Robust Ion Trap Quantum Computation Enabled by Quantum Control
by
Pak Hong (James) Leung
Department of Physics
Duke University
Date:
Approved:
Kenneth Brown, advisor
Stephen Teitsworth
Harold Baranger
Thomas Barthel
Jungsang Kim
Dissertation submitted in partial fulfillment of
the requirements for the degree of Doctor
of Philosophy in the Department of
Physics in the Graduate School
of Duke University
2020 ABSTRACT
Robust Ion Trap Quantum Computation Enabled by Quantum Control
by
Pak Hong (James) Leung
Department of Physics
Duke University
Date:
Approved:
Kenneth Brown, advisor
Stephen Teitsworth
Harold Baranger
Thomas Barthel
Jungsang Kim
Dissertation submitted in partial fulfillment of
the requirements for the degree of Doctor
of Philosophy in the Department of
Physics in the Graduate School
of Duke University
2020 Copyright by
Pak Hong (James) Leung
2020 Abstract
The advent of quantum computation foretells a new era in science and technology, but the fragility of quantum bits (qubits) and the unreliability of gates hinder the realization of functioning quantum computers. For ion trap quantum computers in particular, 2-qubit operations relying on the Mølmer-Sørensen interaction have the greatest error rates. This dissertation introduces frequency-modulated (FM) pulses as a measure to maximize 2-qubit gate fidelity and a way to calibrate gate errors through the measurement of circuit performance.
A key challenge of two-qubit gates in ion chains is unwanted residual entanglement between the ion spin and its motion. Frequency-modulated pulses are developed to achieve such goal. This theoretical advance has led to high-fidelity 2-qubit gates that are robust against small frequency drifts in a 5-ion experiment. Combining frequency and amplitude modulation, numerical calculations suggest that entanglement between an arbitrary pair of qubits are possible in a lattice with up to 50 ions. More recently, long-distance 2-qubit gates have been realized within a 17-ion chain.
Quantum circuit calibration is proposed to improve quantum circuits using feed- back from measurement results. A relationship between the error parameters and measured observables can be established to identify systematic circuit errors. The calibration of a 6-qubit parity check circuit targeting 2-qubit overrotations has been
iv implemented using measurement results from an experimental 15-ion trap. This im- provement is conducive to quantum error correction protocols which involve high- weight stabilizers. A 4-bit Toffoli circuit with an error vector of length 6 is calibrated using a custom circuit simulator, reducing the average error size by a factor of 4. Using linear and quadratic approximation, a 6-bit Toffoli circuit with 12 error parameters is calibrated in the presence of 3 ancilla qubits.
v Contents
Abstract iv
Frequently Used Variables xii
List of Figures xv
Acknowledgements xvi
1 Introduction 1
2 Basics of Quantum Information 9
2.1 Superposition and Entanglement ...... 9
2.2 Quantum Gates ...... 11
2.2.1 Matrix expressions ...... 11
2.2.2 Continuous rotations ...... 12
2.3 Circuit Diagrams ...... 14
2.4 Universal Gate Set ...... 15
2.5 Unitary Evolution ...... 16
2.6 Single-qubit Rotations ...... 17
2.7 Mixed States ...... 19
vi 2.8 Measurements ...... 21
2.9 Fidelity and Trace Distance ...... 23
3 Ion Qubits 26
3.1 State Structure of a Hydrogen-like Ion ...... 26
3.2 Selection Rules for State Transitions ...... 28
3.3 The Longevity of a Qubit ...... 29
3.3.1 Qubit Decay Lifetime (T1)...... 29
3.3.2 Qubit Dephasing Lifetime (T2)...... 31
3.4 Relevant State Transitions ...... 32
3.4.1 State Preparation and Measurement ...... 32
3.4.2 Qubit Transitions ...... 34
3.5 1-D Ion Lattice ...... 35
3.5.1 1-D Ion Distribution ...... 35
3.5.2 Transverse motional spectrum ...... 37
3.6 Mølmer-Sørensen (MS) Gates ...... 38
3.6.1 Hamiltonian for sideband transitions ...... 38
3.6.2 Effective Unitary Evolution ...... 40
3.7 Scaling Ion Traps ...... 42
3.7.1 Surface Trap Architecture ...... 42
3.7.2 Ion Shuttling ...... 43
3.7.3 2-D Ion Lattice ...... 44
3.7.4 Photonic Interconnects ...... 44
4 Optimized MS gates for shorter ion chains (< 20 ions) 46
vii 4.1 Two-qubit Gate Optimization ...... 46
4.1.1 Error due to Residual Entanglement ...... 47
4.1.2 Robustness against Frequency Drifts ...... 50
4.1.3 Power Estimate for Every Pair of Qubits ...... 51
4.2 Modulated Pulses ...... 53
4.2.1 Amplitude Modulation (AM) ...... 53
4.2.2 Frequency Modulation (FM) ...... 54
4.2.3 Phase Modulation ...... 56
4.2.4 Multi-tone MS gates ...... 57
4.3 Short Ion Chains in Experiment ...... 57
4.3.1 5-ion trap experiment ...... 58
4.3.2 17-ion trap experiment ...... 60
5 Optimized MS gates for longer ion chains (> 20 ions) 62
5.1 Trapping Long, Uniform Ion Chains ...... 62
5.1.1 Ideal Trap Shape ...... 63
5.1.2 Motional Spectrum and Resonant Modes ...... 66
5.2 Optimization Methodology ...... 67
5.2.1 Selective Mode Coupling ...... 67
5.2.2 Optimized Pulses ...... 68
5.3 Further Scalability ...... 71
5.3.1 Neighbouring Entangling Regime ...... 72
5.3.2 Arbitrary Entangling Regime ...... 74
5.4 Advanced Pulse Modulation Techniques ...... 75
5.4.1 Parallel 2-qubit Gates within a Short Ion Chain ...... 76
viii 5.4.2 Power-optimal, stabilized 2-qubit Gates ...... 77
6 Quantum Circuit Calibration (QCC): Theory 79
6.1 Calibrating Gates VS Calibrating Circuits ...... 79
6.2 A Mapping from Errors to Coordinates ...... 82
6.3 Calibration Procedure ...... 84
6.4 Measurements, Projection, and Coordinates ...... 86
6.5 Linear Approximations ...... 88
6.6 An Error Model for Trapped Ion Qubits ...... 90
6.7 The Role of Classical Simulation ...... 93
6.8 Example: Toffoli Circuit ...... 94
7 QCC: Application to Multi-qubit Controlled Gates 97
7.1 Controlled Circuits ...... 97
7.1.1 N-CNOT circuit ( N (U)) ...... 98 ∧ 7.1.2 Parity check circuits ( N (U)) ...... 100
7.1.3 Hidden Inverse ConfigurationsL ...... 103
7.2 Experimentally Calibrating a 6(X) Gate ...... 106
7.3 Simulating the Calibration of L3(X) Gate ...... 110 ∧ 7.4 Simulating the Calibration of 5(X) Gate ...... 112 ∧
8 Conclusion 116
Appendices 120
A - FM Optimization with continuous pulses 121
ix B - Circuit Simulator: QRSim 124
B.1 Single and 2-qubit Rotations ...... 125
B.2 Gate sequence ...... 126
B.3 The CuPy module ...... 127
Bibliography 130
Biography 137
x Frequently Used Variables
Modulated two-qubit gates
Ωi(t) Rabi frequency, a measure of laser intensity when the i-th ion is
addressed.
µ(t) Driving frequency relative to the frequency splitting between
ground and excited state.
ωk The k-th sideband frequency.
δk(t) Detuning between the driving frequency and ωk
ηi,k Lamb-Dicke parameter for the i-th ion and k-th motional mode.
More explicitly, the ratio between sideband Rabi frequency and the
carrier Rabi frequency for motional mode k when the i-th ion is
addressed.
τ Single or 2-qubit gate time.
αˆk Spin-dependent motional displacement of the k-th sideband.
βij Entanglement between ions i and j.
xi Quantum Circuit Calibration
~ε The error vector for a circuit, for instance the overrotation errors
for a set of 2-qubit gates.
~r The vector of expectation values of observables of an output state.
For instance, the projection of a qubit onto the X basis, or σ . h xi Varies with ~ε.
N (U) A controlled rotation U with N control qubits and 1 target qubit. ∧ In some literature it is referred to as “(N + 1)-bit Toffoli” .
N (U) A single-qubit rotation U conditioned on the parity of N qubits. L
xii List of Figures
2.1 An illustration of the Bloch sphere...... 22
3.1 State preparation and measurement of a qubit for 171Yb+...... 33
3.2 Three-level state system for Raman transitions...... 34
3.3 A sketch of a typical Paul trap...... 36
3.4 A fluorescent image of a 53-ion chain...... 36
3.5 State transition diagram of a Mølmer and Sørensen gate...... 39
3.6 An illustration of a surface trap...... 43
4.1 The area enclosed by the phase space trajectory, proportional to the entanglement between the addressed qubits...... 52
4.2 FM 2-qubit gates are implemented in a 4-ion trap...... 54
4.3 Robust and non-robust frequency patterns, fidelity plots, and phase space trajectories...... 58
4.4 AM-FM 2-qubit gates on a 17-ion trap...... 61
5.1 Depiction of particular motional modes acting as the means for entan- glement...... 65
5.2 AM-FM gates for a simulated 50-ion setup...... 68
5.3 Residual gate error versus unwanted frequency offset...... 70
5.4 Power required to entangle any pair of qubits in Rabi frequency. . . . 71
xiii 5.5 Sideband driving in the far-detuned and near-detuned limits...... 72
6.1 Black box circuit with variable control (error) parameters...... 82
6.2 The calibration cycle consists of 3 stages: calibration, computation, and transfer of quantum information...... 84
6.3 The circuit component during calibration and computation stages. . . 85
6.4 The error in estimating a single-qubit state angle versus the number of measurements for 1,000 repeated simulations...... 90
6.5 An implementation of the Toffoli gate using native gates in ion traps. 94
6.6 Optimization results from scipy’s BFGS algorithm...... 95
6.7 The fractional error in estimating ~ε varies linearly with the average error size in the error vector...... 96
7.1 Breakdown of the 2(U)...... 99 ∧ 7.2 Breakdown of the 3(U) circuit...... 99 ∧ 7.3 5(X) with 3 scrap qubits ...... 100 ∧ 7.4 An implementation of N (U)...... 101 ∧ 7.5 Parity check operation ( N (X)) ...... 102
7.6 An example of a hiddenL inverse in a CZ circuit...... 104
7.7 Final state fidelities for the controlled-Z rotation circuit plotted as a function of rotation angle θ for regular and conjugated configurations. 105
7.8 Final state fidelities of N (R ( 2θ)) circuit...... 105 z − 7.9 6-qubit parity check circuit,L broken down to Mølmer-Sørensen and single-qubit gates...... 107
7.10 Qubit layout along the ion chain for the 6-qubit parity check experiment.107
7.11 Simulation and experimental results for the parity-check circuit as well as the estimated error vectors...... 108
xiv 7.12 Measured coordinates and actual coordinates versus the input state for the 3(X) circuit...... 111 ∧ 7.13 Estimate error for the length-6 ~ε over 500 trials...... 112
7.14 An improved version of the 5(X) circuit using Peres gates...... 113 ∧ 7.15 Fidelity VS Input state for a randomly generated error vector for the 5(X) circuit...... 113 ∧ 7.16 Linear and quadratic correlations between errors and select measured coordinates...... 115
A.1 A sketch of the displacement diagram for a motional mode...... 122
B.1 A simple example for building a CNOT gate using QRSim...... 125
B.2 The “ideal gates” data type for QRSim...... 128
B.3 The “native gates” data type for QRSim...... 128
B.4 The “noisy gates” data type for QRSim...... 128
xv Acknowledgements
A graduate degree in science is a challenging path, and I want to take this opportunity to thank everyone who has helped me along the way. The first person I would like to thank, of course, is my advisor Ken Brown. When I was still a junior graduate student, he gave me a great headstart on fundamental subjects such as composite pulses and
Mølmer-Sørensen gates, which set my eyes on 2-qubit gate control. Through his leadership and extensive knowledge in theory and experiment, he has nurtured a large group of graduate and undergraduate students and drew many talented individuals into the fold. He is a practical, forward-looking scientist who is always in touch with the newest and never held back by tradition. For many times, he has connected me with experimentalists and made collaboration possible.
Speaking of collaboration, I thank Kevin Landsman and Caroline Figgatt who also pursued PhD in Physics in University of Maryland, as well as then-postdoc Norbert
Linke. They are dedicated experimentalists who embraced modulated 2-qubit gates and are quick to consider and accept my ideas about frequency-modulated gates.
Norbert went on to become a Professor, while Kevin and Caroline found their places in the industry. Their advisor, Chris Monroe, is a leader in ion trap quantum computers in academia and industry and has been very supportive of collaboration between us.
More recently, Laird Egan, another student of Chris, has cooperated with me on
xvi calibrating parity check circuits in a 15-ion trap.
My amazing colleagues at Duke have made this a memorable experience. Swar- nadeep Majumder is a fellow graduate student who also works in quantum control and state tomography, and has embraced the ever-spreading machine learning doctrine in science. While doing inspiring work in spectator qubits, he shows great flexibility and readiness to adopt new technologies. Shilin Huang is a genius in computer science, mathematics, and physics, and truly exemplifies quality Tsinghua education. He has provided me with crucial help in academic subjects despite my endless questions, and have collaborated with me on controlled phase-modulated pulses. Sarah Brandsen, a graduate student in Henry Pfister’s group, has been very helpful in explaining to me the concepts of Quantum State Discrimination and has worked tirelessly despite her responsibilities as a new mother. Daniel Murphy and Catherine Liang have worked with me as undergraduate students on projects related to ion trap gate control and have shown great knowledge and potential in physics and quantum computation.
My degree began at Georgia Institute of Technology, and I have not forgotten about my old companions from there. Mauricio Gutierrez was always a fun and open-minded theorist to work with, with his incredible talent in Quantum Error
Correction. He went on to become a postdoc in Swansea University and a professor in Costa Rica, his home country. Rick Shu was a true master in experimental control who had more than 8 years of experience as a postdoc by the time I left Georgia Tech.
He informed me of everything about control in experimental ion traps, which gave me a background for 2-qubit gates. He also has a knack for organizing spontaneous outdoor trips with group members and friends. Aaron Calvin was an experimentalist through and through and was always willing to answer my questions about molecular
xvii trapping. I remember him for his focus and relative quietness, but his expertise in chemistry never failed to shine through.
The Committee Members for my dissertation defense are themselves inspiring leaders in Physics and beyond. Jungsang Kim is incredibly devoted to ion trap experiments and has come a long way to advance quantum computation. Stephen
Teitsworth, the Director for Graduate Studies in Physics, has helped many graduate students like me navigate their way through their PhD programs. Thomas Barthel professes in many-body physics theory and was my instructor for Graduate Advanced
Physics course. Harold Baranger leads a group in open quantum systems and also has an interest in Quantum Information.
Last but not least, my parents have always been there for me no matter where I am and what I am going through. They respected my decision to pursue a Physics degree, and I hope my work will make them proud.
xviii 1. Introduction
Quantum computation holds tremendous potential in impacting science and engineer- ing as well as cryptography and security. During the 1930s, Alan Turing laid down the principles of a Turing machine, which effectively define the modern computer [1].
They store information in binary units called bits, which can be flipped and corrected during computation, and can be manipulated by any Turing-complete programming languages. Physical computers went from large, unwieldy transistors to widely com- mercially available laptops and gadgets with processing powers in the order of GHz, their capacity trending along Moore’s Law for over half a century. Although comput- ers have evolved rapidly in appearance, Turing’s principles remain unchanged. The advent of quantum computers, however, will change these principles, and hence will be the most disruptive event in computing science. The idea of superposition and entanglement, originally established by physicists, have no correspondence in classical computers, and are likely the source of the coveted exponential speedup of quantum computers. Quantum information is stored in qubits, with an exponentially growing
Hilbert space. A quantum computer with N qubits has 2N probability amplitudes and complex phases, discounting unit normalization and the global phase. To provide some perspective, 300 qubits can store more such amplitudes than there are atoms in the universe. A series of no-go theorems, however, forbid direct extraction of clas-
1 sical information from limited copies of quantum states [2, 3]. The vast potential of quantum computer to store and process information and how we can harness such potential have fascinated scientists for many decades.
The true versatility of a quantum computer remains an active area of research in the scientific and computational community. Experts in complexity theory have been searching for problems where quantum computers truly excel the performance of their classical counterparts. Specifically, the complexity class “BQP” includes algorithms that take quantum computers polynomial time to solve with respect to the problem size. It is established that quantum computers can solve any problems that classical computers can efficiently, but the question remains what kind of problems a quantum computer can solve efficiently that their classical counterparts cannot. The most well- known and sought after quantum algorithm in such category is Shor’s algorithm [4], which shows super-polynomial speedup in factoring large numbers. Quantum com- puters also have a quadratic speedup over classical ones when implementing Grover’s search algorithm, whose goal is to find the input to a black box function that yields one, or equivalently an unstructured search for a marked element [5]. A shorter-term aspect is the simulation of a quantum system, as originally suggested by Richard
Feynmann in 1989 [6]. Methods such as Variational Quantum Eigensolver (VQE) are used to predict ground state energy of any quantum systems, which has been done for a simple model of a water molecule recently [7]. There are even proposals for using specific quantum circuits to solve problems related to Quantum Field Theory [8].
While researchers are still on the hunt for special algorithms and applications for quantum computers, one thing is strikingly clear - we need better qubits, better gates, and better circuits. In many ways, quantum computing is as much an engineering
2 subject as a scientific one, as we are focused on improving quantum computers based on established scientific principles. The conditions for building a tenable quantum computer can be summarized in DiVincenzo’s criteria [9], which stipulate that quan- tum computers must be scalable with long-lived qubits, equipped with a universal set of quantum gates. The qubits’ fragility and overall fickleness have been the bane of researchers who try to make quantum computers a reality. Experimental groups are on a race to make quantum computers larger, faster, and more resilient against inaccuracies and environmental disturbances, and control theorists propose schemes to improve a quantum computer’s physical performance.
Ion traps are among the top competitors for hosting quantum computation. Chap- ter 3 gives a brief summary of the principles as well as state-of-the-art regarding ion trap quantum computers. Unlike macroscopic materials, atoms or ions of the same isotope are identical. Optical qubits with a metastable state as well as hyperfine ground state qubits have been shown to have very long lifetimes [10, 11]. The ground states can be prepared using optical pumping, which allows effective state initializa- tion. Short-lived, transitive states can be used for Raman transitions with variable detuning. Multiple ions can be trapped in crystalline form, where their resonant mo- tion serve as a quantum bus for multi-qubit entanglement, as proposed by Mølmer and Sørensen [12, 13]. Using these principles, universal gate sets have been realized with upwards of 99% fidelity by multiple ion trapping groups during the past 20 years [14, 15, 16, 17]. In particular, collective entanglement of 14 qubits has been reported using fundamental Mølmer-Sørensen interaction [18]. Small algorithms and quantum codes have been demonstrated within the past few years [17, 19]. Although it is predicted that multi-ion entanglement will become less feasible for 1-D traps
3 with hundreds of ions, many proposals for scaling them exist, as will be discussed in section 3.7. As far as hardware architecture is concerned, due to limited connectivity between qubits as the number of qubits increases, it may become necessary to break up quantum computers into separate modules, perform quantum operations within modules, and allow communication between modules if necessary. The scalability of ion traps will heavily depend on technological advancements in trap manufacturing which enable interaction between trapping zones, using either physical shuttling or photonic interconnects [20, 21].
Just like any real quantum computer, ion trap quantum computers are faced with experimental errors and environmental disturbances, and improving multi-qubit gates through the use of modulated pulses is the focus of Chapters 4 and 5. Experimental- ists have been hard at work to upgrade the hardware, but theorists have also made contributions to modelling and optimizing gate processes that can minimize such errors. The field of quantum control attempts to minimize errors by introducing pa- rameters into quantum gates to suppress errors of certain types, such as overrotation errors. For example, composite pulses can be used to cancel out the first-order effect of overrotation errors [22]. This is particularly useful against fluctuations in the EM waves needed to drive quantum gates. When applying Mølmer-Sørensen gates, any residual coupling between the qubits and the ions’ motional state must be minimized, which can be achieved using amplitude and/or frequency modulated gates (AM, FM)
[23, 24]. The application of such control methods have been extensively adopted in experimental groups, leading to an average gate fidelity above 99% [25, 24]. The final quality of gates can be measured using many metrics, such as the fidelity, which quan- tifies the overlap between the ideal and final output states, and the trace distance, a
4 more conservative measure based on the distinguishability between two states. Since the viable width and depth of our quantum circuit depend on the error rates of our gates, the aforementioned prospects are crucial for the competence of traps with tens of qubits or more.
Another practice that can help improve our quantum computer is to calibrate our quantum gates as accurately as resource permits before computing [26], which is the key topic in Chapters 6 and 7. In addition to suppressing errors directly using controlled pulses, we can break up our quantum computer into smaller modules, each of which calibrated to the best of our ability. Given a quantum circuit consisting of a sequence of single and 2-qubit rotations prone to overrotation and/or phase errors, the goal is to determine the error vector to the highest precision with a finite number of measurements on the output state. For example, assuming we are trying to calibrate a Toffoli gate with only 2-qubit overrotations, we can estimate the errors by making three specific measurements on the Bloch sphere coordinates of the output qubits, as demonstrated in section 6.8. To expedite the calibration process, we can use linear and quadratic approximations to characterize the response of the circuit to small error parameters, as we discuss in sections 6.5 and 7.16. Finally, instead of using fixed- basis measurement, we may want to adjust our measurement basis such that the error vector is determined to the highest precision. This problem can be thought of as a continuous version of Quantum State Discrimination (QSD), a well-studied problem in the field of Quantum Information [27]. A variety of numerical techniques relying on locally adaptive protocols and machine learning tools have proved to enhance our success probability in guessing the correct state [28, 29].
