum Computation by Communica- tion. New J. Phys., 8:30, 2006. [SPS05] S.G. Schirmer, I.C.H. Pullen, and A.I. Solomon. Controllability of multi-partite quantum systems and selective excitation of quantum dots. J. Opt. B, 7:S293–S299, 2005. [SS04] S.G. Schirmer and A.I. Solomon. Constraints on relaxation rates for N-level quantum systems. Phys. Rev. A., 70:022107, 2004. [Ste96a] A.M. Steane. Error Correcting Codes in Quantum Theory. Phys. Rev. Lett., 77:793, 1996. [Ste96b] A.M. Steane. Simple quantum error-correcting codes. Phys. Rev. A., 54:4741, 1996. [Ste97] A.M. Steane. Active Stabilization, Quantum Computation and Quantum State Synthesis. Phys. Rev. Lett., 78:2252, 1997.

254 REFERENCES

[Ste02] A.M. Steane. Fast Fault-Tolerant ﬁltering of quantum codewords. quant-ph/0202036, 2002.

[Ste03] A.M. Steane. Overhead and noise threshold of Fault-Tolerant quantum error correction. Phys. Rev. A., 68:042322, 2003.

[Ste07] A.M. Steane. How to build a 300 bit, 1 Giga-operation quantum computer. Quant. Inf. Comp., 7:171, 2007.

[Sto52] G.C. Stokes. Trans. Cambr. Phil. Soc., 9:399, 1852.

[SW76] B.P. Straughan and S. Walker. Spectroscopy. Chapman and Hall; Wiley; Distributed in the U.S by Halsted Press, London, New York, 1976.

[TED+05] J.M. Taylor, H.-A. Engel, W. D¨ur, A. Yacoby, C.M. Marcus, P. Zoller, and M.D. Lukin. Fault-Tolerant Architecture for Quan- tum Computation using Electrically controlled Semiconductor Spins. Nature Physics (London), 1:177, 2005.

[THE+05] M. Trupke, E.A. Hinds, S. Eriksson, E.A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft. Microfabricated high-ﬁnesse optical cavity with open access and small volume. App. Phys. Lett., 87:211106, 2005.

[TJBB06] B. Trauzettel, A.N. Jordan, C.W.J. Beenakker, and M. Buttiker. Parity meter for charge qubits: An eﬃcient quantum entangler. Phys. Rev. B., 73:235331, 2006.

[Tof81] T. Toﬀoli. Bicontinuous extension of reversible combinatorial functions. Math. Syst. Theory, 14:13–23, 1981.

[VBE96] V. Vedral, A. Barenco, and A.K. Ekert. Quantum Networks for elementary arithmetic operations. Phys. Rev. A., 54:147, 1996.

[VC04] F. Verstraete and J.I. Cirac. Valence-bond states for quantum computation. Phys. Rev. A., 70:060302(R), 2004.

[VD04] G. Vidal and C.M. Dawson. Universal quantum circuit for two- qubit transformations with three controlled-NOT gates. Phys. Rev. A., 69:010301(R), 2004.

[vEKCZ99] S.J. van Enk, H.J. Kimble, J.I. Cirac, and P. Zoller. Quantum Communication with dark photons. Phys. Rev. A., 59:2659, 1999.

[VSB+01] L.M.K. Vandersypen, M. Steﬀen, G. Breyta, C.S. Yannoni, M.H. Sherwood, and I.L. Chuang. Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature (London), 414:883, 2001.

255 REFERENCES

[VWW05] J. Vala, K.B. Whaley, and D.S. Weiss. Quantum error correction of a qubit loss in an addressable atomic system. Phys. Rev. A., 72:052318, 2005.

[WB03] C. Wunderlich and C. Balzer. Quantum Measurements and new concepts for experiments with trapped ions. Adv. At., Mol., Opt. Phys., 49:293, 2003.

[WBB+03] D.J. Wineland, M. Barrett, J. Britton, J. Chiaverini, B. DeMarco, W.M. Itano, B.M. Jelenkovi´c, C. Langer, D. Leibfried, V. Meyer, T. Rosenband, and T. Sch¨atz. Quantum Information processing with trapped ions. Phil. Trans. Royal Soc. London A., 361:1349, 2003.

