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. !k The k-th sideband frequency. δk(t) Detuning between the driving frequency and !k 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. α^k Spin-dependent motional displacement of the k-th sideband. βij Entanglement between ions i and j. xi Quantum Circuit Calibration ~" 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 ~". 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 ~" 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.