One prospect worth mentioning concerning the longevity of quantum computa-
5 tion is Quantum Error Correction (QEC). Since quantum states are in superposition during computation, it may seem impossible to take measurements to gain infor- mation about potential errors without disrupting the quantum state. In order to circumvent this, one can use data qubits to encode our information in logical states which are simultaneous eigenstates of several measurement operators called stabiliz- ers [30]. By conducting parity check measurements, we can decode the results and make corrections accordingly, given that the weight of the errors is less than half of the code distance. Additional ancilla qubits are used to extract the parity informa- tion by performing conditional logic with data qubits. However, in order for QEC to suppress the logical error rate, the physical error rate per gate must get below a certain threshold [31], which relies on improved experimental and control methods.
Moreover, decoherence worsens as the number of qubits increases, and the number of qubits that can be managed in a quantum computer is limited. As of 2020, we can only perform coherent operations with a few tens of qubits within an ion trap, which means we cannot afford to encode quantum information into more qubits in the foreseeable future. QEC is therefore perceived as a long-term solution for protecting quantum information from errors.
This dissertation gives a detailed description of cutting-edge techniques for en- hancing the performance of ion trap quantum computers. My main contributions include the following:
1. The adoption of frequency modulation for performant Mølmer-Sørensen gates
(see Chapter 4). Compared with traditional stepwise amplitude-modulated
pulses, continuous frequency modulated pulses show remarkable robustness
against frequency fluctuations in practical experiments.
6 2. The implementation of selective mode coupling for realizing long-range ion-to-
ion entanglement within a large 1-D crystal (see Chapter 5). The common
practice had been to drive sideband transition while being far detuned from
all motional modes, but if instead we drive only a small number of modes, we
can fine-tune our gates such that only 2 out of N ions are entangled despite
their distance, which can be a useful boost relative to computers with only local
connectivity. This can be easily generalized to the 2-D case.
3. The proposal of efficient circuit calibration (see Chapters 6 and 7). Instead
of tweaking gates individually, we estimate error parameters by examining the
circuit’s behavior as a whole. Information is extracted by making strategic
measurements of abstract coordinates of the output state, which reveals the
error parameters within the circuit.
This work is organized into 8 chapters, the first one being this introduction. Chap- ter 2 reviews fundamental concepts of a generic quantum computer and the idea of universal quantum computation. Chapter 3 introduces the ion qubit and state transitions, explains Mølmer-Sørensen interactions from a physical perspective, and discusses the prospects of the scalability of ion trap quantum computers. Chapter 4 gives an overview of existing modulated pulses for multi-qubit Mølmer-Sørensen gates, including frequency modulation, and demonstrate with experimental proof how we can achieve high-fidelity 2-qubit gates in shorter ion chains (< 20 ions). Chapter
5 describes the possibility of maintaining a long, uniform ion crystal (> 20 ions) and how we can entangle an arbitrary pair of ions in it using modulation methods.
Chapter 6 introduces the concept of quantum circuit calibration, where instead of improving gates individually, we attempt to fine tune our circuits using measurement 7 feedback. Chapter 7 presents simulation results of calibrating controlled circuits and parity-controlled circuits, with and without linear and quadratic approximation.
8 2. Basics of Quantum Information
The rules of quantum computing are well established, yet they inherited many as- pects of “quirkiness” from quantum mechanics and can be daunting to understand.
Thankfully, much literature has been written over the past few decades to help be- ginners understand the basics, and the increasing availability of courses in quantum computation in universities and beyond has sparked much interest in the field. The most well-known textbook, written by Nielsen and Chuang [32], is an excellent start- ing point for general knowledge in quantum computation. This chapter gives a brief overview of the fundamental concepts in quantum information that will appear fre- quently throughout this dissertation.
2.1 Superposition and Entanglement
Classical computers store information in bits, which can be read as either 0 and 1, while quantum computers work with qubits, which can be anywhere “between” 0 | i and 1 , meaning they can be measured with a probabilistic distribution. Moreover, | i the two states carry a relative phase of between each other, allowing interference with other qubits. Thankfully, the state of an pure, isolated qubit can be expressed concisely as a complex, linear combination between 0 and 1 with two coefficients: | i | i
9 cos(θ/2) ψ = cos(θ/2) 0 + eiφ sin(θ/2) 1 = (2.1) | i | i | i eiφ sin(θ/2) As shown above, the state can be expressed in bra-ket notation or in vector nota- tion. The state vector ψ is said to be in a “superposition” between 0 and 1 when | i | i | i neither coefficient is zero. The probabilities of measuring states 0 and 1 are cos2(θ) | i | i and sin2(θ) respectively, for θ [0, π]. The complex phase φ [0, 2π) between the ∈ ∈ two qubits is not observed when if we measure in the Z-basis, where our only possible outcomes are 0 and 1. Angles θ and φ are similar to the latitude and longitude of a point on a spherical surface, allowing us to visualize it on a Bloch sphere, which will be discussed in the “Quantum Gates” section.
The probability of measuring any state φ from the qubit is given by the fidelity | i φ ψ 2 (more in section 2.9), where φ ψ denotes the Hermitian inner product, or | h | i | h | i “overlap”, between the two states. They are said to be orthogonal to each other if they have a fidelity of zero, and identical if they have a fidelity of one. 0 and 1 are | i | i orthogonal by definition, and form a basis for the state of a qubit.
In the more general case where there are N qubits, the state as a whole is expressed as:
ψ = c x (2.2) | i x | i x 0,1 N ∈{X} where c is complex and x x0 = δ 0 . The coefficients are constrained by the fact x h | i x,x that the global phase is ignored (e.g. we can set the first coefficient to be real), and the probabilities (magnitude of amplitude squared) must sum to 1. Thus, the state as a whole has 2 2N 2 degrees of freedom. × −
10 2.2 Quantum Gates
2.2.1 Matrix expressions
Analogous to gates in classical computers, quantum gates transform one quantum
state vector to another, and hence can be considered matrix operations. Certain
gates have classical analogs. For example, the X gate corresponds to the NOT gate,
and the CNOT is related to XOR in classical computers. Represented in matrices:
1 0 0 0 0 1 0 1 0 0 X = ,CNOT = (2.3) 1 0 0 0 0 1 0 0 1 0 Similarly, the Toffoli gate, sometimes written as CCNOT, is a bit flip with two control qubits. But apart from bit flips, qubits can undergo phase flips. The phase shift operator has the following matrix form:
1 0 Z(φ) = (2.4) 0 eiφ A phase flip, denoted simply as Z, effectively adds a relative phase of φ = π between 0 and 1 . For example, the phase flip takes + to and vice versa | i | i | i |−i
11 (Z = ). Also, we specifically define: |±i |∓i
1 0 1 0 S = ,T = (2.5) 0 eiπ/2 0 eiπ/4 for the special cases of φ = π/2 and π/4.
In order to express quantum gates more generally, we introduce the Pauli opera- tors:
0 1 0 i 1 0 − X := σx = ,Y := σy = ,Z := σz = (2.6) 1 0 i 0 0 1 − which form an SU(2) Lie algebra. The commutation relations are summarized as
[σi, σj] = 2iεijkσk. Together with the identity, they can be generally denoted as σi for i = 0, 1, 2, 3, or I,X,Y,Z. We may generalize to the n-qubit Pauli group:
σ σ ... σ (2.7) i1 ⊗ i2 ⊗ ⊗ in
A quantum gate can be expressed as a linear sum of n-qubit Pauli matrices (in- cluding identity) acting on individual qubits. For example, the CNOT gate can be written as 0 0 I + 1 1 X, implying that an X flip is only applied to the | i h | ⊗ | i h | ⊗ second qubit if the first qubit is at 1 . | i
2.2.2 Continuous rotations
As described earlier, unlike bits, qubits can undergo bit flips as well as phase flips.
However, quantum gates are in general continuous evolution operators. They can be
12 expressed as unitary matrices, which preserve the norm of a state vector. The inverse
of a unitary operation U is simply its Hermitian conjugate U † (UU † = I), which is
obviously also unitary. Hence all quantum gates are reversible processes. In general,
quantum gates do not commute with each other ([U, V ] = UV VU = 0). − 6 Pauli matrices are unitary and thus valid evolution operators. The same is true
for any S := ~s ~σ, where ~σ = (σ , σ , σ ) and ~s is a real 3-dimensional vector with · x y z θ unit length. They are also Hermitian, meaning that U (θ) := exp( i S) is a unitary S − 2 operator for any θ [0, 2π). It can be shown that ∈
θ θ θ exp( i S) = cos( )I i sin( )S (2.8) − 2 2 − 2 θ θ exp( i S)† = exp(i S) (2.9) − 2 2
US(θ) can be interpreted as “a single-qubit rotation about S by angle θ”. The second identity shows that the inverse of a qubit rotation is simply another qubit rotation but with negative angle. To represent two-qubit rotations, we may simply replace S with a tensor product of two S’s. Note that the identity operator I is sometimes omitted when multiplied with a coefficient.
The operator U(θ) = U(2π + θ) up to an overall negative sign, which can be omitted as a global phase. However, if this is a controlled rotation (e.g. 0 0 I + | i h | ⊗ θ 1 1 exp( i S)), then this will lead to a relative π phase between the control and | i h |⊗ − 2 target qubit, which cannot be neglected.
For convenience we will use a more concise notation to represent specific qubit
13 rotations. For example,
θ Y (θ) := exp( i σ0) (2.10) 0 − 2 y θ XX (θ) := exp( i σ0σ1) (2.11) 0,1 − 2 x x
represent a rotation about the Y axis acting on the 0-th qubit, and a rotation about
XX axis acting on qubits 0 and 1 respectively.
2.3 Circuit Diagrams
Circuits are defined as sequences of single-qubit and multi-qubit gates. In matrix
notation, Uc = ΠiUi. Each quantum gate Ui can be written as a unitary matrix, as
shown previously, which implies that Uc is also unitary. Alternatively, they can be
expressed graphically using circuit diagrams. For example, the CNOT circuit can be
decomposed as follows:
CNOT = Y ( π/2)X ( π/2)X ( π/2)XX (π/2)Y (π/2) (2.12) 0,1 0 − 0 − 1 − 0,1 0 Y (π/2) X( π/2) Y ( π/2) (2.13) • = XX(π/2) − − X( π/2) −
It is important to note that the order of gates in circuit diagrams goes from left to right, whereas unitary matrices in mathematical equations are read from right to left (for left multiplication). The order of gates in a circuit cannot be changed unless individual gates commute with each other.
Most useful quantum gates can be synthesized by a limited gate set. For example,
14 the Hadamard gate H is a rotation about ~s = 1 (1, 1, 0) by angle θ = π, but can √2 be generated by X and Y rotations: H = Y ( π/2).X(π). CNOT gates can be − decomposed into one XX rotation and several single-qubit rotations, as shown in
equation (2.12). We discuss the universal gate set in the next section.
2.4 Universal Gate Set
Multiple quantum gates are needed to achieve universal quantum computation. A
set of quantum gates is called universal if any unitary operator can be approximated
with arbitrary precision by a sequence of these gates. For a quantum computer made
of qubits, the following are a few examples of universal gate sets:
H,S,T,CNOT (2.14) { }
H, S, CCNOT (2.15) { }
exp( iθ σ ), exp( iθ σ ), exp( iθ σ σ ) θ [0, π), φ [0, 2π) (2.16) { − 0 z − 1 φ1 − 2 φ2 φ3 | i ∈ j ∈ }
Consider the universal gate set in (2.14). The subset C = H, S, CNOT is known
as a Clifford group, which normalizes the group of Pauli operators P.
C = U UU † = I,U †PU = P (2.17) { | }
Due to this property, the operations of C can be simulated efficiently with a classical computer and hence C cannot be universal [33]. However, the union of C with any other single-qubit gate, say T = exp( i(π/8)σ ), as shown in (2.14), is − z
15 universal [34]. The gate set in (2.15) is also universal, as proved in [35].
The gate set in (2.16) is in most cases the gates that are actually implemented in
ion traps. When exposed to a driving field, a qubit undergoes a continuous rotation
about a flexible axis (U(θ, φ) = exp( iθσ ) for any θ or φ), as described in subsection − φ 2.2.2. Apparently, the rotation angle and axis can be adjusted by changing the intensity and phase of the applied field. Since the gate sets in (2.14) and (2.15) can be synthesized by (2.16), any qubit system equipped with controllable single and two-qubit interactions are potential candidates for becoming quantum computers. In ion traps, Z rotations are effected by the natural evolution of a qubit, σφ rotations are done by Rabi oscillations driven by a coherent light field (see section 2.6), and
2-qubit rotations are realized indirectly using Mølmer-Sørensen interactions, which will be covered in section 3.6.
2.5 Unitary Evolution
The time evolution of a state vector is described using the Schr¨odingerPicture:
ψ(t) = U(t, t ) ψ(t ) (2.18) | i 0 | 0 i
where U is the unitary time evolution and U(t0, t0) = I.
The Schr¨odingerequation is given by
∂ i~ ψ(t) = Hˆ ψ(t) (2.19) ∂t | i | i
where Hˆ is the Hamiltonian operator, which corresponds to the total energy of the
16 system.
Solving for U gives
i U = exp Htˆ (2.20) − ~ if H is time-independent. Otherwise,
i t U = exp Hˆ (t0)dt0 (2.21) T − ~ t Z 0 where is the time-ordering operator. T
Now consider a Hamiltonian with two parts: Hˆ = Hˆ0 + Hˆ1. It can be readily shown that the Schr¨odingerequation in (2.19) can be transformed to:
∂ ˆ i~ ψI (t) = HI ψI (t) (2.22) ∂t | i | i by putting
ˆ ˆ iH0t/~ iH0t/~ HˆI = e Hˆ1e− (2.23)
ˆ ψ (t) = eiH0t/~ ψ(t) (2.24) | I i | i
This is known as the interaction picture for the quantum state evolution.
2.6 Single-qubit Rotations
A typical use case for the interaction picture is when H0 is time-independent and quickly rotating. Suppose we have a qubit with splitting ω0, where the Hamiltonian
17 ~ω0 is simply Hˆ = σ . Discarding the global phase, we can solve for the unitary and 0 2 z get ψ(t) = c 0 + c eiω0t 1 . In the Bloch sphere picture, the state vector rotates | i 0 | i i | i about the z-axis at frequency ω0. However, oftentimes ω0 is much faster in time scale
as quantum gate operations, and the phase eiω0t becomes inconvenient to keep track of. When the qubit is exposed to a driving field of Rabi frequency Ω and frequency
µ near resonance with ω0, the total Hamiltonian is
ˆ ~ω0 H = σz + ~Ωσx cos(µt) (2.25) 2
Transform the interaction Hamiltonian in equation (2.23) with H0 = ~ω0σz/2 and
H1 = ~Ωσx cos(µt), we get
ω0t ω0t ˆ i 2 σz i 2 σz HI = e ~Ωσx cos(µt)e− (2.26)
iω0t iω0t = ~Ω(e− σ+ + e σ ) cos(µt) (2.27) −
~Ω i∆t i∆t (e σ+ + e− σ ) (2.28) → 2 −
where detuning ∆ = µ ω0, σ+ := 1 0 = X iY , and σ := 0 1 = X + iY . − | i h | − − | i h |
(ω0+µ)t In (2.28), the terms proportional to e± are neglected using the rotating wave
approximation (RWA), since quickly rotating terms have little impact on the unitary
dynamics.
The Hamiltonian reduces to ~Ωσx/2 when ∆ = 0. The result is Rabi oscillations
between 0 and 1 at frequency Ω, where state population is transferred between | i | i the two states periodically. If this driving field is applied to the qubit for a duration
18 of τ, the effective unitary is U = exp( iθσ ), where the rotation angle θ = Ωτ. By − x
changing the applied phase, σx can be generalized to σφ = σx cos φ + σy sin φ. This
represents a right-hand rotation by θ about any axis on the x-y plane in the Bloch
sphere.
2.7 Mixed States
There are certain occasions where the quantum system cannot be expressed as a pure
state wave function. Imagine we initially have a superposed qubit with phase φ, written as ψ = a 0 + a eiφ 1 , where a and a are real. The qubit splitting can | i 0 | i 1 | i 0 1 become perturbed by a small unknown magnetic field, which effectively introduces an uncertainty in φ. Since φ is no longer precise, we cannot describe the state as a pure state anymore. Therefore, ignorance or uncertainty in a quantum system causes the system to become impure, or “mixed”.
There is another very important situation where states are no longer pure: when the system is entangled with the environment. For example, qubits become entangled with each other after the application of multi-qubit gates. As a result, the set of qubits cannot be described by tensor products of pure qubit states. In the case of a 2-qubit system,
ψ = c 00 + c 01 + c 10 + c 11 (2.29) | i 00 | i 01 | i 10 | i 11 | i = (b 0 + b 1 ) (d 0 + d 1 ) (2.30) 6 0 | i 1 | i ⊗ 0 | i 1 | i
for any bi, di in general.
19 We instead describe our state as a density matrix, written as
ρ = p ψ ψ (2.31) i | ii h i| i X where ψ are a set of orthonormal basis states, and p 0 are the probabilities of | ii i ≥ detecting ψ upon projection onto the basis. The expression ψ ψ is the outer | ii | ii h i| product of ψ with itself. By definition, density matrices are positive semi-definite | ii Hermitian matrices with unit trace, and equation (2.31) is known as its spectral decomposition.
How do we describe the quantum state of a subsystem of a larger system? Suppose we have a pure state ψ which consists of two subsystems. We have to perform a | i partial trace over subsystem 2 in order to obtain the density matrix for subsystem 1:
ρ = Tr (ρ ) = Tr ( ψ ψ ) (2.32) 1 2 whole 2 | i h |
Likewise ρ2 can be obtained by tracing out subsystem 1. When the two subsystems
are entangled, ρ and ρ are mixed states, and ψ ψ cannot be expressed as ρ ρ 1 2 | i h | 1 ⊗ 2 due to the correlations between the two subsystems. Any observable with projection
operator P that only acts on subsystem 1 has expectation value:
ψ P I ψ = Tr (P ρ ) (2.33) h | ⊗ | i 1 1
Hence, the expression in (2.32) after partial trace over subsystem 2 is used to
represent subsystem 1 when subsystem 2 is not observed.
20 The unitary evolution for density matrices is given by
ρ(t) = U(t, t0)ρ(t0)U †(t, t0). (2.34)
which is described by the von Neumann equation:
∂ i~ ρ(t) = [H,ˆ ρ(t)] (2.35) ∂t
Every qubit can be treated as a subsystem of set of qubits and the environment they interact with. The state of a qubit can be succinctly written as
1 ρ(~r) = (I + ~r ~σ), ~r 1. (2.36) 2 · | | ≤
The qubit is pure if ~r = 1, and mixed otherwise. ρ can be visualized with a | |
vector ~r on the Bloch sphere. Its coordinates ~r = (rx, ry, rz) represent the expectation values of the Pauli operators ~σ = ( σ , σ , σ ), which are simply the averaged h i h xi h yi h zi probabilities of measuring , or + ρ + ρ . |±ii h |i | ii − h−|i |−ii
2.8 Measurements
Measurements lie at the heart of axioms of quantum mechanics, according to most definitions. When an observer probes a quantum state ψ , it collapses to another | i state φ known to the observer, with a probability given by the Born rule: φ ψ 2. | i | h | i | If the Hilbert space is finite, we can define a set of basis states φ , such that {| ii} φ φ = I. We can further impose the orthonormality condition: φ φ = δ , i | ii h i| h i| ji ij aP very commonly used assumption that reflects the experimental reality where state 21 |0
y x
|1
Figure 2.1: An illustration of the Bloch sphere using qutip, showing two pure states (brown and green) and one mixed state (blue). The vectors are exactly ~r in equation 2.36.
detection is often mutually exclusive.
The probability of measuring P from a state ψ is ψ P ψ , and Tr(ρP ) for i | i h | i| i i a mixed state ρ. As described in section 2.7, any density matrix has a spectral decomposition of orthogonal states. If our measurement projections Pi coincide with the spectral decomposition of a density matrix ρ, then the probability of measuring
Pi is exactly the eigenvalues of ρ.
Given a state ψ and projection operators P , the post-measurement state | i { m} after measuring outcome i is given by
Pi ψ ψi = | i (2.37) | i ψ P ψ h | i| i p For density matrices,
PiρPi ρi = (2.38) Tr(ρPi)
For qubit measurements, the measurementp outcomes are the two orthogonal states,
0 and 1 , referred to as the computational basis. To change the measurement basis, | i | i 22 we simply apply a single-qubit rotation before the measurement. Once measured, the state collapses to either basis state and the remaining information stored in the previously superposed state is lost, including the relative phase between 0 and 1 . | i | i However, if multiple copies of such a state is available, we can measure both ampli- tudes and phases of the state up to statistical uncertainty using multiple measurement bases.
The generalization of projections is positive-operator valued measures (POVMs), which describe the impact of measurements of a larger system to a subsystem.
POVMs do not satisfy the orthonormality condition in general. An example can be found in section 2.2.6 in [32].
2.9 Fidelity and Trace Distance
It is natural to ask the closeness or distance between two states in the same Hilbert space. One popular metric is the fidelity:
F ( ψ , φ ) = ψ φ 2 (2.39) | i | i | h | i | F (ρ, σ) = Tr √ρσ√ρ (2.40) q
It follows that ψ = φ if and only if F ( ψ , φ ) = 1. If F ( ψ , φ ) = 0, then | i | i | i | i | i | i they can be distinguished using a measurement basis including ψ ψ and φ φ . | i h | | i h | The same goes for density matrices.
In order to characterize gates instead of specific states, the gate fidelity is defined as the worst-case scenario for the fidelity of a positive trace-preserving quantum process
23 compared to the intended circuit operation U [32]: E
F (U, ) := min F (U ψ , ( ψ ψ )) (2.41) E ψ | i E | i h | | i
Another popular metric is the trace distance:
1 1 D(ρ, σ) = Tr( ρ σ ) = λ (2.42) 2 | − | 2 | i| i X where λ are the eigenvalues of ρ σ. For a single qubit, the trace distance is exactly i − 1 ~r ~r where ~r are the vectors in the Bloch sphere (c.f. equation (2.36)). In 2 | 1 − 2| i relation to fidelity, it can be shown that
D(ρ, σ) 1 F (ρ, σ) (2.43) ≤ − p
Equality holds when ρ and σ are pure, in which case the quantity 1 F goes as − D2. Hence, trace distance is considered a conservative estimate for the likeness of two states, whereas fidelity is considered optimistic.