[WBL02] L.-A. Wu, M.S. Byrd, and D.A. Lidar. Eﬃcient Universal Leakage Elimination for Physical and Encoded Qubits. Phys. Rev. Lett., 89:127901, 2002.

[WJMK01] A.G. White, D.F.V. James, W.J. Munro, and P.G. Kwiat. Explor- ing Hilbert Space: Accurate characterization of quantum information. Phys. Rev. A., 65:012301, 2001.

[WKN01] J. Wrachtrup, S. Ya. Kilin, and A. P. Nizovtsev. Quantum computation using the 13C nuclear spins near the single NV defect center in diamond. Optics and Spectroscopy, 91(3):429, 2001.

[WLHN05] L.F. Wei, X. Li, X. Hu, and F. Nori. Eﬀects of dynamical phases in Shor’s factoring algorithm with operational delays. Phys. Rev. A., 71:022317, 2005.

[WMI+98] D.J. Wineland, C. Monroe, W.M. Itano, D. Leibfried, B.E. King, and D.M. Meekhof. Experimental Issues in coherent quantum state manipulation of trapped atomic ions. Journal of research NIST, 103:259, 1998.

[Woo98] W.K. Wooters. Entanglement of Formation of an Arbitrary State of Two Qubits. Phys. Rev. Lett., 80:2245, 1998.

[WS05] G. Wendin and V.S. Shumeiko. Superconducting Quantum Cir- cuits, Qubits and Computing. quant-ph/0508729, 2005.

[WSB+04] A. Wallraﬀ, D.I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Ma- jer, S. Kumar, S.M. Girvin, and R.J. Schoelkopf. Circuit quantum electrodynamics: Coherent coupling of a single photon to a Cooper pair box. Nature (London), 431:162, 2004.

[WZ82] W.K. Wooters and W.H. Zurek. A Single Quantum Cannot be Cloned. Nature (London), 299:802, 1982.

256 REFERENCES

[Yao93] A.C. Yao. Quantum Circuit Complexity. Proc. 34th Ann. Symp. FOCS, page 352, 1993.

[YPA+03] T. Yamamoto, Yu. A. Pashkin, O. Astaﬁev, Y. Nakamura, and J.S. Tsai. Demonstration of a conditional gate operation using superconducting charge qubits. Nature (London), 425:941, 2003.

[YR03] N. Yoran and B. Reznik. Deterministic Linear Optics Quan- tum Computation with Single Photon Qubits. Phys. Rev. Lett., 91:037903, 2003.

[Zal98] C. Zalka. Fast Versions of Shor’s quantum factoring algorithm. quant-ph/9806084, 1998.

[ZMD06] S-L. Zhu, C. Monroe, and L.-M. Duan. Trapped ion quantum computation with transverse phonon modes. Phys. Rev. Lett., 97:050505, 2006.

[Zur01] W.H. Zurek. Decoherence, Einselection and the Quantum Origins of the Classical. quant-ph/0105127, 2001.

[ZVSW03] J. Zhang, J. Vala, S. Sastry, and B. Whaley. Geometric theory of nonlocal two-qubit operations. Phys. Rev. A., 67:042313, 2003.

[ZW05] J. Zhang and K.B. Whaley. Generation of quantum logic operations from physical Hamiltonians. Phys. Rev. A., 71:052317, 2005.

[ZZH97] M. Zukowski, A. Zeilinger, and M.A. Horne. Realizable higher- dimensional two-particle entanglement via multiport beam splitters. Phys. Rev. A., 55:2564, 1997.

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Minerva Access is the Institutional Repository of The University of Melbourne

Author/s: Devitt, Simon

Title: Quantum information engineering: concepts to quantum technologies

Date: 2007-11

Citation: Devitt, S. (2007). Quantum information engineering: concepts to quantum technologies, PhD thesis, Physics, University of Melbourne.

Publication Status: Unpublished

Persistent Link: http://hdl.handle.net/11343/39351

File Description: Quantum information engineering: concepts to quantum technologies

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