Unitary evolution preserves both fidelity and trace distance. But for quantum processes in general (trace-preserving maps for density matrices), trace distance con- tracts, whereas fidelity converges to 1.
For an intended quantum gate U, the gate error can be characterized as
F (UρinU †, ρout) or D(UρinU †, ρout) (2.44)
Since a quantum circuit consists of a series of gates i Ui, it is not clear whether F or D is better for evaluating individual gates. In the worstQ case scenario where errors 24 accumulate coherently, the trace distance would be the appropriate metric. However, if the errors are incoherent in nature, or we expect the gates scramble the direction of the errors, F would be much more suitable for evaluating circuit performance.
25 3. Ion Qubits
This chapter introduces trapped ions as our qubits. First, we give an overview of the state structure of our qubit, the selection rules for state transitions, and consider the factors that contribute to the qubit’s longevity. We describe the procedures for state preparation, measurement, and relevant transitions of ion qubits using laser interaction. Then we consider the ion lattice’s collective motion and how they can be used for Mølmer-Sørensen interaction, which is crucial for multi-qubit entanglement.
Finally, we discuss the prospects of a scalable architecture using trapped ions, and conveying quantum information from one module to another. Much of this chapter will take the ground hyperfine states of 171Yb+ as an example as they are the main qubit used in our labs.
3.1 State Structure of a Hydrogen-like Ion
Macroscopically, ions behave like point charges that only have electrostatic interac- tion with other ions and external electric fields, which allows us to trap them in a crystalline structure. But in fact, ions have complex internal state structures, with each state having a unique set of quantum numbers. Characterizing these states is crucial to our choice for the qubit.
26 A typical choice for atom species for the ion qubit is one with 2 valence electrons, such that it has hydrogen-like behaviour when ionized. For example, an Yb atom has electronic configuration [Xe] 4f 146s2, and when ionized, has only 1 valence electron in the 6s subshell. The other electrons combined with the positive nucleus behave like a
+2 charge which the valence electron orbits in the outermost shell. In most settings, the state of the ion can be solely described by the orbital which the valence electron occupies. Of course, electrons from inner shells can be excited to outer shells, forming alternative excited states, but they play a very limited role in this work.
To dissect the state structure of a qubit, the first point of consideration is electronic spin-orbit coupling, which is the contribution of L~ S~ to the Hamiltonian. The · relevant quantum numbers are the azimuthal quantum number L, spin quantum number S, and total electronic angular momentum quantum number J, where J =
L S , L S +1, ..., L+S. We use the convention 2S+1L to denote quantum states | − | | − | J due to different L, S, J combinations (2S + 1 is also called multiplicity).
The next step is to consider spin-spin coupling between the electron and nucleus, which is the contribution of I~ J~ to the Hamiltonian. The nuclear spin I interacts · with the electronic spin J, such that the total angular momentum quantum number
F = I J , I J +1, ..., I +J. Under the weak field limit, the states with non-zero F | − | | − | will split into 2F + 1 levels with magnetic quantun numbers m = F, F + 1, ..., F F − − with an energy splitting proportional to the size of the magnetic field.
With so many possibilities for these quantum numbers, which of these states should we choose as our qubit? In section 3.3, we discuss the main factors that determine a qubit’s longevity. In section 3.4, we cover the most important state transitions between the qubit states and other excited states.
27 3.2 Selection Rules for State Transitions
The most direct way to realize a state transition is through electric dipole inter- action. Transitioning one state to another entails the absorption or emission of a photon, whether it is spontaneous or stimulated. Since the photon carries an angu- lar momentum of magnitude ~, there must also be corresponding changes in angular momentum states. One can deduce a set of selection rules for the initial and final quantum numbers by computing the transition dipole moment integral
µ = ψ∗µψˆ (3.1) h i f i Z whereµ ˆ is the dipole moment operator (for instance, qxˆ) and ψi and ψf are the wavefunctions for the initial and final states respectively.
In braket notation, for a transition from initial state i to final state f , the | i | i transition dipole matrix element d f xˆ i is non-zero only if the change in or- if ∼ h | | i bital quantum number L is 1. The following encapsulates the selection rules for ± transitions within a generic atom or ion
∆L = 1 (3.2) ± ∆J = 0, 1, (0 0 forbidden) (3.3) ± → ∆F = 0, 1, (0 0 forbidden) (3.4) ± → ∆m = 0, 1 (3.5) F ±
No quantum number can change by more than 1. The forbidden 0 0 transition → 28 can be understood classically by noting that only a non-zero angular momentum vector can have a net change of ~ by changing its direction without changing its magnitude.
3.3 The Longevity of a Qubit
In theory, we can assign any two energy levels in an atom as our qubit and transition between the two levels using EM waves with the appropriate wavelengths. However, we must choose states with a long coherence time, which is critical to the functionality of a quantum computer. Here we cover the two main aspects that determine the qubit’s longevity: the decay and dephasing lifetimes, often denoted as T1 and T2.
3.3.1 Qubit Decay Lifetime (T1)
We consider an isolated two-state system, where we use 0 to denote the ground state | i and 1 the excited state. If a qubit or a set of qubits with state ρ undergoes an error | i process where there is a probability p such that the qubit becomes flipped from 0 | i to 1 and vice versa, we can describe the evolution of state with the so-called bit-flip | i channel:
(ρ) = (1 p)ρ + pXρX (3.6) E −
But for a qubit suspended in vacuum, there is practically zero likelihood that a qubit will be excited from 0 to 1 by absorbing a stray photon, and it should only | i | i decay from 1 to 0 by emitting a photon with probability p. In this scenario, the | i | i
29 error channel is
(ρ) = E ρE† + E ρE†, (3.7) E 0 0 1 1 where E = 0 0 + 1 p 1 1 , (3.8) 0 | i h | − | i h | p E1 = √p 0 1 = √pσ (3.9) | i h | −
t/τ Naturally, p is a function of time and can be written as 1 2− where τ is the − half-life of the excited qubit. Thus, a typical definition of T1 is τ, or a quantity proportional to τ.
Since the primary function of a qubit is to preserve its quantum state, we want
τ to be as long as possible. This is determined by a set of “selection rules”, which dictate whether a state can be directly transitioned to another using electric dipole interaction, as well as other EM interactions. If we choose two states whose electric dipole interaction is “forbidden”, then the excited state will decay to the ground state at a very slow rate. We have introduced electric dipole selection rules in section 3.2.
But if their transition is “forbidden”, how do we perform quantum gates that
flip between 0 and 1 ? We can do so by using higher-order EM interactions such | i | i as magnetic dipole interactions, or by using a third intermediate state as a bridge between the two qubit states. The latter procedure is detailed in 3.4.
30 3.3.2 Qubit Dephasing Lifetime (T2)
The second factor to be considered is the stability of the qubit’s phase. Consider the phase damping channel where the phase can be flipped with a probability of p.
ρ0 = (1 p)ρ + pZρZ (3.10) −
After a long enough period of time, the phase information will become completely scrambled by the environment. Indeed, ion qubits typically have shorter T2 than T1, meaning that phase is in most physical settings more prone to decoherence than the amplitude.
The primary cause for the loss of phase coherence is the variability of the qubit energy splitting with the surrounding magnetic field. A single qubit has Hamiltonian
H = ~ω0σz/2, where ω0 is the transition frequency between 0 and 1 . When | i | i exposed to a fluctuating magnetic field, ω0 also fluctuates, causing an uncertainty in the phase of the qubit.
In section 3.1, we introduced the quantum numbers L, S, J, I, and F . In the weak-
field limit, F and m are good quantum numbers, and states with m = 1 have F F ± energies that are first-order sensitive to the magnetic field. Thus, qubits consisted of these states are more susceptible to dephasing than those consisted of mF = 0.
Qubit dephasing is currently the limiting factor for the coherence time of hyperfine qubits. Experiments report qubit lifetimes from 20 minutes to 10 hours, depending on the ion temperature and surrounding magnetic field [36]. Whichever states we choose for our qubit, it must be isolated from stray fields as much as possible. The more stable the ambient magnetic field is, the longer the lifetime of our ion trap quantum
31 computer.
3.4 Relevant State Transitions
Due to the complex state structure of the atom, it is crucial to manipulate the qubit states reliably, while respecting the selection rules for atomic state transitions. In this section, with the state structure and selection rules in mind, we explain the transitions needed for the state preparation, measurement, and transitions of ion qubits.
3.4.1 State Preparation and Measurement
Manipulating modern ion traps often requires comprehensive utilization of coherent laser fields (see Figure 3.1). An atom beam can be ionized by a two-photon transition: we apply a light tone that excites a valence electron from the ground state to an excited state, and another tone from the excited state to the continuum. The ions can then be Doppler cooled using counter-propagating lasers red-detuned from a state transition. They can be further cooled to the quantum regime using red sideband transitions, where the ions are excited but with a phonon removed, then allowed to naturally decay to the ground state.
Once a trapped and cooled ion lattice, we first need to prepare the ions in the ground state 0 . The most common procedure involved is called optical pumping, | i where the excited state 1 are pumped to excited states which quickly decay to the | i ground state according to selection rules. It is possible that an excited state decays to other metastable states (states with higher energies than the ground states but have relatively long lifetimes), so we must ensure that all of them are driven to short-lived
32 excited states.
3 3 D[3/2]1/2 (a) D[3/2]1/2 (b) F=0 F=0 2.2095(11) GHz F=1 2.2095(11) GHz F=1 es
es stat
stat F=1
1/2 F=1 2 S 2 2 P P & 1/2 1/2 o F=1 t F=1 2.105 GHz F=0 2.105 GHz 1/2 F=0 S F=0 2
o t
2 2 D3/2 D3/2 F=2 F=2 0.86(2) GHz 0.86(2) GHz F=1 F=1
t
pumping ec
det optica l
2 2 S |1 S |1 1/2 > 1/2 > F=1 F=1 12.643 GHz 12.643 GHz |0 |0> F=0 > F=0 Figure 3.1: We use 171Yb+ as an example to demonstrate state preparation and measurement of a qubit [37]. (a) We can optically pump the state to 0 = 2 | i F = 0, mF = 0 by exciting 1 = F = 1, mF = 0 to P1/2, and from the metastable |2D to 3D[3/2]i , which are| i “allowed”| to spontaneouslyi decay to 0 . (b) State 3/2 1/2 | i population in 1 is detected by driving it to 2P1/2,F = 0, which scatter photons with frequency equal| i to the energy difference between them. Photons scattered from the transition from 3D[3/2] to 1 are also taken into account. 1/2 | i
State measurement is accomplished by driving transitions between one of the qubit
states, say 1 , and a short-lived excited state. Photons will be scattered continuously | i in the process due to spontaneous decay, and be detected by a setup ideally with high
numerical aperture. In this scenario, 1 is called the “bright” state as it leads to | i detection of photons, whereas 0 is called the “dark” state. Contrary to optical | i pumping, we should ensure that the excited states do not decay to any state other
than 1 , and if they do, excite it to another short-lived state that decays back to the | i bright state, forming a closed transition cycle that does not involve 0 . | i
33 δ F = 1 2 F = 0 P1/2 369.5 nm
F = 1
2 S1/2 F = 0
Figure 3.2: Three-level state system for Raman transitions, reproduced from [38]. A state 0 can be transitioned to 1 and vice versa by applying two light tones with | i | i frequencies ωb and ωr where the detuning ωb ωr equals the energy difference ω0 between 0 and 1 . The transition rate is inversely− proportional to the frequency | i | i difference δ between the applied frequencies ωr or ωb and the energy of the excited state ω . To minimize unwanted transition to p , we must ensure that δ Ω , Ω p | i b r
3.4.2 Qubit Transitions
Consider a qubit consisting of 0 = F = 0, m = 0 and 1 = F = 1, m = 0 in an | i | F i | i | F i 171Yb+ ion. Since electric dipole transitions are forbidden, and they are insensitive
to magnetic field disturbances, they are a desirable choice for our qubit. The energy
difference between these two states is 12.6 GHz, a frequency within the microwave
spectrum. Due to the small splitting of the energy levels, qubit lifetimes can be as
long as a few minutes and even a few hours, given sufficient isolation from stray
magnetic fields [36].
It is possible to transition between the qubit states directly using microwaves, but
this means the whole trap will be exposed to the microwave due to its long wavelength.
In order to address individual ions arranged in a lattice, we can instead apply a pair
of tightly-focused lasers with a detuning equal to the splitting of the qubit. This
process is known as stimulated Raman transitions, illustrated in Figure 3.2.
34 This leads to effective Rabi oscillations between 0 and 1 , with an effective Rabi | i | i frequency proportional to ΩrΩb/δ. δ must be set to be much greater than Ωb and Ωr to isolate the coherent interaction between 0 and 1 and prevent direct transition | i | i to p . Otherwise, the state may decay from p randomly to either 0 or 1 or to any | i | i | i | i lower-energy state, resulting in gate errors. A full derivation for stimulated Raman transitions can be found in [39].
3.5 1-D Ion Lattice
This section discusses the 1-D ion lattice in static equilibrium and its motional spectrum due to Coulomb interactions. We then describe the general formalism of
Mølmer-Sørensen (MS) Gates and how they can be realized in a scalable architecture.
3.5.1 1-D Ion Distribution
Earnshaw’s theorem states that it is not possible to trap point charges in a static equilibrium with only electrostatic fields, but this can be circumvented by creating an average potential minimum with quickly alternating fields. In a typical 1-D ion trap, tight radial confinement is realized using a combination of dc and rf fields, whereas weak axial confinement is achieved by dc electrodes. To this end, the Paul trap is widely adopted for trapping short ion chains with a quadratic axial potential
(see Figure 3.3). More recently, the surface trap has grown in popularity owing to its ability to generate anharmonic potentials, which allow us to trap tens of ions with relatively uniform density (see section 3.7.1).
If the analytical form of the axial confinement is available and given by U(z), the
35 Figure 3.3: A sketch of a typical Paul trap from [10]. Four blade-shaped electrodes on the side provide radial confinement, whereas the rods at the end provide axial confinement. equilibrium ion positions can be calculated by minimizing the total electric potential energy Vtotal using standard optimization techniques:
k q2 V = U(z ) + 0 (3.11) total i z z i i For a uniform 1-D ion lattice, U(z) is given by (see section 5.1.1) L2 V (z) ρ ln (3.12) uniform ∼ 0 L2 z2 − ! where ρ0 is the linear charge density of ions and L is the total length of the ion chain. Uniform ion separation is considered a “desideratum” as it makes individual ad- dressing more convenient from a hardware perspective. It also makes motional modes more uniform, a fact that we exploit to perform long-distance 2-qubit gates for longer ion chains in chapter 5. In typical experiments, however, ion chains tend to be less dense towards the ends due to strong Coulombic repulsion (see Figure 3.4). Figure 3.4: A fluorescent image of a 53-ion chain from [40]. Typical 1-D chains have varying ion density, which is highly dependent on the axial trapping potential. 36 We note that the ion chain will “buckle” into a 2-D zigzag formation unless the radial confinement is stronger than electrostatic repulsion: kq2 mω2 > (3.13) x n3d3 n=1,3,5,... X 3.5.2 Transverse motional spectrum To deduce the motional spectrum of transverse modes, we perform the harmonic ap- proximation around the equilibrium positions zi of the ions. The Coulombic repulsion between ions i and j is approximated as 2 2 2 1 k0q ∆xi,j UCoulomb = k0q 1 2 . (3.14) 2 2 ≈ ∆zi,j − 2∆zi,j ∆zi,j + ∆xi,j ! q where ion-to-ion separation ∆z = z z . The Hamiltonian can then be evaluated i,j i − j as follows: p2 1 1 H = i + mω2x2 + k q2 (3.15) 2m 2 x i 0 r r i i 37 where the harmonic interaction potential V in the x-direction is given by 2 1 2 k0q mωx 3 , if i = j 2 − s=i 2∆zi,s Vi,j = 6 (3.19) 2 P k0q 3 , if i = j 2∆zi,j 6 1 is diagonalized to D = mω2δ . This yields the motional frequencies ω and k,l 2 k kl k motional modes ~y. Concrete numerical results for shorter (N < 20) and longer (N > 20) ion chains are presented in the next two chapters. 3.6 Mølmer-Sørensen (MS) Gates The collective motion of the ions can be utilized as a quantum bus for multi-qubit entanglement, as suggested by Mølmer and Sørensen [12, 13]. When evaluating the Hamiltonian and unitary evolution of the ions during a Mølmer-Sørensen (MS) gate, both the internal and motional state need to be quantized, which can be described by a Jaynes-Cummings-type Hamiltonian [41]. This section explains how multi-qubit entanglement is realized through simultaneous application of blue and red-sideband interactions. 3.6.1 Hamiltonian for sideband transitions Suppose ions indexed i is addressed with a frequency tuned to the qubit’s energy splitting ωz plus the motional frequency ωk. This triggers the blue sideband transition where the internal and motional states are simultaneously excited or de-excited. The 38 Figure 3.5: State transition diagram of a Mølmer and Sørensen gate, reproduced from [42]. 2-qubit rotation is achieved by the combination of the simultaneous excitation of the red and blue sidebands, regardless of the starting number of phonons n. Hamiltonian is given by N 1 i iθk(t) i iθk(t) HBSB = ηikΩi(t)(σ+ak† e + σ ake− ). (3.20) 2 − i X Xk=1 t where θ (t) = (µ(t0) ω )dt0 is the cumulative phase between the driving frequency k 0 − k R µ(t) and the k-th motional frequency ωk, and ηik is the Lamb-Dicke parameter, which is the ratio between the sideband and carrier Rabi transition strength, dependent on both the addressed ion i and motional mode number k. Similarly, a frequency tuned to the energy splitting minus the motional frequency ωk triggers the red sideband transition, whose Hamiltonian is N 1 i iθk(t) i iθk(t) HRSB = ηikΩi(t)(σ+ake− + σ ak† e ). (3.21) 2 − i X Xk=1 These Hamiltonians are also called anti-Jaynes-Cummings and Jaynes Cummings Hamiltonians, as they have the same spin-bosonic interaction as isolated atoms in an optical cavity. They behave similarly as driven two-level systems, which have Rabi oscillations between 0, n and 1, n 1 . However, a completely different dynamic | i | ± i 39 occurs when two sideband Hamiltonians are combined (see Figure 3.5), assuming that their driving intensities Ω(t) and detuning from the carrier δk are the same: N 1 i iθk(t) iθk(t) H = η Ω (t)σ (a e− + a† e ) (3.22) MS 2 ik i x k k i X Xk=1 The total Hamiltonian now consists of two parts: one that only acts on spin, and another that only acts on motion. This allows the unitary to be solved analytically, as shown in the next subsection. 3.6.2 Effective Unitary Evolution The resulting unitary is most easily solved using Magnus expansion [43], where the exponent has a series expansion with integrals of nested commutators. ∞ U = exp( Ωn(t)) (3.23) n=1 Xt Ω1 = i H(t0)dt0 (3.24) − 0 Z 0 1 t t Ω2 = [H(t0),H(t00)]dt0dt00 (3.25) −2 0 0 Z Z0 00 i t t t Ω = ([H(t0), [H(t00),H(t000)]] (3.26) 3 6 Z0 Z0 Z0 + [H(t000), [H(t00),H(t0)]])dt0dt00dt000 etc. (3.27) i j By noting that σx commute with σx and [a, a†] = 1, we realize that the expansion terminates after the second term, so only the terms in (3.24) and (3.25) remain. The 40 unitary is therefore exactly expressed as: U = D(α ˆk(t))E(t) (3.28) Yk The first-order terms ΠkDk result in a state-dependent displacement operator, acting on motional modes k: D(α ˆ (t)) = exp(ˆα (t)a† αˆ† (t)a ) (3.29) k k k − k k t 0 1 i iθk(t ) αˆ (t) = η σ Ω (t0)e dt0 (3.30) k 2 ik φ i i 0 X Z This state-dependent displacement, if not suppressed, leads to qubit decoherence. Details on how unwanted motional displacement can be minimized and the conse- quences of not doing so are deferred to the next chapter. To simplify analysis, we can keep track of the displacement of a specific eigenstate ofα ˆk: t 0 1 iθk(t ) α (t) = η Ω (t0)e dt0 (3.31) k 2 | ik| i i 0 X Z which can serve as the upper bound for the magnitude of displacement for motional mode k. Due to the complex nature of αk, we can make 2-D plots of the so-called “phase space trajectories” (PSTs) as a function of time for all relevant modes, with the axes being the real and imaginary parts of αk. Let’s further assume that only two ions i and j are addressed with the blue and red sideband Hamiltonians. The second term of the expansion in (3.28) is the desired 41 two-qubit rotation: t t0 i i j Eˆ(t) = exp σ σ η η Ω (t)Ω (t) sin(θ (t0) θ (t00))dt0dt00 −2 φ φ ik jk i j k − k i Specifically, to maximally entangle two qubits (namely, to transform a product state to a Bell state) in a given gate time τ, we require that Eˆ(τ) = exp( i(π/4)σi σj ). − φ φ The next chapter provides full examples of how pulses can be optimized such that displacement is minimized and the appropriate 2-qubit rotation is achieved. 3.7 Scaling Ion Traps 1-D ion traps cannot be scaled infinitely, since 2-qubit interaction becomes progres- sively difficult as the total number of ions N increases. This can be seen from the fact that the Lamb-Dicke parameters ηik scale down as 1/√N, as well as other hardware limitations. In this section, we give a quick survey of the prospects of scalable ar- chitectures of trapped ion quantum computers, and conveying quantum information from one module to another through either physically shuttling ions between trapping zones or transferring quantum information through photonic interconnects. 3.7.1 Surface Trap Architecture The traditional 4-rod Paul trap design has proved difficult to scale. Due to the macroscopic size of the electrodes, there is limited freedom to adjust the electric potential along the axis of the ion trap, and no mechanism is available to shuttle the ions into or out of the trap. 42 (a) (b) trap location inner DC oxide RF top silicon inner DC gold silicon substrate 1 mm Figure 3.6: An illustration of a surface trap, reproduced from [36]. The electrodes allow better control over ion density along the length of the chain, and it is possible to address individual ions with lasers from above/under the trap. The surface trap offers an alternative architecture which greatly enhances scala- bility and control of trap potential. Tens of d.c. electrodes with width comparable to ion chain lengths are arranged in the axial direction to enable the shuttling, separa- tion, and recombination of long ion chains. In certain designs, a central slot between the electrodes allows tightly focused laser beams to pass through. When the elec- trodes are switched on, a line of ions can be trapped directly above slot, and can be addressed by lasers parallel or perpendicular to the trap surface (see Figure 3.6). 3.7.2 Ion Shuttling Using the surface trap design, ions can be shuttled from one trapping zone to the next, with their internal state intact. Certain zones can be designated for single and two-qubit gate processing, while others can be reserved for quantum memory. Since ions are subject to heating in the process, we can physically relocate them from one set of ion qubits to another, and cool down the ions that do not carry any quantum information. This is compatible with the implementation of quantum error correction using surface code, where the parity of local 2-D stablizers can be checked with 2- 43 qubit gates [20, 44]. Thus, multi-zone ion trapping equipped with shuttling is a very promising architecture for scalable quantum computing. 3.7.3 2-D Ion Lattice It is possible to trap ions on a 2-D surface and address the ions perpendicularly. Under tight confinement in the x-axis and loose confinement in the y and z-axis, ions can be arranged in a hexagonal lattice, with transverse motional modes used as a bridge for multi-qubit interactions. Suppose we have a region of qubits arranged in a uniform hexagonal pattern with separation d. The transverse mode frequencies would be given by [45] 2 e 1 cos(π(k1α + k2β)/N) ωk1,k2 = 1 3 2 − 2 2 3/2 (3.33) v − 4πε0d mω (α + β + αβ) u x α,β u X t for k , k = 0, 1, ..., N 1. The sum over integers α and β represent the forces acting 1 2 − on a specific ion by neighbouring ions. We can then optimize 2-qubit pulses for these motional frequencies in a similar way as we do for 1-D ion traps. 3.7.4 Photonic Interconnects It has been proposed that quantum information be transferred from one trap to the next with photons [21]. A large number of ion qubits can be divided into small modules known as elementary logical units, each of which capable of implementing the universal gate set on its qubits. The quantum information of a qubit can be coupled with the polarization or excitation of a photon field inside a single-mode 44 fiber, which then propagates to a different trapping region to couple with a set of the qubits there. This process must be accompanied with excellent optics and is otherwise subject to photon losses as well as limitations of detection efficiency. 45 4. Optimized MS gates for shorter ion chains (< 20 ions) This chapter is dedicated to the implementation of high-fidelity Mølmer-Sørensen gate for small numbers of ions. Originally, MS gates consist of a simple, constant-frequency pulse, which limits the quality of gate due to residual motion. Since then, a variety of pulse modulation methods have been invented to resolve complex sideband spectra to maximize fidelity. In the following sections, we will explain the theory behind optimizing 2-qubit gates, introduce the most well-known pulse modulation techniques, and discuss the implementation of high-fidelity 2-qubit gates using frequency as well as amplitude modulation. My main contribution is in section 4.3 ans subsection 4.2.2. 4.1 Two-qubit Gate Optimization With the ability to modulate the electromagnetic force driving the quantum gates, we can parametrize its amplitude and frequency to minimize the motional displacement of ions at the end of the gate. The amplitude can be scaled linearly such that the two qubits are maximally entangled. This section describes how one can quantify the errors due to residual entanglement with the motion, minimize it using suitable optimization algorithms, and estimate the power needed to maximally entangle target 46 qubits. 4.1.1 Error due to Residual Entanglement As discussed in chapter 2, sideband transitions lead to a state-dependent displacement operator D(ˆα (t)) = exp(ˆα (t)a† αˆ† (t)a) acting on both the internal and motional k k − k space of the ions. As a function of time, the displacement is written as: t 0 1 i j iθk(t ) αˆ (t) = (η Ω (t0)σ + η Ω (t0)σ )e dt0 (4.1) k 2 i,k i φ j,k j φ Z0 t Note the cumulative phase θ(t) = 0 δk(t0)dt0, which is simply δkt for constant frequency. Thus, in the case where bothR the amplitude Ω and detuning is stepwise constant,α ˆk can be analytically solved. Since the goal of optimization is to minimizeα ˆk for all eigenstates of the spin operators, we only need to keep track of one of the eigenstates, say ++ , and | iφ minimize its displacement. We are interested in estimating the error contribution due to small displacements αk. The Mølmer-Sørensen gate can be described by ΠkDk(ˆαk)E(β). Since the entan- glement operator E commutes with the displacement operators Dk, we can single out the effect of D if E ψ is the target final state. | i To estimate the error contribution due to the displacement operator D, we can make a first-order expansion with respect to a small αk for motional mode k 1 2 D (ˆα ) 1 + (ˆα a† αˆ† a ) + (ˆα a† αˆ† a ) (4.2) k k ≈ k k − k k 2 k k − k k With advanced experimental techniques such as sideband cooling, ion trap motion 47 can be cooled to near-zero phonon number. Let’s suppose that there is only one motional mode and the initial motion is at the vacuum state for simplicity. Plugging in the approximation in equation (4.2), the state impacted by this operator is given by ρ0 = Tr (D ψ, 0 ψ, 0 D†) (4.3) mot | i h | = Tr ( ψ, 0 ψ, 0 +α ˆ ψ, 1 ψ, 1 αˆ† + ...) (4.4) mot | i h | k | i h | k = ψ ψ +α ˆ ψ ψ αˆ† + ... (4.5) | i h | k | i h | k where the motional trace is done in the number state basis: ∞ Tr = n n (4.6) mot h | · | i n=0 X To find the fidelity loss, we calculate the probability of the state ψ transitioning | i i to a new state ψ0 = σ ψ by a single bit flip due to operator D. In practical | ii φ | i scenarios, our initial state ψ lies in the computational basis (e.g. 00 ) such that | i | i ψ0 ψ , which is also the worst-case scenario for gate fidelity. We also need | ii ⊥ | i to take into account displacements on multiple modes (k = 1, ..., N). Recall from i j equation (4.1), the spin-dependent displacementα ˆk is given by Ii,kσφ + Ij,kσφ where 1 τ iθ (t) I = η Ω (t)e k dt. By noting that D(ˆα ) 1 + (ˆα a† αˆ† a ), the i,k 2 i,k 0 i k k ≈ k k k − k k R Q P 48 probability for transitioning to ψ0 is | ii P ( ψ0 ) = ψ0 ρ ψ0 (4.7) | ii h i| | ii 2 = ψ0 αˆ ψ (4.8) | h i| k| i | Xk i j 2 = ψ0 I σ + I σ ψ (4.9) | h i| i,k φ j,k φ| i | Xk 2 = Ii,k (4.10) k X Hence, when the Mølmer-Sørensen gate is performed on ions i and j, the gate fidelity can be estimated as 2 2 ε := 1 F I + I . (4.11) − ≈ i,k j,k k k X X Roughly speaking, the fidelity decreases quadratically with sum of motional dis- placements. Note that if Ωi = Ωj = Ω, Ii,k and Ij,k are only off by a ratio between Lamb-Dicke constants ηi,k and ηj,k. Obviously, ε is zero if the integrals Ii,k and Ij,k are zero. Since it is easier to think of the displacement as a scalar quantity, we simply keep track of t 0 1 iθk(t ) α := ( η + η ) Ω(t0)e dt0, (4.12) k 2 | i,k| | j,k| Z0 which is the upper bound for the amount of displacement for the eigenstates ofα ˆk, and we consider α = 0 for all k ε = 0 (4.13) k ⇒ to be the sufficient condition for suppressing motional displacement-induced errors for any state ψ . | i 49 4.1.2 Robustness against Frequency Drifts The standard optimization criterion for a modulated pulse is given by (4.13), meaning that the motional displacements for all modes should be zero, such that the displace- ment operator acts trivially on our state. However, this condition is prone to motional frequency drifts, which are often observed in realistic ion traps. It is possible to find a pulse robust against frequency drifts denoted as δ1 up to the first order, where δ1 is assumed to be constant for the duration of the pulse. Suppose a pulse with frequency δk(t) already satisfies αk(τ) = 0. A simple by parts integration gives τ iθk(t)+iδ1t αk(t) Ω(t)e dt ∼ 0 Z τ iθk(t) (1 + iδ1t)Ω(t)e dt ≈ 0 Z τ iθk(t) = iδ1 tΩ(t)e dt (4.14) 0 Z t τ t 0 τ 0 iθk(t ) iθk(t ) = iδ1 t Ω(t)e dt0 Ω(t0)e dt0dt 0 0 − 0 0 h Z i Z Z = iδ 0 τα 1 − k,avg Since Ω(t) represents the speed of the trajectory and eiθk(t) the direction, α | | k,avg can be interpreted as the time-averaged position of the trajectory, and we should make sure that it is located at the origin in order to null out αk(t) up to the first order in δ1. We conclude that the condition for a pulse robust against frequency drifts is τ t 0 iθk(t ) min α , α Ω(t0)e dt0dt = 0, k = 1, .., N (4.15) | k,avg| k,avg ∼ Z0 Z0 If we further assume that the pulse is symmetric around t = τ/2 (i.e. Ω(τ t) = − 50 Ω(t) and δ (τ t) = δ (t)), then the minimization of α automatically guarantees k − k k,avg that αk(τ). Thus, we only have to focus on the condition in (4.15). A proof of this fact is provided in Appendix A. For optimization, we simply define our cost function to be τ t 2 0 iθk(t ) Cost = Ω(t0)e dt0dt (4.16) k Z0 Z0 X τ t 2 τ t 2 = Ω(t0) cos θk(t0)dt 0dt + Ω(t0) sin θk(t0)dt0dt (4.17) 0 0 0 0 Xk Z Z Z Z whose minimization guarantees the condition in (4.15). 4.1.3 Power Estimate for Every Pair of Qubits The entanglement operator is given by Eˆ(β ) = exp( iβ σi σj ) (4.18) ij − ij φ φ i j t t0 iσφσφ = exp ηi,kηj,kΩi(t)Ωj(t) sin(θk(t0) θk(t00))dt0dt00 − 2 0 0 − Xk Z Z (4.19) For maximal entanglement, we set βij = π/4. The single-qubit Rabi frequency Ωi and Ωj is scaled with a constant such that βij = π/4 is satisfied. Due to the dependence of Lamb-Dicke constants on the ions involved, it is expected that Ωi and Ωj can be calibrated freely for every ion pair before the two-qubit gates are performed. This gives us an estimate of the power required to entangle any two qubits in an ion chain given an optimized pulse with fixed gate time. 51 It is helpful to note that the quantity βij is proportional to the area enclosed t iθ(t0) by the phase space trajectory. Recall that the displacement α(t) Ω(t0)e dt0, ∼ 0 where Ω(t)eiθ(t) is the velocity of the trajectory at time t. For a durationR of time dt, the area swept by the trajectory with respect to the origin (yellow strip in Figure ??) is proportional to 1 α(t) Ω(t)dt sin(φ) (4.20) 2 1 iθ(t) = Ω( t)dt Im e α∗(t) (4.21) 2 t 1 iθ(t) iθ(t0) = Ω(t)dt Im Ω(t0)e − dt0 (4.22) 2 Z0 ! 1 t = dt Ω(t)Ω(t0) sin θ(t) θ(t0) dt0 (4.23) 2 − Z0 where φ is the angle between α(t) and eiθ(t). Now we see that this is proportional − to the integrand in equation (4.19). Thus the double integral for βij represents the total entanglement, proportional to the power squared. α(t) φ β(t) Figure 4.1: The area enclosed by the phase space trajectory α(t) is proportional the double integral β(t), which represents the total entanglement received by the 2 qubits. When optimizing for pulses, we ought to choose pulses with larger areas such that less overall power can be used to achieve maximal entanglement between the qubits. 52 4.2 Modulated Pulses As explained in section 3, the coupling between internal and motional states during Mølmer-Sørensen (MS) gates introduces considerable complexity to the dynamics of ion qubits. As a result, a large variety of pulse modulation techniques have been proposed to maximize 2-qubit gate performance. This section will discuss four major modulation methods, including their principles and experimental performance. 4.2.1 Amplitude Modulation (AM) Stepwise AM pulse was originally proposed by S. L. Zhu et al. [23] for minimizing motion at the conclusion of MS gates in large 1-D ion traps. For a total gate time of τ, the amplitude Ω(t) is divided into constant segments with duration ∆τ, each of which is allowed to vary such that the displacement nulling condition (4.13) is satisfied. Assuming that the frequency is constant, the displacement for the ++ | i eigenstate is τ iδkt αk,++(τ) = (ηik + ηjk) Ω(t)e dt (4.24) Z0 M 1 iδkm∆τ iδk(m 1)∆τ = (η + η ) Ω (e e − ) (4.25) ik jk iδ m − k m=1 XM 1 2 iδk m ∆τ δk∆τ = (η + η ) Ω e − 2 sin (4.26) ik jk δ m 2 k m=1 X where M is the number of steps for the AM. To minimize displacement, we demand αk = 0 for all k, regardless of the eigenstate. In vectorized notation, the condition 53 can be summarized as AΩ~ = 0 where δ ∆τ Ω cos δ m 1 ∆τ sin k for k = 1, ..., N m k − 2 2 (4.27) δ ∆τ Ω sin δ m 1 ∆τ sin k for k = N + 1, ..., 2N m k − 2 2 which nullifies complex displacements for all N motional modes. Discrete AM pulses have been widely used for demonstration of fundamental al- gorithms [17] and prospects of fault-tolerant computing [46]. 4.2.2 Frequency Modulation (FM) Figure 4.2: FM 2-qubit gates are implemented in a 4-ion trap in [38]. (a) Discrete FM pulse sequence, consisting of 20 constant-length segments. (b) Phase-space tra- jectories for the motional modes, with matching colors of the motional frequencies in (a). Note that a change in detuning represents a change in trajectory curvature. It is possible to modulate the frequency in such a way the motional displace- ment is suppressed for multiple modes. Assuming the amplitude Ω is constant, the 54 displacement is expressed as τ iθk(t) αk,++(τ) = (ηik + ηjk)Ω e dt (4.28) Z0 As shown in the above equation, to introduce time-dependence to the detuning iδkt iθk(t) t δk(t), the complex phase e is replaced with e where θk(t) = 0 δk(t0)dt0. Since the curvature of the phase space trajectory is inversely proportionalR to the detuning δk(t) from the motional mode frequency ωk, we can modulate the detuning in such a way the trajectories for relevant motional modes return to zero. In [24], we establish FM pulses as a competent method for suppressing residual motional displacement, supported by experimental results. A continuous FM pulse with duration 90µs is used for 2-qubit gates in a 5-ion trap experiment, reaching 98.3% fidelity. Crucially, we demonstrate that the pulse can be optimized such that it is robust against small drifts in the motional frequency ωk by imposing additional restrictions for δk (see section 4.1.2). Details of implementation are discussed in subsection 4.3.1. Discrete FM pulses have also been used to achieve high-fidelity MS gates. In [38], a 20-segment FM pulse (see Figure 4.2) is used to suppress residual entanglement with motion, achieving 99.5% 2-qubit gate fidelity for a 2-ion trap and 99.3% 2-qubit gate fidelity for a 4-ion trap. Under this setting, it is estimated that motional dephasing 3 3 contributes only 1.2 10− to the error budget, compared to 2.7 10− due to optical × × phase instability. 55 4.2.3 Phase Modulation The use of phase modulation (PM) to suppress residual motion is proposed in [47]. By introducing discrete changes in the phase, phase space trajectories undergo abrupt turns at pre-determined time intervals. Again, a closed path represents successful decoupling between the qubits and the motional mode. In the original proposal, N 2 the optimal solution has an analytical expression and has 2 − pulses, where N is the number of motional modes to be suppressed. Fortunately, a numerical approach results in a pulse with linear complexity, and has shown to achieve 99.4% fidelity in 2-ion trap experiment [48]. It is worth noting that phase modulation is a type of Quadrature Amplitude Modulation (QAM), meaning that a pulse with constant amplitude but varying phase can be decomposed to a sum of an in-phase component and a quadrature (out of phase by π/2) component with modulated amplitudes: A sin(ωt + φ(t)) = B(t) sin(ωt) + C(t) sin(ωt + π/2) (4.29) This is readily verified by decomposing the compound angle on the left hand side, and is true for both continuous and discrete modulation. 56 4.2.4 Multi-tone MS gates We can perform the same optimization using Multi-tone Mølmer-Sørensen (MTMS) gates, as proposed by Haddadfarshi et al. [49]. Consider the Hamiltonian ˆ HMTMS = ~δS(a†f ∗(t) + af(t)) (4.30) N f(t) = cj exp(ijδt) (4.31) j=1 X where the phase f(t) has a Fourier series expression with frequencies jδ for j = 1, ..., N and with amplitudes cj. To remove unwanted motional displacements, we minimize c the integral of H and find that this reduces to N j = 0. MTMS j=1 j To protect the pulse from detuning errors, we evaluateP the displacement due to an unwanted frequency shift ∆: N cjδ 2π∆ α(τ) = e δ 1 (4.32) i(jδ + ∆) − j=1 X By expanding α(τ) in ∆ assuming ∆ δ, we can obtain the conditions for a pulse robust against detuning errors. Using this method, a 3-tone MTMS pulse has been applied to a 2-ion trap in [50], which 99.4% fidelity. 4.3 Short Ion Chains in Experiment When dealing with shorter ion chains (< 20 ions), optimal AM and FM 2-qubit gates can be found with standard convex optimization procedures. In the following subsections, we provide concrete examples of how high-fidelity 2-qubit gates can be 57 (a) (b) 1 Robust 3.2 Non-robust 0.9 Robust Non-robust Hz) ( M 3.1 o pu lation P c y 0.8 y e n 1 2 ari t - P req u F 3 3 e n 0.7 4 E v 5 2.9 0.6 0 20 40 60 80 6 4 2 0 2 4 6 − − − Time ( µs) Detuning O set (kHz) (c) 0.3 0.1 0.1 (d) 0.3 0.1 0.1 Mode 1 Mode 2 Mode 3 Mode 1 Mode 2 Mode 3 Robust Mode 4 Mode 5 Mode 4 Mode 5 Non-robust 0.1 0.06 0.1 0.06 Figure 4.3: (a) Frequency-modulated pulses optimized for minimum displacement only (blue-dashed) and robustness against frequency drifts (purple). The dots rep- resent the degrees of freedom used and are allowed to move along the y-axis during optimization. The pulse is symmetric and is formed by connected sinusoidal curves. (b) The even-parity populations at the end of the 2-qubit gates versus a detuning offset deliberately introduced to test the robustness of both pulses. It is obvious that the robust version significantly outperforms the non-robust version over the range of detuning offsets tested. (c) and (d) Phase space trajectories for the 5 motional modes with no sideband offset and with -1 kHz sideband offset. The non-robust pulse is sen- sitive to detuning errors, whereas the robust pulse has first-order tolerance against detuning errors. optimized and evaluated for traps with fewer than 20 ions. 4.3.1 5-ion trap experiment In [24], 5 171Yb+ ions are trapped in a 1-D lattice in a Paul trap with an aver- age separation of 5µm and a radial frequency of about 3.045MHz. The qubit is 58 2 formed from the hyperfine splitting between the ground states of the S1/2 manifold: 0 = F = 0, m = 0 and 1 = F = 1, m = 0 . Qubit rotations are realized using | i | F i | i | F i Raman transitions with tightly focused, counter-propagating laser beams. Motional phase closure is ensured by applying frequency modulation to Mølmer-Sørensen gates, which has a typical duration of 100 to 200 µs. The frequency pattern is constructed as follows. We set a series of turning points (blue and purple dots in Figure 4.3(a)) at equal time intervals and connect each pair of neighboring points with cosine curves, leading to an oscillatory shape. Only the first half is parametrized since the pulse is symmetric in time. Our algorithm adjusts the vertical positions of the extreme points until the cost function reaches a local minimum. The cost function we seek to minimize is the sum of squares of time-averaged displacements of the phase space trajectories of the 10 nearest-detuned modes. For the robust pulse, we do not just minimize residual displacement for all modes, but also the time-averaged value of the displacement, such that the first-order dependence of the displacement on unwanted frequency offsets δ1 is zero, as described in section 4.1.2. As a result, the error = N α (τ) 2 will scale as δ4, making the E k=1 | k | 1 gate robust against small frequency drifts ofP the trap. We compare the performance of pulses that are robust against frequency drifts (satisifying condition (4.15)) and those that are not (only satisfying (4.13)) in Figure 4.3(b). A significantly higher fidelity is observed over a wide frequency range when the robustness condition is satisfied (see Figure 4.3). For the robust pulse, the fidelity stays above 95% for a range of 8 kHz offset, whereas the non-robust pulse only has a tolerance window of 1 kHz for 90% fidelity. The maximum fidelity achieved is 98.3%, excluding SPAM errors. By simulation, 59 4 the loss of fidelity due to unwanted motional coupling should be lower than 10− for the robust, optimized pulse. The remaining errors can be attributed to other factors such as laser intensity fluctuations and Stark shifts. 4.3.2 17-ion trap experiment Another experiment has been done with an ion chain of 17 171Yb+ where we demon- strate high-fidelity entanglement over longer distances [45]. In this setup, only ions with indices 5, 7, 9, 11, and 13 can be addressed, since the middle ions have a more uniform distribution, and the minimum separation between the laser beams is about 5 µm. The ions from index 5 to 13 have an average separation of 2.5 µm. A multi- channel photo-multiplier tube (PMT) is used to measure the state of individual ions. Both AM and FM are applied to optimize the Mølmer-Sørensen gate. The driving frequency is tuned closest to 4-th motional sideband, and the gate is optimized such that the residual motion for the nearest 10 sidebands are minimized. In Figure 4.4, only the highest 6 mode trajectories are plotted as most of the 2-qubit coupling occurs in the 4-th motional mode. A pulse with detuning robustness is used to entangle ions 5 and 13, as well as ions 7 and 9, achieving fidelities 97(1)% and 95(1)% respectively. The error is comparable to reported state-preparation-and-measurement (SPAM) errors. 60 Figure 4.4: (a) FM and AM patterns for the 2-qubit gate for the 17-ion experiment. The smooth dependence on time allows for quick decay of residual motion as the detuning between the sideband and the driving frequency increases. (b) The phase space trajectories for the top 6 motional modes. The displacement αk is the greatest for the 4-th motional mode since it has the smallest detuning, which therefore has the highest contribution for entanglement. 61 5. Optimized MS gates for longer ion chains (> 20 ions) In the previous chapter, we discussed the principles of Mølmer-Sørensen gates, their optimization for minimum motional displacement, as well as recent experimental re- sults for small numbers of ions. But as ion traps grow in size, the gate time increases linearly with time, and it may become prohibitively difficult to carry out long-distance entanglement. We also note that it may not be practical (or necessary) to develop pulses optimized for all motional modes, as they become less relevant as detuning increases. This chapter describes how 2-qubit gates can be optimized for ion chains as long as 50 ions as well as the limitations for scaling long-range 2-qubit interactions in an ion trap quantum computer, as detailed in [51]. 5.1 Trapping Long, Uniform Ion Chains It is desirable to trap ions in a uniform 1-D lattice. Not only does it make individ- ual addressing more convenient, it also leads to a more predictable motional mode structure that facilitates arbitrary 2-qubit entanglement within the lattice. We begin our discussion for the prerequisites for trapping a long ion chain with almost uniform density and the resulting the motional modes and frequencies. 62 5.1.1 Ideal Trap Shape We find the conditions for an electric field that leads to a 1-D formation. The radial confinement (x and y-axis) is achieved by a combination of d.c. and r.f. fields [10]. They are assumed to be uniform, harmonic, and much stronger than axial confine- ment (z-axis). In order to trap large numbers of ions (N 10), highly anharmonic terms are required in the axial potential. We also need to ensure that all ions are sufficiently separated from their neighbors such that they can be addressed individu- ally using high-precision laser beams. Previous work with anharmonic traps has used a parameterized form for the potential, then sought to minimize the variance in ion separation [52, 53]. Here we opt for a different approach: we first assume a continuous ion distribution, then integrate to find the potential required to generate it. We investigate the idealistic case of having N = 50 evenly spaced ions across the whole chain, with an average separation ∆z 3µm. In the continuous limit the ≈ chain can be modelled as a uniform charge density ρ0 = q/∆z. We find an analytical expression for the effective electric field acting on each ion due to Coulombic forces: z ε L − kρ0dz0 Erep(z) = 2 L − z+ε !(z0 z) Z− Z − (5.1) 1 1 = kρ 0 L z − L + z − ! where L is the half length of the ion chain, ε is half the average ion-to-ion separation, and ρ0 = q/∆z is the linear charge density. We integrate again to find the effective 63 trap potential required to maintain the shape of the lattice L2 V (z) = rkρ ln (5.2) trap 0 L2 z2 − ! where r 1 is an additional scaling factor for adjustment (blue curve in Figure ≈ 5.2(a)). We have to consider the constraint of the Laplace equation, due to the increasing second derivative of Vtrap(z) towards the edges. We have to consider the constraint of the Laplace equation, which will weaken the radial confinement of the ions due to the increasing second derivative of Vtrap(z) towards the edges. A quick 2 check shows that qV 00 (z) mω for the vast majority of ions and result in only a trap x 10% drop in the radial frequency for the outermost ion. Thus, the edge effects are ∼ ignored in this paper, but could ideally be compensated in one radial direction with additional DC fields. To avoid infinite potential walls, we let this expression be valid only for z < sL | | where s is slightly smaller than 1 such that there is a finite upper bound for the electric field, Emax. The minimum for Emax is the electric field experienced by the ions at the edges of the chain, given approximately by N kq π2 kq E = 260V/m (5.3) edge (n∆z)2 ≈ 6 ∆z2 ≈ n=1 X for an average separation ∆z = 3 µm. This should be well within the maximum field ∼ 10 V/100 µm = 105 V/m that can be generated by typical microfabricated ion traps. Tight laser beamwidths of about 1.5 µm have been realized in past experiments [17], allowing us to address any ion with very small crosstalk errors with precision beam steering. 64 We calculate the equilibrium positions z of N ions due to such potential (blue { i} dots in Figure 5.2(a)). We initialize the ion crystal at even ion separations slightly smaller than the expected ∆z. We repeatedly evaluate the total force acting on each ion, and move them fractionally in that direction in order to minimize electric potential energy. The results show that the equilibrium ion separation averages to about 2.9µm with less than 5% variation from minimum to maximum. (a) (b) (c) Figure 5.1: Depiction of how we may choose particular motional modes as the means for entanglement. (a) Ideal trap potential from equation (5.2) for r = 0.95 and the corresponding distribution of ions found by gradient descent of electric potential energy. A minimum trap depth of 1.52 meV is required to trap all 50 ions. (b) The middle section of the transverse motional frequencies, showing ω22 to ω31 (solid red lines) and the approximate driving frequency µ = ω 3.7 kHz (blue dashed line), 0 26 − for radial trapping frequency ωx = 3.07 MHz and average ion separation = 2.9µm. (c) The 25th, 26th and 27th transverse motional modes (normalized). The uniformity of the 26th mode makes it useful as a channel for arbitrary 2-qubit entanglement. 65 5.1.2 Motional Spectrum and Resonant Modes For any ion distribution in general, we can find the collective transverse vibrations of the ion chain by computing the xixj dependence of the Hamiltonian (Figure 5.1(b) and (c)). This is done by expanding electric potential and inter-ion repul- sion around their equilibrium positions z , based on the Lamb-Dicke approximation { i} ( ~ n + 1 λ < ∆z). The radial potential V (x) = 1 mω x2 is assumed to 2mω 2 trap 2 x q q be constant along the z-axis, whereas the axial potential Vtrap(z) is given by equation 5.2. Note that xizj couplings vanish up to the first order, allowing us to calculate the longitudinal and transverse motional modes separately. As shown in section 3.5.2, the xixj terms can then be diagonalized, giving us the transverse motional modes ˆ N Xk = i=1 ukixˆi and resonant frequencies ωk [54], for k from 1 to N. Using the new basisP Xˆ , the total potential energy is now equivalent to a collection of N { k} non-degenerate harmonic oscillators with no phonon hopping, and the motion can be characterized by coherent displacements of these oscillators in their respective rotating frames [42]. We let the common mode frequency ωx be 3.07 MHz and calculate the higher order modes using the z previously obtained where ∆z 2.9µm. As expected, { i} ≈ the resonant frequencies are unevenly spaced, with the lowest frequency being 2.45 MHz. The motional modes appear to be standing waves with increasing wavenumbers, which can be explained by the periodicity of a uniform charge density. Figure 5.1(b) shows the middle part of the spectrum and Figures 5.1(c) shows the the 25th to 27th transverse modes explicitly. 66 5.2 Optimization Methodology 5.2.1 Selective Mode Coupling The essence of the methodology used in [51] is to exploit the uniformity of the middle mode shown in Figure 5.2. Since the ion separation is roughly constant, every ion in the chain sees a similar potential due to the neighbouring ions. As a result the motional modes have a periodic structure within the lattice. This method is “scalable” in the sense that all ions are equally involved in the motion, and by coupling the ions to this motion we expect to entangle any pair of ions with a laser intensity that does not scale with overall distance, which is indeed the case in our simulation. The fact that “one optimized pulse works for all” makes it a very practical technique for arbitrary 2-qubit entanglement, since the only parameter that needs to be adjusted for different ion pairs is the overall laser power (see Figure 5.4). The driving frequency is chosen to be near resonant with ω26 since all ions are excited to a similar degree in the 26-th mode, making it an ideal quantum channel for multi-qubit entanglement. We note that the sideband splitting near this mode is at 18 kHz, less dense than most other parts of the spectrum, which allows us to resolve the motional modes with shorter gate times. Another important advantage of using such a high-order mode is the significantly lower heating rates due to trap noises, since typical trap electrode sizes are much larger than the average separation between neighboring ions. 67 5.2.2 Optimized Pulses The first step is to set the shape of Ω(t). For the sake of comparison, we assume two intensity profiles, pulse A and B (Figures 5.2(a) and (b)), and perform the same optimization with frequency to minimize state-dependent motion. Pulse A has a time dependence of (sin(πt/τ))1.5, whereas Pulse B consists of 3 steps connected by cosine functions. We note that Pulse A is “smoother” than B in the sense that it has a lower maximum rate of change in intensity, which should make it more resilient against frequency offsets. In both cases, the smoothness suppresses excitation of far-detuned modes, such that they only contribute to a small fraction of . E (a) (b) (c) (d) (e) (f) Figure 5.2: (a) and (b) are the relative Rabi strength Ω(t) applied over a gate time of 500 us, denoted as Pulse A and B respectively. (c) and (d) are the corresponding frequency patterns (µ(t) µ0 where µ0 = ω26 3.7 kHz) that minimize residual ion motion. The blue dots− are allowed to move− vertically during optimization. Note that the pulses are set to be symmetric in time. (e) and (f) are the resultant phase space trajectories from the 24-th to 28-th motional modes. Note that the detuning determines the curvature of the phase space trajectories, whereas the Rabi strength determines trajectory speed as well as curvature. 68 For both pulses, we need to initialize the frequency pattern and then seek a final pattern that minimizes ion motion. In this example, we choose a reference frequency µ ω 3.7 kHz, such that the phase space trajectory of the 26-th mode closes 0 ≈ 26 − for the desired gate time. We then modify the driving frequency µ(t) around µ0 periodically to minimize the displacement for the 10 nearest-detuned modes (α22 through α31). Finally, we calculate the total error due to motion in all 50 modes and 4 confirm that < 10− , and test the gate’s robustness against frequency offsets. E The frequency profile is built the same way as we do in section 4.3, with a symmet- ric, evenly segmented pattern. We apply FM to satisfy the robustness condition in 4.1.2. Figures 5.2(c) and (d) show the optimized frequency patterns for Pulse A and B, both consisting of 8 oscillations and thus 15 turning points. But since the pulse is set to be symmetric, there are only 8 degrees of freedom (8 blue dots) for frequency modulation. The oscillatory amplitude is about 2 kHz or less, much smaller than the average sideband splitting of 18 kHz. The initial gate error (when µ(t) = µ0) due to 4 3 residual motion is 9.4 10− for Pulse A and 6.3 10− for Pulse B, and the final × × 6 5 error is 2.4 10− and 3.0 10− after optimization with µ(t). The corresponding × × trajectories of the 5 nearest modes are plotted in Figures 5.2(e) and (f). The fact that they are centered around the origin confirms that the pulse suppresses final mo- tion and is robust against slow frequency drifts (small change in overall curvature of trajectories). A log-log graph is plotted in Figure 5.3 of the additional gate error versus E − E0 constant frequency offsets δ ( = when δ = 0). It is readily seen that Pulse A 1 E E0 1 (blue circles) has a greater tolerance against frequency offsets than B (green triangles), especially for lower error thresholds. The slopes found by linear regression is 5.95 ± 69 0.29 for Pulse A and 4.01 0.16 for Pulse B, which are equal to or larger than the ± predicted 4. We conclude that Pulse A is more robust against frequency errors than Pulse B, as expected. ) 0 10 2 ( t e s f f o 3 10 o t e u d r o 4 r 10 r e e t a g Pulse A a 5 r 10 t Pulse B x E 100 130 180 240 320 420 550 700 900 Frequency offset 1 (Hz) Figure 5.3: Rabi frequency Ωmax required to entangle any pair of qubits (Ωmax corresponds to relative Rabi strength = 100 in Figure 5.2(a)). It ranges from 2π 109 kHz to 541 kHz for Pulse A, and from 2π 97 kHz to 263 kHz for Pulse B. × × The colored graph in Figure 5.4 shows the power or maximum Rabi strength required to entangle any pair of ions in the chain (Ωmax such that βij = π/4). All pairs can be entangled with power 2π 541 kHz for Pulse A, and 2π 263 ≤ × ≤ × kHz for Pulse B, implying that the latter pulse has higher coupling efficiency. This is partly due to a higher overall detuning of the optimized frequency pattern from ω26 for Pulse A than for Pulse B (see Figures 5.2(c) and (d)), which leads to a greater enclosed area by the phase space trajectory for Pulse B (Figures 5.2(e) and (f)). The required power does not increase as a function of distance. Instead, it alternates between low and high, and averages to roughly 2π 150 kHz for long distances. × The required power is also higher towards the edges of the ion chain. To summarize, 70 for the same gate time and degrees of freedom, Pulse A consumes less power, but Pulse B has higher tolerance against frequency errors. This flexibility with the initial conditions allows trade-off between robustness and power efficiency. No. No. No. No. No. No. Figure 5.4: Rabi frequency Ωmax required to entangle any pair of qubits (Ωmax corresponds to relative Rabi strength = 100 in Figure 5.2(a)). It ranges from 2π 109 kHz to 541 kHz for Pulse A, and from 2π 97 kHz to 263 kHz for Pulse B. × × 5.3 Further Scalability In the limit of large numbers of ions, we may need to think about choosing between multiple strategies for generating entanglement over short and long distances. This will depend on how the driving frequency is tuned relative to the motional structure of the ion traps. This section discusses two different regimes of Mølmer-Sørensen interactions. 71 (a) (b) Figure 5.5: The two sideband driving regimes: far-detuned (a) and near-detuned (b), defined by comparing the detuning between the applied frequency (blue dashed) and the range of the sidebands (red). 5.3.1 Neighbouring Entangling Regime Traditionally, Mølmer-Sørensen gates are achieved with the driving frequency being far-detuned from all motional modes (see Figure 5.5(a)). This way, motional dis- placement αk for any motional mode k is suppressed, and the two-qubit rotation βij accumulates linearly with the duration τ. The additional benefit is that the time τ needed to entangle a pair of ions is independent of overall lattice size N, providing a predictable way of entangling neighbouring ions. The frequency is set to be far-detuned from all modes compared to the range of the spectrum itself. In this limit, the motional displacement αk is small due to inverse relationship with detuning (max(α ) 1/δ ), and can be further suppressed using k ∼ k pulse modulation, as described in the previous sections. Suppose ions are aligned along the z-axis and have a uniform separation of d. Then the resonant modes can be assumed to have a periodic form: z eiωt cos(kz + ∼ φ). By making the harmonic approximation for Coulomb interactions, the resonant 72 frequencies are predicted to be 2k q2 P 1 ω2 ω2 = 0 (1 cos(pk ∆z)) (5.4) x − n m∆z3 p3 − n p=1 X where kn is the wavenumber for the n-th resonant mode, and P neighbouring ions are taken into account. Since the ion chain is finite, we can further put k = n/((N n − 1)∆z) if ion separations are uniform. As long as the detuning is much greater than ωN 1 ω0 , the system is considered to be in the far-detuned regime. | − − | It can be shown that ions can be coupled with constant power and gate time with respect to N, meaning that the physical overhead for the 2-qubit gate is indepen- dent of the total number of ions in the chain. For neighbouring ions, the sign of entanglement add constructively, whereas for non-neighbouring ions the total entan- glement diminishes quickly due to cancellations. In the limit where N , effective → ∞ Ising coupling between ion pair decays as 1/d3 where d is the distance between the ions. Thus, it is undesirable to entangle non-neighbouring ions with this method, but rather transfer the state from one qubit to the next like a game of “telephone” using neighbouring SWAP gates before performing entangling the qubits. This also enables the possibility of parallel two-qubit gates, where multiple near- neighbouring ions are entangled in a long ion chain (see Figure). “Crosstalk” errors between different qubit pairs decay as r3 where r is the ratio between the distance within the ion pairs and the distance between the pairs. This result is generalized to the 2-D case with a hexagonal lattice [45]. 73 5.3.2 Arbitrary Entangling Regime As suggested in [51], any pair of ions within a long ion chain can be entangled effec- tively as long as one sideband transition is coupled more strongly than the others. This is made possible if the inverse of the gate time is much smaller than the average splitting between adjacent motional frequencies (1/τ δ δ ), as illustrated in k − k+1 Figure 5.5(b). In the k-th motional mode of an ideal, uniform ion chain, the ions participate in resonance with spatial dependence cos(κ x + φ) where κ is the ∼ k k wavenumber, no matter which part of the ion lattice they are in, such that the decay of 2-qubit coupling with respect to distance d is suppressed. The scaling of parameters with N can be summarized as follows: 1 1 1 η N − 2 , δ N − , τ N, Ω N − 2 (5.5) ik ∼ k ∼ ∼ ∼ The decay of Lamb-Dicke parameter is due to the increase in inertia of the ion chain as a whole. The spectral density increases linearly with N if ion separation ∆z is constant (see equation 5.4); as a result the gate time has to scale linearly with N in order to close out the phase space trajectory of ωk. The motional displacements in other modes can be closed out using pulse optimization techniques described in section 5.2. From equation (3.32), since the 2-qubit rotation has the following dependence: β η η Ω2τ/δ , (5.6) ij ∼ ik jk k 1 we conclude that Ω scales down as N − 2 for maximal entanglement. From the reasoning above, we expect long-distance 2-qubit gates to be N times 74 slower than neighbouring 2-qubit gates. One may try to compare the time taken be- tween the two strategies for 2-qubit entanglement, depending on the distance between the qubit pair. Suppose only neighbouring 2-qubit gates are allowed in a 1-D qubit structure. A series of SWAP gates are required to enable entanglement between 2 qubits with distance d: the state of a qubit is transferred towards the other qubit, entangled with it, and then transferred back. Since every SWAP gate involves 3 2-qubit gates, a maximum of 6(d 1) + 1 2-qubit gates are needed. Thus, long-distance 2-qubit gates − (t N) are faster than the neighbouring 2-qubit gates when the distance is greater ∼ than 1/6 of the ion chain. This gives an improvement in overall speed and circuit errors. However, as discussed in section 3.7, as quantum computers scale up in size, it is likely that it will be divided into smaller modules that are connected by shuttling and photonic interconnects (N is at most 102 to 103 in the near term). It is also possible to minimize the number of 2-qubit gates involved depending on the connectivity of the quantum computer [55]. 5.4 Advanced Pulse Modulation Techniques This section will briefly discuss two advanced optimization methods proposed recently for 2-qubit gates in ion traps [56, 57]. They are closely related to the pulse modulation techniques described in the previous sections and have great potential in enhancing the scalability of multi-qubit entanglement in ion traps. 75 5.4.1 Parallel 2-qubit Gates within a Short Ion Chain It is highly desirable to develop optimization techniques for parallel 2-qubit gates within a short ion chain, where pair-to-pair entanglement is not negligible. In [56], multiple 2-qubit gates are executed in parallel a 13-ion trap with 11 addressable qubits with an average fidelity of 90 to 93%. This is made possible by using a stepwise AM pulse optimized for both displacement and entanglement for more than 2 addressed ions. Let ion index be labelled as m, segment index as k, and mode as p. The motional displacement in gate time τ is summarized as follows: τ α(m)(τ) = η(m) dtΩ(m)(t) cos(µt)eiωpt = 0 (5.7) p − p Z0 Nseg Nseg Ω(m) dt cos(µt)eiωpt = Mˆ Ω~ (m) = 0 (5.8) ⇒ k Xk=1 Xk=1 where we used a vectorized Ω~ to express the stepwise time dependence of Ω. Thus ˆ [i] the null space of M, denoted as Ωnull, satisfies the above condition. The entanglement factor is expressed as: N seg k kτ/Nseg kτ/Nseg (m,n) (m) (n) χ = Ωk Ωl dt2 dt1 (5.9) (k 1)τ/Nseg (l 1)τ/Nseg k=1 l=1 Z − Z − X XN 4η(m)η(n) sin[ω (t t )] cos(µt ) cos(µt ) (5.10) − p p p 2 − 1 1 2 " p=1 # X = (Ω~ (m))T Dˆ (m,n)Ω~ (n) (5.11) where Dˆ (m,n) contains the segment-wise double integrals. We then set χ(m,n) = θ(m,n) to specify the pairs of ions that need to be entangled. 76 To ensure diagonalizability, we symmetrize Dˆ and solve for the reduced matrix Vˆ (m,n): Sˆ(m,n) := [Dˆ (m,n) + (Dˆ (m,n))T ]/2 (5.12) ˆ (m,n) ~ [i] T ˆ(m,n)~ [j] Vi,j = (Ωnull) S Ωnull (5.13) We choose the eigenvector with the maximum absolute eigenvalue in order to (m) minimize the norm of Ω~ needed to entangle the qubits. The Nseg required to satisfy the above conditions is 2N + N 1 where N is the number of participating EASE − EASE qubits. 5.4.2 Power-optimal, stabilized 2-qubit Gates R. Bl¨umelet al. [57] provided a further generalization of [56], albeit from a different starting point. The pulse is decomposed in the Fourier basis with respect to the gate time τ: g(t) = An sin(2πnt/π) (5.14) n X For a single 2-qubit gate, the displacement and entanglement conditions are: τ NA iωpt sin(2πnt/τ)e dt = MpnAn = 0 (5.15) 0 n=1 Z X N τ t2 ηi ηj dt dt g(t )g(t ) sin[ω (t t )] = A~T SA~ = χ (5.16) p p 2 1 2 1 p 2 − 1 ij p=1 0 0 X Z Z where χij is the intended 2-qubit rotation for ions i and j, and Mˆ and Sˆ are the respective first and second-order integrals. The vectorized A~ represents the Fourier 77 amplitudes An. Thus far, the pulse g(t) assumes a general form where amplitude and frequency can both have time dependence. The pulse g(t) can then be “demodulated” to the form: g(t) = Ω(t) sin ψ(t) (5.17) t where ψ(t) = 0 µ(t0)dt0 is the cumulative phase of the applied frequency. This represents an AMR pulse if the detuning is kept constant, and an FM pulse if Ω is kept constant. The condition for robustness against frequency drifts in ωp is also generalized to higher orders by nulling the k-th derivative of displacement integrals with respect to δp, a generalization of the robustness condition (4.15) mentioned in [24]: ∂kα τ k = tkg(t)eiωptdt = 0 (5.18) ∂ωk p Z0 t where θ(t) = µ(t0) ω dt0. We can then search in the corresponding null space for 0 − k the smallest normR of g(t) in order to minimize the power used. 78 6. Quantum Circuit Calibration (QCC): Theory The natural progression from the improvement of gates is the improvement of cir- cuits. Rather than fully characterizing a set of quantum gates, we can take strategic measurements of the output of a circuit to evaluate its response to specific types of error, ideally with the help of classical simulation. The act of correcting circuit errors using a mapping between circuit errors and circuit output coordinates is referred to as Quantum Circuit Calibration (QCC) in this research. This chapter describes the relevant concepts of QCC, with and without classical circuit simulation. 6.1 Calibrating Gates VS Calibrating Circuits Characterizing a quantum gate set is a fundamental task in quantum computation. The most well-known tomographic methods include Quantum Process Tomography (QPT) [58] and Gate Set Tomography (GST) [59]. The latter requires a greater overhead in number of measurements but is more reliable when dealing with SPAM errors, and can distinguish between individual gate errors and errors on the overall gate set. Both are within the limits of modern quantum computing technology and have been compared in simulation and experiment [60, 61]. 79 Any quantum gate can be understood as a quantum channel Λ acting on a quan- tum state ρ. One way of expressing it is using Pauli transfer matrices: d2 Λ(ρ) = χj,kPjρPk (6.1) j,kX=1 N where the dimension d = 2 , Pj and Pk are Pauli operators acting on N qubits, and χjk are complex coefficients. This way, it is possible to fully characterize a limited gate set. An important weakness of tomography, however, is the lack of scalability. Tomog- raphy is not the optimal way to measure common error types such as overrotation errors. Given a specific error model, the sheer number of measurements needed is very large compared to a handful of error parameters. A second problem is the indirect correspondence between tomography and experimental control. Given the transfer matrices, it is not straightforward to deduce how we should adjust physical parameters to correct for the gate errors. To give a rough estimate of gate performance, one can make use of Randomized Benchmarking (RB), where a randomized sequence of Clifford gates are carried out and the final fidelity is measured, which yields the average error rate of the gate set as a whole [62]. Due to its scalability and relative ease of implementation, this has been widely used in experiment to quantify gate performance, for example to compare between a variety of composite pulses [63]. Unfortunately, RB only provides a metric for overall gate fidelity, not a way to correct the errors. An alternative to characterizing a gate set is to calibrate on the circuit level. In practice, we tend to carry out the same types of circuits repeatedly (e.g. a frequently 80 used subroutine), rather than a random sequence of gates from a universal gate set. Given a set of experimental controls, there is an optimal combination of controls such that the circuit output fidelity is maximized. And in most cases, we are only concerned about parts of the output of circuit, not the entire quantum state. In light of the above, we develop a strategy which we name Quantum Circuit Calibration (QCC). The motivation for QCC is three-fold: 1. We focus on experimental control - parameters that can be adjusted with phys- ical apparatus, denoted as ~ε. When ~ε = 0, the circuit is subject to errors. 6 2. The circuit to be calibrated is fixed, but the input state can be varied. It is carried out N times for each input state, which effectively gives us N copies of the final state that can be measured. 3. We only examine the impact on a limited number of observables Oˆ that we { } care about, such that they do not grow exponentially with the number of qubits. The expectation values of such observables are called “coordinates” and denoted as ~r. We may try to visualize these principles in Figure 6.1. A quantum circuit executed on any physical platform is subject to gate errors, which can be partly compensated by adjusting experimental controls (twisting the knobs on the black box) based on feedback from the output states. The experiment is repeated N times, such that the coordinates ~r are measured with statistical uncertainty and ~ε can be found approxi- mately by repeated experiment or simulation. Note the definition for controls and coordinates can be adjusted: we can expand the definition of ~ε to include more error types, as well as ~r if we require more in- 81 Figure 6.1: A circuit, naturally composed of a sequence of gates, can be imagined as a tunable black box, whose controls ~ε can be calibrated based on a set of output parameters ~r. formation from the output state. We can simulate the relationship between the two using an error model that best describes the experimental environment. This gives us a high-level picture of circuit performance that is adaptive to our physical setup, and correct errors by strategically observing the circuit output. In the next section, we discuss in detail the relationship between errors and circuit output. 6.2 A Mapping from Errors to Coordinates We understand that unitary operations are norm-preserving, one-to-one mappings from a quantum state to another. We can represent the action of our unitary operators as: U ψ = ψ (6.2) C | ii | f i Since quantum circuits consist of single and 2-qubit unitaries, they are themselves unitary operations. The following mappings are equivalent: U ψ U1 ψ U2 ... Ui ψ i+1 ... Un ψ (6.3) | 0i −→| 1i −→ −→| ii −−→ −→| ni 82 Qn U ψ i=1 i ψ (6.4) | 0i −−−−→| ni To maximize our circuit performance, we can choose to either improve every pro- cess in (6.3), or calibrate the circuit (6.4) as a whole. Established methods, such as QPT and GST discussed in the previous section, adopt the former approach. QCC, on the other hand, focuses on the latter. A common metric for circuit performance is state fidelity, which quantifies the “closeness” between the final state with and without errors: d 2 2 F = ψf () ψf (0) = bi∗ci (6.5) | h | i | i=0 X where d is the dimension of the Hilbert space. Although fidelity is a comprehensive metric, it is inefficient in terms of classical simulation and experimental measurements due to the exponential number of coefficients involved. The key idea of QCC is to describe the circuit not as a mapping between an initial and final state, but between an error vector and a set of “coordinates”, which are expectation values of a set of observables U : ~ε ~r (6.6) → The coordinates ~r have a trajectory in the newly defined parameter space (pro- jected from a much larger Hilbert space) that moves with ~ε. For example, the Bloch sphere coordinates of a qubit can be shifted as a result of gate errors. In the case where the mapping is not injective, we will have trouble determining ~ε from ~r. But 83 this can often be solved by generalizing to “abstract coordinates” by taking into ac- count multiple input states ψ . Section 6.4 goes into further detail in this regard. | initi Strictly speaking, experimental control ~εexp and experimental errors ~ε can be fur- ther separated, such that ~ε can never be directly compensated. However, in the majority of this dissertation, we focus on the special case where ~εexp is the same as ~ε, meaning that we’re not concerned about error types that are beyond our control. Thus, once we found ~ε, it is straightforward to apply ~ε0 = ~ε to improve our cir- − cuit. For example, in order to correct amplitude errors of laser fields with which we manipulate our qubits, we simply adjust the laser strength through experimental equipment. 6.3 Calibration Procedure Calibration Transfer of Q. Info Computation Figure 6.2: The calibration cycle consists of 3 stages: calibration, computation, and transfer of quantum information. The effectiveness of QCC hinges on the integration of the calibration process in- side the calibration cycle (see Figure 6.2). Since we expect the circuit performance to improve from calibration, we normally wish to carry out the desired computation im- mediately afterwards. The calibrated quantum system may only be part of a quantum computer, as quantum supremacy can be achieved with at least hundreds of qubits [64], and the scalability of quantum computer will largely depend on modularization. 84 Uf Figure 6.3: During calibration stage (left), the circuit output is “post-processed” such that circuit measurements are most sensitive to errors. On the other hand, we want the computational circuit (right) to be as robust to the same errors as possible. Hence, the quantum information stored in the qubits will have to be transferred from one module to the next through either teleportation or physical shuttling of qubits. The cycle repeats until the computational task is completed. It is worth emphasizing that the quantum circuits during calibration need not be identical with that during computation. As shown in Figure 6.3, both circuit stages may share the same main computational part, but the circuit for calibration can be processed by an additional component Uf to increase error sensitivity. Effectively, we attempt to change the measurement basis such that the measurement results ~r change most sensitively with the error parameters ~ε. This operation can be as simple as a series of single-qubit rotations. After error correction, the circuit performance during computation should also improve as a result. We now develop a procedure that corrects circuit errors using repeated measure- ments of the circuit output. The steps are as follows: 1. Simulate the mapping: ~ε ~r between an error vector and observables (coordi- → nates) based on a physical error model. 2. Take measurements of coordinates ~r using preferred technique e.g. fixed-basis measurement. 85 3. Fit the “measured” coordinates using a circuit simulator e.g. using least squares method: 2 min (rj(~ε) rj,meas) (6.7) ~ε − j X 4. Compare the estimated ~εest and actual error vector ~ε. There are occasions that prohibit us from using a circuit simulator, for example when the Hilbert space is exceedingly large (e.g. when more than 50 qubits are involved) or when the error model is uncertain. Theoretically, it is possible to carry out gradient approximation to solve for the errors purely by experiment. The steps are outlined as follows: 1. Attain the derivatives ∂~r/∂~ε by measuring the circuit’s response to changes in ~ε. 2. Measure coordinates ~r using preferred technique and invert the gradient rela- tionship L to obtain error ~ε. 3. Compensate for the errors ~ε ~ε ~ε using experimental control. → − est 4. Repeat step 2 to confirm improvement from correction. More details on taking linear approximation of circuit’s response to errors are described in section 6.5. 6.4 Measurements, Projection, and Coordinates Consider a quantum system divided in to N subsystems, such that our measure- ment operators (projections) only act on these subsystems separately, which can be 86 expressed as: P = j j I , where exc(i) = 1, ..., i 1, i + 1, ...N (6.8) i,j | i h | ⊗ exc(i) { − } where Pi,j are projection operators acting on the i-th subsystem for the j-th outcome. It is helpful to define the auxiliary state: N ρ ( ψ ) = ρ , where ρ = Tr ( ψ ψ ), (6.9) aux | i i i exc(i) | i h | i O such that we can ignore the correlations between the subsystems. Indeed, the expec- tation values corresponding to the P are { i,j} P = ψ P ψ = Tr ( j j ρ ) (6.10) h i,ji h | i,j| i i | i h | i Thus ψ behaves like ρ if we restrict to measurements within subsystems. We | i aux call these expectation values “coordinates”. In the specific case where the N subsystems are qubits, an obvious choice for the coordinates would be the Bloch sphere coordinates: 1 ρ(~r) = (I + ~r σ~ ), ~r 1 (6.11) 2 · | | ≤ We denote ~r = ( σi , σi , σi ) for qubit i. Hence, we have a mapping from a i h xi h yi h zi 2N Hilbert space to a 3N dimensional parameter space. Thus, our redefined space grows linearly with N and can be managed with limited measurement resources. But oftentimes this map alone is not injective and thus insufficient to determine ~ε for only one input state. Many input states may be needed to characterize the impact 87 of U(~ε). We can define “abstract coordinates” where the input state, subsystem, and the coordinate within the subsystem. For example: (input = 000 , qubit = 2, coordinate = σ ) (6.12) | i h xi 6.5 Linear Approximations The idea of circuit calibration hinges on the fact that observables ~r has a trajectory with ~ε around ~ε = 0 where the circuit is free from errors. It is reasonable to assume that when ~ε is small in size, ~r varies almost linearly with ~ε r (~ε) r (~ε = 0) = L ε + O(ε2) (6.13) i − i ij j j X where the correlation matrix Lij = ∂ri/∂εj is the gradient or Jacobian of the measured parameters. Disregarding the second-order terms we have: L~ε ∆~r(ε) (6.14) ≈ This gives a first-order characterization of the circuit’s behaviour for small ~ε as far as the error and measured parameters are concerned. Without loss of generality, by defining ~r(~ε) = 0, ∆~r and ~r are one and the same. This method is readily adopted in experiment for tuning control parameters to maximize circuit performance in real time. For example, Cerfontaine et al. [65] have used this method to calibrate two-qubit gates in a system of singlet-triplet qubits. The resultant gate fidelities are improved and upper-bounded by incoherent and high- 88 frequency error types. Intuitively, we may choose ~r such that it is linearly independent with and has the same dimension as ~ε, allowing us to solve for the latter by simply inverting a square correlation matrix. We will demonstrate the use of linear approximation for calibrating a 3-qubit Toffoli gate in section (6.8), in the case where we have the exact values of the coordinates. But in reality, we can only measure ~r with limited accuracy since we are given a finite number of copies of states N, which are normally obtained by performing the same circuit N times. Moreover, it is not optimal to randomly pick and choose a handful of measurement results and ignore others just so we can uniquely determine ~ε. In reality, it is often the case that we measure more coordinates than needed but with statistical uncertainty. Since qubit measurements are quantized and binary, we can only have a statistically averaged ~rmeas. Given N copies of the state, the uncertainty in determining a state angle decays as 1/√N (see Figure 6.4), as we would expect from any statistically sampled quantity. In order to get to an accuracy of 0.01 radians, thousands of measurements are required. As a result, equation (6.14) over-determines ~ε with an approximate ~rmeas. To deal with this scenario, the most straightforward solution is to find a pseudo-inverse of the correlation matrix L. Specifically, we use the Moore-Penrose (MP) inverse of Lij for left-multiplication to approximate ~ε + + 1 ~ε ~ε = L ~r , where L = (L†L)− L† (6.15) ≈ MP meas The MP inverse is optimal for minimizing the 2-norm of the discrepancy between 89 0.044 0.031 0.022 theta error 0.016 0.011 500 1000 2000 4000 8000 # of measurements Figure 6.4: The error in estimating a single-qubit state angle ∆θ VS the number of measurements N for 1,000 repeated simulations. ∆θ goes almost strictly as 1/√N. ri and j Lijεj: P min L~ε ~r 2 = L~εMP ~r 2, (6.16) ~ε || − || || − || Thus, linear approximation allows us to solve for ~ε quickly provided that L is obtained beforehand through either simulation or experiment. In section 7.3, we will use equation (6.15) to find the errors instead of repeated circuit simulations (i.e. brute force). 6.6 An Error Model for Trapped Ion Qubits The reason for the prominence of ion trap quantum computers is their high-fidelity 4 quantum gates. Nowadays, trapped ion qubit operations reach 10− error levels for 3 single-qubit gates and 10− for 2-qubit gates (refer to Chapters 3-5). Despite their reliability and relatively high connectivity, 2-qubit gates are still the limiting factor 90 for circuit success rate, and much circuit optimization has been done to minimize the number of 2-qubit gates in a circuit [66]. Still, errors in ion traps are straightforward to model, and we give an overview of the errors involved in modern day ion trap quantum computers and how we can model them. Let us first consider single-qubit errors, which can be divided to amplitude-type and phase-type errors, referring to applied driving fields that effect the gates. When the amplitudes has errors, qubits are said to overrotate or underrotate due to errors in the rotation angle θ. This is known as overrotation errors. θ θ + ∆θ U = exp i σ exp i σ (6.17) − 2 φ → − 2 φ It is assumed that the relative error in θ is constant, meaning that ∆θ changes linearly with θ. This can be caused by laser intensity fluctuations, errors in the total gate time, and the slow position drifts of the ions. Nevertheless, this can be corrected by simply tuning the Rabi frequency for the particular ions. Quantum gates can also be impacted by phase errors, where the rotation axis φ deviates from intended: θ θ U = exp i σ exp i σ (6.18) − 2 φ → − 2 φ+∆φ It is assumed the absolute error ∆φ is constant. Systematic phase errors can be caused by erroneous synchronization between different gates at different stages of the circuit. This can be corrected by tuning the phase of the laser applied, or more often, adjusting the waiting times between gates. The above error models and the corresponding measures can be easily extended 91 to 2-qubit (Mølmer-Sørensen) gates. Combining them, our circuit can be fashioned as: U (θ , φ~ ) U (θ + ∆θ , φ~ + ∆φ~ ) (6.19) i i i → i i i i i i i Y Y θi pi qi where Ui(θi, φ~i) = exp( i σ σ ) are single and 2-qubit gates subject to both − 2 φi,1 φi,2 amplitude and phase errors. It is apparent that 2-qubit gates suffer overrotation and phase type errors as well, and in fact disproportionately so. In our simulations, we focus on the impact of 2-qubit gate overrotations and try to mitigate them through calibration. Other causes for limited gate fidelity are numerous, including: 1. Crosstalk error: individual laser addressing is imperfect and may accidentally excite/de-excite neighbouring ions. 2. Entanglement between qubit state and ion motion: motional state may not re- turn to zero after Mølmer-Sørensen gates, despite measures detailed in chapters 4-5. This is worsened by sideband frequency drifts and motional heating. 3. State leakage errors: excited ions may decay to states outside the qubit space, even if that is “forbidden” by selection rules. 4. Background magnetic field noise: although qubit energy levels can be cho- sen such that they are first-order insensitive to magnetic fields (e.g. hyper- fine ground states), they still have limited lifetime when subject to magnetic fluctuations. It is not easy to take into account these errors without overburdening classical simulation. Luckily, they can be suppressed by improving experimental equipment, 92 such as laser control and magnetic field shielding. 6.7 The Role of Classical Simulation Classical simulation of the circuit enables the mapping: ψ ψ or ρ ρ , (6.20) | initi → | finali init → final which allows us to evaluate any observable r = Oˆ = Tr(ρOˆ) and probability p of h i i detecting state ψ . This establishes a two-way relationship between our error vector | ii and observable parameters: ~ε ~r (6.21) ←→ The mapping from left to right is obvious from (6.20). In order to go from right to left, we can carry out repeated simulation and search for ~ε that best approximates the resultant ~r, min ~r(~ε) ~rmeas (6.22) ~ε || − || which may be time-consuming for larger circuits. Alternatively, we can find the Jacobian matrix ∂~r/∂~ε for small ~ε by simulation and compute its pseudoinverse to find ~ε, as discussed in section (6.5). Though not scalable or even necessary in theory, classical simulation is very helpful with smaller circuits and makes the calibration process significantly easier. The details of the circuit simulator, which models single and 2-qubit rotations and coherent-type errors, are discussed in Appendix B. 93 Figure 6.5: An implementation of the Toffoli gate using native gates in ion traps suggested by [66]. Note that the rotation angles in XX gates here are off by a factor π of two. For example, “XX 8 ” in fact represents a π/4 XX-rotation. 6.8 Example: Toffoli Circuit We will apply QCC to a Toffoli circuit as an example. Suppose we want to carry out a Toffoli gate in an N-qubit ion trap quantum computer. The native gates are single-qubit rotations and Mølmer-Sørensen 2-qubit gates (XX-type rotations). A breakdown of the Toffoli is shown in Figure 6.5. We assume that 2-qubit gates suffer overrotation errors, which we can correct through physical control. θ + ∆θ R (θ + ∆θ) = exp i σ σ (6.23) XX − 2 x x We also assume that the fractional error ∆θ/θ is time-independent and only de- pends on the qubit pair (i.e. 1 and 2, 2 and 3, 1 and 3). Thus, we have an error vector of length 3. The Bloch sphere coordinates, i.e. σ , σ , σ can be freely h xi h yi h zi measured for all 8 input states for all 3 qubits in the computational basis, which are then transformed to the spherical coordinates ~x = (r, θ, φ). Thus the total coordinate vector ~r has a length of 8 3 3 = 72. × × The simulated calibration procedure is outlined as follows: 94 1. Generate a random error vector ~ε of dimension 3. 2. Compute the output state from the input state ( ψ = U ψ ) | f i | ii 3. Choose 3 coordinates ~r with a linearly independent relationship with ~r 4. Perform multivariate minimization of ~r(~ε ) ~r(~ε) || est − || Using this method, ~εest can be estimated with very high precision for any given ~ε. Many numerical optimization techniques are readily available; here we use Broyden- Fletcher-Goldfarb-Shanno (BFGS) algorithm provided by scipy. See Figure 6.6 for sample results. Figure 6.6: Optimization results from scipy’s BFGS algorithm. The error vector is solved with very high precision if the exact coordinates are given. We turn our attention to the linear approximation between ~r and ~ε. By introduc- ing a small ∆~ε, we can approximate the gradient of ~r with respect to ~ε (i.e. ∂~r/∂~ε), also called the correlation matrix. Exactly 3 correlations are found randomly and automatically such that we have an invertible square matrix. 95 As we will see, the relationship between ~r and ~ε is not perfectly linear. The error in estimating ~ε is expected to go as square of the size of ~ε: ~ε ~ε O( ε 2). In other g − ∼ | | words, the fractional estimate error is expected to grow linearly with error size. This is confirmed by results from repeated “experiments” as shown in Figure 6.7. 0.020 0.015 0.010 0.005 Fractional estimate error 0.000 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 Average gate error size Figure 6.7: The fractional error in estimating ~ε varies linearly with the average er- ror size in the error vector. The combined result of 50 repetitions with randomly generated error vectors is used for each data point. 96 7. QCC: Application to Multi-qubit Con- trolled Gates Multi-qubit controlled circuits are a key component to introducing non-trivial corre- lations between multiple qubits. In particular, (N + 1)-bit Toffoli gates perform a bit-flip when the first N qubits match a specific bit string, and N +1-bit parity check gates perform a bit-flip conditioned on the parity of N qubits. They play a signif- icant role in quantum computation and are a key component in algorithms such as the Bernstein-Vazarani algorithm and Grover Search, which have been implemented in experiment recently [67, 17]. In practice, these gates are synthesized by single and 2-qubit rotations, which are realistically subject to amplitude and phase-type errors. We will study how we can calibrate these components with limited measurement resources. 7.1 Controlled Circuits It is firmly established that a universal gate set must consist of single-qubit operations and at least one entangling gate, such as the CNOT (XOR) or Toffoli gate. We can generalize the CNOT gate into two categories: state-controlled rotations ( N (U)) and ∧ parity-controlled rotations ( N (U)). We examine the roles of such gates in quantum L 97 computation and their decompositions into single and 2-qubit gates, specifically XX rotations. 7.1.1 N-CNOT circuit ( N (U)) ∧ A multi-qubit controlled operation applies a rotation U to a register y only when register x satisfies a specific bit string s. ψ = c x y (7.1) | i x,y | i | i x 0,1 N y 0,1 ∈{X} ∈{X } U (s) ψ = c x y + c s (U y ) (7.2) C | i x,y | i | i s,y | i y | i x 0,1 N /s y 0,1 y 0,1 ∈{X} ∈{X } ∈{X } This operation is essential to multi-partite entanglement and allows us to introduce non-trivial correlations within N qubits. This gate has use in quantum algorithm such as the Grover Search (for amplitude amplification). It turns out this operation enables the mapping between any quantum states in an (N + 1)-qubit system. An algorithm is proposed in [68] for generating an arbitrary qubit state using multi-qubit controlled rotations synthesized with 2-qubit gates. We can change s in (7.2) to any bit string by simply applying X-bit flips on the appropriate qubits before and after the circuit. Henceforth, we focus on the case where s is a string with all 1’s, and call the circuit N (U). ∧ There exist natural physical interactions which can be used to implement multi- control operation. For example, Rydberg atoms can be “blockaded” from excitation if one of the surrounding atoms is excited, which leads to a massive controlled operation. For ion traps, it is possible to do this with 4N global Mølmer-Sørensen gates, where all-to-all entanglement is applied to all ions in the same trap, according to [69]. 98 But realistically, global MS gates suffer high error rates due to long duration and complexity of the motional mode spectrum. It is very likely that high-fidelity circuits will be achieved with up to 2-qubit gates in the foreseeable future. = U V V † V Figure 7.1: Breakdown of the 2(U). ∧ = U V V † V V † V V † V Figure 7.2: Breakdown of the 3(U) circuit. ∧ Barenco et al. summarizes in [70] how one can synthesize N (U) with 2-qubit ∧ gates. Figures 7.4 and 7.2 show how one can build N (U) with 2-qubit rotations ∧ without ancilla qubits. This method is not scalable, however, as the number of controlled rotations grows exponentially with the number of control qubits, whereas the rotation angle of V diminishes exponentially. Fortunately, if one is allowed to use at least N 2 scrap qubits, the scaling becomes − linear. We can simulate a N (X) by stacking 4(N 2) Toffoli gates symmetrically, ∧ − connecting the control, target, and scrap qubits, as illustrated in Figure 7.3. From the first 2N 3 gates, it is readily observed that an X flip is performed on the last − N qubit only when the first N qubits are at 1 ⊗ . The last 2N 5 gates return the | i − scrap qubits to their original state. 99 1 2 3 4 5 = 6 7 8 9 Figure 7.3: 5(X) with 3 scrap qubits, adapted from [70]. ∧ The circuit decompositions shown above are by no means optimal in terms of total number of gates. An simplified version of N (X) is proposed in [71] and will be ∧ elaborated in section 7.4. A framework proposed in [72] also seeks to lower 2-qubit gate count for any Toffoli-type circuit. A more optimized version of this gate has been experimented in [67] with an ion trap to demonstrate Grover search algorithm with 4 qubits. Finally, we can carry out any N (U) by sandwiching a simple controlled U with ∧ two N (X) operations, given an ancilla qubit initiated at 0 , as shown in Figure ∧ | i θ (7.4). If the rotation axis anti-commutes with X (i.e. U = exp( i (~r ~σ)) where − 2 · ~r = (0, cos γ, sin γ)), then the ancilla set at 0 is no longer needed. | i 7.1.2 Parity check circuits ( N (U)) Another important variant of controlledL gate is the parity-controlled rotations. Con- sider a bit-flip X that is applied conditioned on the parity of N control qubits 100 c x y c x y f (s)) (36) x,y | i | i → x,y | i | ⊕ x i x 0,1 n y 0,1 x 0,1 n y 0,1 ∈{X} ∈{X } ∈{X} ∈{X } f (s) = x s mod 2 (37) x · r = (input = 000 , qubit = 2, coordinate = σ ) i | i h xi x0 H H | i • x1 H H | i • x2 H H | i • x3 H H | i • x4 H H | i • x5 H H | i • 0 | i • • • • • • x | 0i • x | 1i • x | 2i • x | 3i • x | 4i • x | 5i • 0 | i U,X =0 { } (38) −−−−−−→ • • • • • • • • • • • • • • • • • • • • = VX=XV † • • • −−−−−−−→ • • • • • • • • • • • • 0 0 0 • U U V † V N Figure 7.4: An implementation of (U). If5U = exp( iθA/2) and A anti-commutes with X, then the ancilla at 0 can∧ be skipped altogether.− Note that V 2 = U. | i ( N (X)): L c x y c x y f (s)) (7.3) x,y | i | i → x,y | i | ⊕ x i x 0,1 n y 0,1 x 0,1 n y 0,1 ∈{X} ∈{X } ∈{X} ∈{X } f (s) = x s mod 2 (7.4) x · The circuit has a straightforward breakdown in terms of CNOTs and single-qubit gates, as shown in Figure 7.5, which has O(N) number of 2-qubit gates. This operation is related to Bernstein-Vazarani algorithm where an oracle applies the following operation on the target qubit y y x s . This is realized by | i → | ⊕ · i applying CNOT gates where si = 1. In the case where ~s is just ~1 we return to our special case of parity checks. In order to generalize from N (X) to N (U) for any single-qubit rotation U, we can use a similar approach as FigureL 7.4 whereL we sandwich a controlled-U with two N (X) operations with the help of a fixed scrap qubit, or simply adjust the rotation L 101 c x y c x y f (s)) (36) x,y | i | i → x,y | i | ⊕ x i x 0,1 n y 0,1 x 0,1 n y 0,1 ∈{X} ∈{X } ∈{X} ∈{X } f (s) = x s mod 2 (37) x · r = (input = 000 , qubit = 2, coordinate = σ ) i | i h xi x0 H H | i • x1 H H | i • x2 H H | i • x3 H H | i • x4 H H | i • x5 H H | i • 0 | i • • • • • • x | 0i • x | 1i • x | 2i • x | 3i • x | 4i • x | 5i • 0 | i N U,X =0 Figure 7.5: Parity check operation ( (X)), effectively an X flip{ conditioned} on the (38) parity of the first N qubits for N = 5. A measurement in the −−−−−−→Z basis is done on the ancillary qubit to reveal their parityL • • • • • axis of the target qubit using• Euler angle identities.• • • • Parity checks are an extremely• important concept• in the• field of Quantum Error• • Correction (QEC) as they allow us to detect and correct errors without interrupting • • • • • the computational process. As discussed= in Chapter 1, an encodedVX state=XV † is a super- • • • −−−−−−−→ • • position of the simultaneous +1 eigenstates of a set of commuting operators called • • • • • stabilizers [30], and parity checks essentially “confirm” their eigenvalues and suggest • • • • • a correction according to the outcome (syndrome) of the measurements. 0 0 0 • The usefulness of parity checks relies on the Markovian error model, meaning that U U V † V the occurrences of errors are statistically independent. For example, the Pauli error channel can be summarized as: 5 (ρ) = (1 p p p )ρ + p XρX + p Y ρY + p ZρZ (7.5) E − X − Y − Z X Y Z It is clear that parity checks can detect single-qubit errors but not 2-qubit errors. One may assume that the probability of n single-qubit errors happening goes as 102 O(pn). Thus, by imposing stabilizer restrictions and conducting the appropriate parity checks, we can increase the distance of the code to n and suppress n-th order leading error terms. There is still a possibility that a single-qubit error can happen to the ancilla qubit, which can be mitigated with techniques such as the use of flag qubits [73]. Since parity checks are synthesized by multiple single and 2-qubit gates, calibrat- ing them before conducting actual error correction will be crucial to its effectiveness. We will do so using experimental data in section 7.2. 7.1.3 Hidden Inverse Configurations Since controlled gates are often synthesized by single and 2-qubit gates, they tend to have many configurations based on their circuit decompositions, and to optimize circuit performance, one can compare multiple configurations of the same circuit and find one such that coherent errors cancel each other most effectively. If a sequence of unitaries i U(θi, φi) with rotation angles θi and axis φi is self- inverse, then Q n n Ui(θi, φi) = Un i( θn i, φn i) (7.6) − − − − i=0 i=0 Y Y which readily follows from identities for Hermitian conjugation. However, this equal- ity in general does not hold when θi and φi are subject to errors ~ε. Thus, choosing between regular (left hand side) and inverted (right hand side) configuration is non- trivial if we wish to cancel out coherent errors most effectively. We call the inverses of gates that are self-inverse without errors “hidden inverses”. This is particularly relevant to controlled gates due to their self-inverse property. If we observe Figure 7.4, we find that there are two Toffoli-type gates flanking the 103 a) CNOT/ CNOT CNOT † | i • • RZ(✓) | i b) CNOT/ † ⇡ ⇡ ⇡ ⇡ ⇡ ⇡ ⇡ CNOT CNOT ⇡ ⇡ 0 RY ( 2 ) RX(⇡) RY ( 2 ) RX( 2 ) RY ( 2 ) 0 RYRY(( 2 )2 ) RXRX( 2(⇡)) RY ( 2 ) RY ( 2 ) RX(⇡) | i XX( ⇡ ) | i XX• ( ⇡ ) • ···x5 4 4 0 RY ( ⇡ ) RX(⇡) RX( ⇡ ) RZ(0 ) RY ( ⇡ ) RXRX( ⇡(⇡)) RZ(✓) RY ( ⇡ ) RX(⇡) | i 2 2 | i 2 2 ··· 2 CNOT CNOT ⇡ ⇡ ⇡ ⇡ ⇡ ⇡ RY ( 2 ) RX( 2 ) RY ( 2 ) RY ( 2 ) RX( 2 ) RY ( 2 ) | i • • ⇡ ⇡ XX( 4 ) XX( 4 ) RZ(✓) ≡ RX( ⇡ ) RZ(✓) RX( ⇡ ) | i 2 2 † CNOT CNOT ⇡ ⇡ ⇡ ⇡ ⇡ ⇡ RY ( 2 ) RX( 2 ) RY ( 2 ) RY ( 2 ) RX( 2 ) RY ( 2 ) | i • • XX( ⇡ ) XX( ⇡ ) 4 4 RZ(✓) ≡ RX( ⇡ ) RZ(✓) RX( ⇡ ) | i 2 2 Figure 7.6: An example of a hidden inverse in a CZ circuit. The second CNOT can be inverted without changing the circuit as a whole. However, the two circuit config- urations react differently in the presence of errors, specifically coherent overrotation errors, as shown in Figure 7.7. controlled-U operation. If U = exp( iθS/2), and we assume S,X = 0 without loss − { } of generality, then N N N (U) = (X)V † (X)V (7.7) ∧ ∧ ∧ N N N (U) = (X)V † (X)V (7.8) M M M where V 2 = U. The trailing V is sometimes omitted for brevity. It is easy to see that if V is identity and the second controlled gate in (7.7) and (7.8) is inverted, then the whole circuit reduces to identity, even in the presence of coherent errors (θ θ +∆θ i → i i and φ φ + ∆φ ). This implies that choosing the inverted configuration is more i → i i advantageous when V is a small rotation, and less advantageous otherwise. We called the circuit “conjugated” if the second controlled gated is inverted, and “regular” if it is the same as the first. As an example, let’s consider the N (U) circuit where U = R ( 2θ) (see Figure z − 7.6 for the 2-qubit case). The circuitL can be expressed as N N Ucirc = CNOTi,N+1Rz,N+1(θ) CNOTN+1 i,N+1 (7.9) − i=1 i=1 Y Y 104 (a) (b) Figure 7.7: Final state fidelities for the controlled-Z rotation circuit are plotted as a function of rotation angle θ for regular (orange) and conjugated (blue) configurations. The circuit is subject to overrotation errors of -10% in (a) and +30% in (b). The conjugated circuit performs better than the regular version for the majority of rotation angles. 1.0 N=2, Conjugated N=2, Not Conjugated 0.9 N=3, Conjugated N=3, Not Conjugated 0.8 N=4, Conjugated Fidelity N=4, Not Conjugated 0.7 N=5, Conjugated N=5, Not Conjugated 0.6 N=6, Conjugated N=6, Not Conjugated /2 0 /2 Rotation angle N Figure 7.8: Final state fidelities of (Rz( 2θ)) circuit for θ [ π, π] according to equation (7.9) for regular and conjugated configurations− for N∈ =− 2, 3, 4, 5, 6, where L each native gate except Rz(θ) has +20% overrotation. Again the conjugated config- uration outperforms the regular version, showing the scalability of hidden inverses. 105 where each CNOT in the first series of CNOTs is decomposed as CNOTi,j = Ry,i(π/2)XXi,j(π/4) (7.10) R ( π/2)R ( π/2)R ( π/2) x,i − x,j − y,i − The CNOTs in the second series can take the form of either 7.10, or the following: CNOTi,j = Ry,i(π/2)Rx,j(π/2) (7.11) R (π/2)XX ( π/4)R ( π/2) x,i i,j − y,i − The choice of CNOT decomposition described above leads to different perfor- mances of the circuit when each gate is subject to overrotation. Figures 7.7 and 7.8 show the performance of the circuit for π < θ < π, assuming that every single-qubit − and two-qubit gate in equation (7.10) and (7.11) suffers the same overrotation. We observe that the conjugated configuration outperforms the non-inverted version con- siderably for a majority of θ values, and has the best performance when θ is close to zero. 7.2 Experimentally Calibrating a 6(X) Gate We demonstrate the calibration of the measurement of aL weight-6 stabilizer, which involves the 6(X) circuit. The circuit consists of 6 2-qubit gates connecting the data to the bareL ancilla qubit to be measured at the end, as shown in Fig 7.9. We assume that the 2-qubit gates are subject to overrotation errors, so the error vector has dimension 6. The 6-qubit parity check circuit is a key component of the Bacon-Shor (BS) sub- 106 c x y c x y f (s)) (36) x,y | i | i → x,y | i | ⊕ x i x 0,1 n y 0,1 x 0,1 n y 0,1 ∈{X} ∈{X } ∈{X} ∈{X } f (s) = x s mod 2 (37) x · r = (input = 000 , qubit = 2, coordinate = σ ) i | i h xi x0 H H | i •MS x1 H H | i •MS x2 H H | i •MS x3 H H | i •MS x4 H H | i •MS x5 H H | i • MS 0 | i • • • • • • Figure 7.9: 6-qubit parity check circuit, broken down to Mølmer-Sørensen and single- qubit gates. Each MS gate is assumed to have overrotation errors. Data qubits 1st, 2nd ancilla qubits Figure 7.10: Qubit layout along the ion chain for the 6-qubit parity check experiment. system code with distance 3 [74, 75]. Although originally meant to only have 2-qubit Continuum limit for charge density: parity checks, we can opt to perform 6-qubit parity checks instead if long-distance Electric field acting on a charge due to surrounding charges: gates are available (e.g. in ion traps), so that fewer ancilla qubits are needed. Sim- kqzˆ 1, if zi > zn En = 2 , wherez ˆ = (38) ulation results show that BS code has lower− overhead(zi zn and) higher pseudothreshold1, if z < z i=n ( i n X6 − − compared to traditional surface codes [76].1 Let ∆zi = (zi+1 zi 1). Define charge density ρz=z = q/∆zi 2 − − i The circuit is done in a 15-ion trap by L. Egan et al. in University of Maryland. The electric field due to a point charge at z = zi can be approximated as As shown in Figure 7.10, ionsthat labelleddue to an 5, even 6, 7, cloud 9, 10, of 11charge form centered the data at qubits,z = zi with and charge density ρz=zi . ions labelled 2 and 14 are used as the ancillaz fori+∆ thezi/2 first and second round stabilizer kρz=zi dzzˆ 1, if z > zn En 2 , wherez ˆ = (39) ≈ − (z zn) 1, if z < z i=n zi ∆zi/2 ( n measurement respectively. For each parityX6 Z check,− a total− of 64 input strings are used− for the data qubits, whereas theFurther, ancilla we is can fixed approximate to be at state the charge0 . At density the endρ ofz to the be a completely smooth | i circuit, the ancilla qubit is measured in the computational basis. 107 5 Single stabilizer measurement 0.95 0.90 0.85 0.80 Correct ancilla prob. 0.75 simulation experiment 0 10 20 30 40 50 60 Input string (a) Single-round stabilizer measurement First-round stabilizer measurement Second-round stabilizer measurement 0.90 0.9 0.85 0.80 0.8 0.75 0.70 0.7 0.65 Correct ancilla prob. 0.6 Correct ancilla prob. 0.60 simulation simulation experiment 0.55 experiment 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Input string Input string (b) Two-round stabilizer measurement Final error vectors 0.15 single measurement first-round measurement 0.10 second-round measurement 0.05 0.00 0.05 Over-rotation error 0.10 0.15 0 1 2 3 4 5 Gate number (c) Optimized error vectors Figure 7.11: Assuming that the six 2-qubit gates are subject to overrotation errors, we attempt to fit the simulated results against the measurement result using the method of least squares. The error bars for the measurement indicate 1 standard deviation, and the lines between discrete data points are merely used to guide the eye. (a) Correct ancilla probability of a single-round weight-6 stabilizer measurement. (b) Correct ancilla probability when the weight-6 stabilizer is measured twice on two different ancilla qubits. (c) The final error vectors for all 3 experiments. They match in overall shape, as we expect. 108 Two sets of stabilizer measurements are made. First, a single-round stabilizer measurement is done on the data qubits (Figure 7.11a). Next, two rounds of stabilizer measurements are done on the data qubits, using the ancilla qubits labelled 2 and 14 respectively (Figure 7.11b). For each round of stabilizer measurement, we find the average success probability of getting the correct parity over 500 repetitions. Thus, the coordinate vector ~r is: σ if x = 1 mod 2 h zi j ij ri = (7.12) L 1 σ if x = 0 mod 2 − h zi j ij L where ~xi is the 6-bit binary representation of input i. Recall the minimization prob- lem: 2 min (ri(~ε) ri,meas) (7.13) ~ε − i X We try to search for the optimal ~ε by simulating the circuit iteratively with dif- ferent ~ε such that 7.13 is achieved. The fit for measurement success probabilities in Figures 7.11a and 7.11b, showing a high degree of agreement between experiment and simulation for both measured coordinates and optimized error vector. A negative offset is added to the fidelities for all inputs to account for incoherent errors as well as high-frequency noises. The points with the lowest fidelities are matched, implying that 2-qubit overrotations are indeed the dominant error type. The error vectors are then plotted in Figure 7.11c. The overall “shapes” for the 3 experiments are similar with each other, which suggests the 2-qubit gate noise is fluctuating slowly relative to circuit duration. For the two-round experiment, we see 109 an overall decrease of about 4.4% in overrotation going from the first round to the second round. Indeed, this is confirmed by the observation that the Rabi frequency drops by somewhere between 2% and 6% over a time period of 1 millisecond, which is slightly shorter than the duration of one round of stabilizer measurement. 7.3 Simulating the Calibration of 3(X) Gate ∧ We simulate the circuit performance of 3(X) (c.f. circuit diagram in Figure 7.2), ∧ assuming XX gates are subject to overrotation. The length of the error vector ~ε is 6 since six 2-qubit gates are needed to connect any pair of 4 qubits. A total of 24 input states are used and all 4 qubits are measured at the conclusion of the circuit in the Y -basis. It turns out that the first qubit does not respond to the errors, so only the results from the second, third, and fourth (labelled 1, 2, and 3) are recorded. Thus the coordinate vector ~r has length 48. To imitate the experimental procedure, we find the final state ψ (~ε) and generate | f i measurement results from N = 500 repetitions, with an uncertainty of roughly 0.044. The measurement average ~rapprox and the actual expectation values ~r are shown in Figure 7.12. As one may expect, coordinates that are less sensitive to errors (e.g. qubit 1) tend to be eclipsed by statistical uncertainty and contribute less to error prediction. In order to speed up the fitting process, we apply linear approximation to find the error vector using the pseudoinverse technique described in section 6.5. The relevant coordinates are collected and the correlation matrix is set up by approximating its Jacobian ∂~r/∂~ε by small steps in ~ε. The pseudoinverse matrix characterizes the circuit’s response to small errors, so we only need to multiply the pseudoinverse 110 Measured coordinates for qubit 1 0.3 Actual coords. Measured coords. 0.2 0.1 0.0 Coordinate value 0.1 0.2 0 2 4 6 8 10 12 14 Input string Measured coordinates for qubit 2 0.6 0.4 0.2 0.0 0.2 Coordinate value 0.4 Actual coords. Measured coords. 0.6 0 2 4 6 8 10 12 14 Input string Measured coordinates for qubit 3 0.3 Actual coords. Measured coords. 0.2 0.1 0.0 0.1 Coordinate value 0.2 0.3 0 2 4 6 8 10 12 14 Input string Figure 7.12: Measured coordinates and actual coordinates plotted against input state for the 3(X) circuit, which is repeated 500 times to emulate the experimental pro- cedure.∧ The error bars correspond to 1 standard deviation of uncertainty for the measured values. We can see that the second qubit is most sensitive to 2-qubit gate errors, followed by the third and first qubit. Low sensitivity means that statistical uncertainty is comparable to if not larger than the coordinate size itself, which is less helpful for predicting the error vector. 111 Estimate error for 6 * 500 coordinates 0.05 0.00 0.05 0.10 0 500 1000 1500 2000 2500 3000 Figure 7.13: Estimate error for the length-6 ~ε over 500 trials, with a standard devia- tion of 0.0180. The standard deviation for ~ε itself is 0.0724. matrix with the coordinates ~r to find the errors ~ε. The estimate error δ~ε for ~ε is plotted in Figure 7.13 for N = 500 trials. The standard deviation in δ~ε is 0.0180, about 4 times smaller than the error size itself. It is expected that the residual error δ~ε will have a much smaller impact on the circuit since state fidelity varies as the square of coherent errors. 7.4 Simulating the Calibration of 5(X) Gate ∧ We build the 5(X) circuit using the method in [71], as shown in Figure 7.14. As- ∧ suming that 2-qubit gates have overrotation errors that do not vary in time, the error vector ~ε has length 12. A total of 29 bit-string inputs are used and all 9 qubits are measured in the Y -basis. Figure 7.15 shows the fidelities as a function of the input string. Steps of 16 are observed, meaning that the initial input for the last 4 qubits does not affect the output state fidelities. It is clear from the plot that the majority of input/output states have unit fidelity and thus do not reveal any information about the errors ~ε. This insight helps us reduce calibration overhead, since our current goal is to evaluate and correct circuit errors. 112 x1 x1 x2 x2 + + + x3 VV V x3 V V V (i) (ii) Figure 1: Structure of the Peres gate and its inverse Before we can describe our improved design, we have to introduce the Peres gate. The Peres gate P (x1, x2, x3) [7, 8] is equivalent to the trans- formation produced by a Toffoli gate T OF (x1, x2, x3) followed by a CNOT gate T OF (x1, x2). A four elementary quantum transformations realization i+1 1 i + of Peres gate is illustrated in Figure 1(i), where V = 2 i−1 and V is its − inverse. Denote the four gates used in the proposed constructio n as A,B,C and D. Trivial analysis shows that the inverse Peres gate can be achieved by 1 1 1 1 a circuit D− C− B− A− (Figure 1(ii)), consisting of the inverses of the gates used for construction of Peres gate. From the point of view of Toffoli-CNOT realization, the inverse Peres gate will act as a CNOT T OF (x1, x2) followed by the Toffoli gate T OF (x1, x2, x3). = Figure 2: Circuit for (m + 1)-bit Toffoli (illustrated for m = 5) x1 x1 We suggest thatx in2 construction of Lemmax2 7.2 in [1] the Peres gate or + + + its inverse are usedx everywhere3 VV insteadV x of3 theV moreV expensiveV Toffoli gate. This is illustrated in Figure(i) 2, where each of the (ii) pairs Toffoli-CNOT and FigureCNOT-Toffoli 7.14: Circuit is adiagram Peres gate for the or its 5-control inverse. Toffoli circuit (top), as proposed by [71], whichTo prove has aFigure thatrelatively such 1: Structurecircuit lower overhead realizes of the an than Peres (m that+1)-bit gate shown and Toffoli its in Figureinverse one can 7.3. inspect Each it Peres circuitor simplycomponent notice or that its inverse a pair of(dashed identical rectangles) CNOTs can be movedbroken together down tousing 4 2-qubit gates (bottom), compared to the Toffoli gate which can be broken down to 5 2-qubit theBefore moving we rule can from describe [4] and our thus, improved be canceled design, out. we Therefore, have to introduce this circuit the gates. Note that V 2 = X. Peres gate. The Peres gate P (x1, x2, x3) [7, 8] is equivalent to the trans- formation produced by a Toffoli gate2T OF (x1, x2, x3) followed by a CNOT gate T OF (x1, x2). A four elementary quantum transformations realization i+1 1 i + of Peres gate is illustrated in Figure 1(i), where V = 2 i−1 and V is its − inverse. Denote the four gates used in the proposed constructio n as A,B,C and D.1.000 Trivial analysis shows that the inverse Peres gate can be achieved by 1 1 1 1 a circuit0.975D− C− B− A− (Figure 1(ii)), consisting of the inverses of the gates used for construction of Peres gate. From the point of view of Toffoli-CNOT 0.950 realization, the inverse Peres gate will act as a CNOT T OF (x1, x2) followed by the Toffoli0.925 gate T OF (x1, x2, x3). Fidelity 0.900 0.875 0.850 0.825= 0 100 200 300 400 500 Input string Figure 7.15: Fidelity VS Input state for a randomly generated error vector with maximum size of 0.3. Apparently, fidelities are exactly 1 for most input strings, and do not dependFigure on the2: Circuit input on for the (m last+ 1)-bit 4 qubits Toffoli (scrap (illustrated and target for qubits).m = 5) We suggest that in construction of Lemma 7.2 in [1] the Peres gate or its inverse are used everywhere instead113 of the more expensive Toffoli gate. This is illustrated in Figure 2, where each of the pairs Toffoli-CNOT and CNOT-Toffoli is a Peres gate or its inverse. To prove that such circuit realizes an (m+1)-bit Toffoli one can inspect it or simply notice that a pair of identical CNOTs can be moved together using the moving rule from [4] and thus, be canceled out. Therefore, this circuit 2 We proceed to search for linear correlations between circuit errors ~ε and coordi- nates ~r, which consist of spherical coordinates of the Bloch sphere of each qubit. As tabulated in Figure 7.16, the results indicate that only 4 out of 12 error parameters have linear dependence on any measured coordinates. It is clear that we will need to go further and look for quadratic relationships. 2 We perform an exhaustive search for coordinates that vary with O(εi ) in order to solve for the remaining ε . We find 8 out of 12 error parameters with which the { i} coordinates vary quadratically, 5 of which cannot be solved from linear correlations. It appears that all coordinates that vary linearly with ~ε are angular coordinates, whereas those that vary quadratically are radial coordinates, which makes sense since the output qubits are pure and disentangled when ~ε = 0. The relevant coordinates and correlation matrices are shown in Figure 7.16. 2 For quadratic dependence, since we have solved for εi not εi, there is a sign problem where we cannot tell between ε . This can be tackled by choosing a sign, correcting ± i for εi, and measuring the coordinates again. If the coordinates get worse, then we know that we have corrected in the wrong direction and can fix εi retrospectively. It appears that there are still 3 error parameters that have no impact on any measured coordinates. By reviewing the sequence of gates in the circuit, we realize that there are 3 gates that cancel each other completely (including their overrotation) and are not needed in the first place. This completes the characterization of the circuit’s response to a small ~ε. 114 Params with linear dependence: [0, 1, 4, 9] Params with quadratic dependence: [0, 3, 4, 6, 7, 9, 10, 11] Linear correlations: Measurement basis (trial, qubit, coordinate) = [[480, 7, 2], [496, 8, 1], [500, 7, 1], [496, 1, 1]] Correlation matrix = [[-1.57079633 0. 0. 0. ] [ 0. -3.14159265 0. 0. ] [ 0. 0. 3.14687869 0. ] [ 0. 0. 0. -0.7854805 ]] Quadratic correlations: Measurement basis (trial, qubit, coordinate) = [[464, 7, 0], [497, 5, 0], [496, 6, 0], [368, 5, 0], [116, 8, 0]] Correlation matrix = [[ -4.93031069 0. 0. 0. 0. ] [ 0. -4.93031068 0. -4.93031068 -1.23341973] [ -4.93031068 -4.93031068 -19.66744322 -4.93031068 -1.23341973] [ 0. 0. 0. -4.93031068 -1.23341974] [ 0. 0. 0. 0. -0.61679411]] Figure 7.16: Numerical results from approximate circuit calibration. A random set of coordinates sensitive to errors are chosen to form square matrices with the error 2 parameters. We search for members of ~r that vary with O(εi), then O(εi ). 115 8. Conclusion Quantum computation is a vast and complex topic of great interest to those in both academia and industry. From black-box algorithms to error correction, and from confirming physical principles to demonstrating quantum supremacy, the challenges faced by quantum computers are fascinating and their potential impact on the world are far and wide. We have tried to make a contribution towards making trapped ion quantum computers better, with specific emphasis on 2-qubit gates, which will be the bottleneck for quantum computers in the foreseeable future. We believe the niche that we occupied gives quantum computers a better chance at disrupting the world of science and computation. We have added frequency modulation (FM) to the ion trap control toolbox by showing that it can be used to suppress residual state-dependent motion. The opti- mization for the pulse is nothing but the minimization of the displacement with the constraint of the entanglement factor equal to π/4. The key generalization is that the cumulative phase equals the integral of detuning with respect to time, instead of the product of detuning and time. Importantly, we have also derived the condition for robustness against small, static drifts in motional frequencies up to the first order. 4 From simulation, the error rate can be suppressed to below 10− . Experimentally, we have achieved 98.6% 2-qubit gate fidelity in a 5-ion trap [24], and up to 97% fidelity 116 for a 17-ion trap [45]. Most recently, 99.49% and 99.30% 2-qubit gate fidelities have been observed in a 2-ion chain and 4-ion chain respectively [38]. We have shown that arbitrary 2-qubit gate coupling is possible for up to N = 50 ions using typical modern ion trap setups [51]. The key is to achieve near-resonance with one motional mode and be far-detuned from the others, leading to non-locality in the entanglement factor. Under this assumption, the gate time is expected to scale linearly with N. This complements the contribution by Y. Wu in [45], which states that if the detuning is much larger than the span of the motional spectrum, multiple pairs of neighbouring or near-neighbouring ions can be entangled simultaneously with very large N, with a gate time independent of N but scaling as d3 where d is the distance between the ions. Together, we have generalized the idea of short and long- range entanglement within a large ion lattice, depending on the detuning between the driving frequency and resonant motional modes. We have put forward the idea of circuit calibration, where circuit errors ~ε are corrected once a sufficient number of output parameters called coordinates ~r have been measured. The mapping between ~ε and ~r is established using a circuit simulator that simulates an appropriate error model. For smaller circuits, we have successfully found ~ε by minimizing the discrepancy between the simulated and measured coordinates. Realistically, measured quantities are subject to statistical uncertainty due to the limited number of measurements available, and ~r usually over-determines ~ε. For small ~ε, the circuit response can be approximately characterized by the Jacobian ∂~r/∂~ε. This relationship can be inverted using the Moore-Penrose pseudoinverse, which minimizes the 2-norm of the difference between the measured and expected coordinates. 117 Calibration has been applied to multi-qubit controlled gates, the natural extension of the CNOT gate. They are categorized into two main types: controlled gates N (U) and parity-controlled gates N (U), which find widespread use in oracle-based ∧ algorithms and quantum error correction.L They can be synthesized by XX-rotations, the native 2-qubit gate for ion traps, the details of which can be found in [70, 66]. Assuming that XX gates suffer overrotation, we have calibrated the 6(X) gate ⊕ against experimental results, showing remarkable agreement for low-fidelity outputs. We have simulated and calibrated 3(X) using linear approximations, reducing the ∧ error size from 0.0724 to 0.0180. In this setting, the measured ~r is subject to statistical uncertainty for N = 500 whereas the Jacobian ∂~r/∂~ε is approximated by small steps. Finally, we calibrated the 5(X) using the synthesis in [71], and affirmed that ~ε can ∧ only be found if quadratic terms are included. Looking into the future, I expect to see modulated pulses being widely used in ion trap experiments, including frequency modulation. The race to high-fidelity gates and fault tolerance will necessitate total suppression of residual ion motion with the help of modulation. Higher powers will be used in order to shorten gate time, which means we have to suppress Stark shift and off-resonant excitations. The incorporation of composite pulses will mean that over-rotation errors are no longer a first-order effect but a second or higher-order effect, but with an altered rotation axis, which should be taken into account. In the end, multi-qubit gates will still be bottlenecked by high- frequency noises, motional decoherence, and SPAM errors. These can be alleviated by improved experimental technique and upgraded equipment. Quantum circuit calibration (QCC) has significant potential to improve quantum computers in the short term and in the long term. The results in Figure 7.12 suggest 118 that coherent errors can be calibrated effectively in a 7-qubit circuit. QCC comple- ments gate set tomography by sacrificing thoroughness for scalability, as long as we can identify the dominant coherent errors. The lack of scalability of circuit simula- tors may become an impediment in larger circuits, but if we can divide a quantum computers into subnetworks that can communicate with each other effectively, as dis- cussed in section 3.7, modular circuit calibration can make quantum computers much more scalable and effective. From the perspective of quantum error correction, fault- tolerant quantum detection protocols are only useful when error detection success rate is high enough, and calibration may be key to achieve that. 119 Appendices 120 A. - FM Optimization with continuous pulses In [24], we showed that it is possible to curb residual ion displacement using frequency modulation (FM) alone. Here we explain the reasoning behind the engineering of the pulse. Given a set of sideband frequencies ωi (from experiment or simulation), we need to minimize the displacement Ωeiθdt. This process is done by allowing frequency to change in time and optimizingR it shape. The most popular way to do this is to divide it into equal segments with constant frequency. The more motional modes there are, the more frequency steps we need to minimize the displacements. This was the approach taken in [23, 47], but with amplitude and phase modulation instead of frequency. The approach taken in [24], however, is to use smooth, oscillatory pulses with several extreme points instead. One reason for using a smooth pulse as opposed to a step pulse is to accommodate imperfect experimental control. When introducing a sudden change to a physical system, the parameter in question tends to suffer damping or overshoot, which can be avoided by using a smooth pulse. At the end, the generated pulse is subject to digitization and thus not perfectly smooth, but 121 Figure A.1: Displacement diagram for a motional mode. For a symmetric trajectory, the time-averaged position αavg (blue dot) must lie on the symmetry line (dashed line). If αavg lies at the starting point, then the end point must also be at the starting point. smaller step intervals provide better control of the pulse shape nonetheless. Another very important benefit of using smooth modulation is the rapid decou- pling as detuning δk increases. This can be understood by noting that a step function in amplitude Ω has a slowly diminishing Fourier coefficients ( 1/δ ), whereas smooth ∼ k pulses have a much more localized Fourier transform. In [51], where the trapped ion crystal has up to 50 motional modes, a smooth amplitude profile is used such that highly detuned modes vanish quickly with detuning, whereas the near-detuned modes 4 are decoupled using an optimized FM pattern. An error of the order of 10− can be achieved by suppressing the 10 nearest motional modes. A set of extreme points are introduced at constant time intervals, and cosine curves are used to connect them such that the slope at each of these points is zero. This yields a frequency profile that is smooth everywhere (continuous first derivative). We also assume that the pulse is symmetric in time (Ω(τ t) = Ω(t), µ(τ t) = − − 122 µ(t)) such that we only need to focus on the first half of the pulse. The first rea- son for this is that it gives us half as many variables to work with, thus reducing computational cost. The second reason is that given this symmetry condition, the robustness condition in (4.15) guarantees (4.13), thereby further simplifying the opti- mization process. This is most easily understood if we draw a symmetric line through the trajectory (see Figure A.1). The double integral expression in (4.15) represents the geometric average of the trajectory. If the time-averaged position of a symmetric trajectory lies at the origin, then so does the end of the trajectory. In general, this is not true for asymmetric pulses. 123 B. - Circuit Simulator: QRSim Using the numpy package, we have managed to create a custom, light-weight circuit simulator called “QRSim” which treats every gate as a single or 2-qubit rotation. “QR” stands for qubit-wise rotation, which is a specific type of unitary operations acting on a 2N Hilbert space. The main goal of our simulator is to evaluate the impact of coherent errors, such as overrotation and phase errors, on any circuit component, which helps us implement circuit calibration as demonstrated in Chapters 6 and 7. A sample code for the circuit simulator is shown in (Figure B.1). We import QR- Sim in order to instantiate a “Circuit” object, which stores the state vectors as well as the gate sequence that comprises the circuit. To add gates to a circuit, we simply chain on the corresponding gate methods while passing in the relevant parameters. When the “compute” method is called, the gates are “compiled” internally to final- ize the state angles, and the final states are found given the initial states and gate sequence. In the following sections, we discuss the data structure and computational process of the simulator as well as its main features. 124 N = 2 circ = Circuit(N) circ.init_state = zero_state(N) circ.runs = 100 circ.Z_is_native = True t = np.pi / 2 circ.Y(0, t).XX(0, 1, t).X(0, -t).X(1, -t).Y(0, -t) final_states = circ.compute() Figure B.1: A simple example for building a CNOT gate using QRSim. The “zero state” helper method simply creates a zero state vector for the purpose of initialization. B.1 Single and 2-qubit Rotations This section explains how single and 2-qubit rotations are applied to the state vectors in our simulator. Our circuit can be broken down to a sequence of single and 2-qubit rotations: U = U , where U = cos(θ/2)I i sin(θ/2)S (B.1) circ i i − i Y where S = σ σ is a 2-qubit Pauli operator. i ⊗ j Given a state ψ , we evaluate the U ψ with two steps: | ii i | ii Find ψ0 = S ψ (fast); • | i | i Perform the linear operation: cos(θ/2) ψ i sin(θ/2) ψ0 (slow). • | i − | i The first step is made efficient by thinking of our state vector ψ as a rank N | i tensor, with 2 dimensions in each rank. For ψ = cx x where x is a N-bit string, | i x | i the complex state amplitudes cx can be stored a multi-rankP array C such that cx = C[x0][x1]...[xN ] := C[x] (B.2) 125 This way, a bit flip is a simple relabelling of the coefficients up to a complex phase. For a general bit flip operator σφ = (cos φ)σx + (sin φ)σy acting on the j-th qubit, the new coefficients are given by 0 iφ 1 Ci+1[xj ] = e− Ci[xj ] (B.3) 1 iφ 0 Ci+1[xj ] = e Ci[xj ] (B.4) 0 1 where xj and xj are any bit strings with the j-th bit being 0 and 1 respectively. The iφ relabelling is done using the method “numpy.roll”, whereas the multiplication of e± is done with the help of “numpy.swapaxes”. This can be easily generalized to 2-qubit bit flips. For a Z-type rotation, the new coefficient after a phase flip is described by 0 0 Ci+1[xj ] = Ci[xj ] (B.5) C [x1] = C [x1] (B.6) i+1 j − i j The second step involves linear operations of two vectors with 2N coefficients, making it the slower of the two computational steps. B.2 Gate sequence While the rank-N tensor stores the state, another data type involved is the gate sequence for the circuit. They are stored in 3 separate properties: Circuit.ideal gates: the original gate sequence in the circuit, including H and • CNOT. 126 Circuit.native gates: the circuit broken down to native gates in ion traps. Gates • such as H and CNOT are decomposed as sequences of single-qubit and XX-type rotations. Circuit.noisy gates: the same native gates but with different rotation angles • and axes, which will be used to compute the final states in parallel. This is important for certain scenarios such as when the circuit is subject to random noises, or when we have multiple input states for the same circuit. The “ideal gates” are compiled to “native gates”, which are in turn converted to “noisy gates”, possibly with the addition of random noise from an error distribution. Depending on the circumstances, we may need to edit “noisy gates” manually be- fore the circuit is run. This allows processing of multiple states with the same gate sequence but different rotation parameters. We show some sample gate data in Figures B.2, B.3, and B.4 for a controlled-Z rotation as illustrated in Figure 7.6 in subsection 7.1.3, with a 10% underrotation for all native gates except the middle Z rotation. The circuit is repeated 300 times with the Z rotation varied from π to π. The state fidelities are computed simultaneously − and then plotted as a function of φ in the first plot in Figure 7.7. B.3 The CuPy module CuPy allows parallel computation with GPU acceleration, specifically for NVIDIA CUDA GPUs. CuPy offers the most often used mathematical operations in numpy with the same syntax as numpy. Its library is referred to as “cupy” in the following. We utilize cupy to speed up the circuit operations described in the previous two 127 [['Y', 0, 1.413716694115407, None], ['XX', [0, 1], 1.413716694115407, None], ['X', 0, -1.413716694115407, None], ['X', 1, -1.413716694115407, None], ['Y', 0, -1.413716694115407, None], ['Z', 1, 0, None], ['Y', 0, 1.413716694115407, None], ['X', 1, 1.413716694115407, None], ['X', 0, 1.413716694115407, None], ['XX', [0, 1], -1.413716694115407, None], ['Y', 0, -1.413716694115407, None]] Figure B.2: The “ideal gates” for the conjugated CZ gate (c.f. Figure 7.6). The four entries indicate the gate type, the qubit(s) involved, the rotation angle, and whether it has a custom rotation axis. Note that all the π/2 rotations have a 10% underrotation. [[['S_phi'], [0], 1.413716694115407, [1.5707963267948966]], [['S_phi', 'S_phi'], [0, 1], 1.413716694115407, [0, 0]], [['S_phi'], [0], -1.413716694115407, [0]], [['S_phi'], [1], -1.413716694115407, [0]], [['S_phi'], [0], -1.413716694115407, [1.5707963267948966]], [['Z'], [1], 0, [0]], [['S_phi'], [0], 1.413716694115407, [1.5707963267948966]], [['S_phi'], [1], 1.413716694115407, [0]], [['S_phi'], [0], 1.413716694115407, [0]], [['S_phi', 'S_phi'], [0, 1], -1.413716694115407, [0, 0]], [['S_phi'], [0], -1.413716694115407, [1.5707963267948966]]] Figure B.3: The “native gates” has a slightly different data format from “ideal gates”. The fourth entry specifically states the rotation axis. [['Z'], [1], [array([-3.14159265, -3.12057866, -3.09956466, -3.07855066, -3.05753666, -3.03652267, -3.01550867, -2.99449467, -2.97348067, -2.95246667, -2.93145268, -2.91043868, -2.88942468, -2.86841068, -2.84739669, -2.82638269, -2.80536869, -2.78435469, -2.7633407 , -2.7423267 , ..., 2.7423267 , 2.7633407 , 2.78435469, 2.80536869, 2.82638269, 2.84739669, 2.86841068, 2.88942468, 2.91043868, 2.93145268, 2.95246667, 2.97348067, 2.99449467, 3.01550867, 3.03652267, 3.05753666, 3.07855066, 3.09956466, 3.12057866, 3.14159265])], [array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., ..., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])]] Figure B.4: The gate element in “noisy gates” representing the Z rotation with 300 rotation angles from π to π but the same rotation axis. The other gates are omitted since they have the same− rotation parameters for all 300 repetitions. 128 sections. In essence, when “Circuit.compute” is called, cupy takes in numpy array, shuffles the data to the GPU, performs parallel computation, and shuffles the result back to the CPU. The method “cupy.array” is called to convert a numpy array storing the states to a cupy array in the GPU. The circuit operations are carried out the same way as we do in numpy. Once complete, “cupy.asnumpy” is called to switch the cupy array back to a numpy array. Our QRSim library uses cupy optionally, depending on whether the system detects a compatible GPU. This is done by an attempt to import cupy, and if it fails, implying that cupy cannot utilize the GPU, the numpy methods are imported instead. Detailed documentation for CuPy can be found on the official cupy website https://cupy.chainer.org/. 129 Bibliography [1] A. M. 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From January to August 2014, he studied in University of California Berkeley as an exchange student and worked as an undergraduate researcher in Hartmut Haeffner’s lab, simulating laser optics. In August 2015, he was admitted to Georgia Institute of Technology to pursue a PhD degree in physics. He began to focus on control the- ory with ion trap quantum computation under his advisor Kenneth Brown. He was the recipient of Amelio Physics Award and obtained a Master’s degree in physics in December 2016. In January 2018, he relocated to Duke University with his advisor, becoming a transfer student. In close collaboration with Jungsang Kim and Chris Monroe’s groups, his work has been published in at least 4 papers and he presented his latest results in multiple conferences across the U.S. such as APS March Meeting and SQuInT. 137