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Towards the First Practical Applications of Quantum Computers Towards the First Practical Applications of Quantum Computers by Kevin J. Sung A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Computer Science and Engineering) in the University of Michigan 2020 Doctoral Committee: Professor Christopher Peikert, Co-Chair Adjunct Professor Yaoyun Shi, Co-Chair Dr. Ryan Babbush, Google Professor John P. Hayes Professor Emeritus Kim Winick Kevin J. Sung [email protected] ORCID iD: 0000-0001-6459-6374 ©2020 Kevin J. Sung To my family ii Acknowledgments This thesis was made possible by my supportive advisors Yaoyun Shi and Christopher Peikert, my friends, and my family. I would like to thank the Google AI Quantum team for hosting me while I performed much of the research in this thesis. I cannot express how lucky I feel to have had ac- cess to their state-of-the-art quantum computing hardware. The experiments on this hardware that I present in this thesis were the result of large team-wide collaborations. iii TABLE OF CONTENTS Dedication ....................................... ii Acknowledgments ................................... iii List of Figures ..................................... vii List of Tables ...................................... xiii List of Algorithms ................................... xv List of Appendices ................................... xvi Abstract ......................................... xvii Chapter 1 Introduction ..................................... 1 1.1 Quantum computing in the NISQ era....................1 1.1.1 The promise of quantum computing................1 1.1.2 Quantum computing in the present and near future........1 1.1.3 Challenges of NISQ computing..................2 1.2 Overview of results.............................3 1.2.1 Using models to improve optimizers for variational quantum al- gorithms..............................3 1.2.2 Quantum approximate optimization on a superconducting qubit processor..............................4 1.2.3 Preparing Slater determinants and fermionic Gaussian states...4 1.2.4 Hartree-Fock on a superconducting qubit processor........5 1.2.5 Generating certified random numbers on a superconducting qubit processor..............................5 1.3 Outline of thesis...............................6 1.3.1 Works appearing..........................6 2 Using Models to Improve Optimizers for Variational Quantum Algorithms .. 7 2.1 Introduction.................................7 2.2 Background.................................9 2.3 Problems studied and cost models..................... 10 2.3.1 Problems studied.......................... 10 iv 2.3.2 Cost models............................ 14 2.4 Optimization strategies........................... 17 2.4.1 Choice of optimizers........................ 17 2.4.2 Model Gradient Descent and Model Policy Gradient....... 19 2.4.3 Hyperparameter selection..................... 20 2.5 Results.................................... 21 2.5.1 The case of p = 1 and no gate errors................ 22 2.5.2 The case of p = 5 and no gate errors................ 24 2.5.3 The impact of rotation errors at p = 5 ............... 25 2.6 Conclusion................................. 26 3 Quantum Approximate Optimization on a Superconducting Qubit Processor . 28 3.1 Introduction................................. 28 3.2 The QAOA................................. 30 3.3 Compilation and problem families..................... 32 3.4 Comparisons with prior work........................ 35 3.5 Energy landscapes and optimization.................... 36 3.6 Hardware performance of QAOA...................... 38 3.7 Conclusion................................. 41 4 Preparing Slater Determinants and Fermionic Gaussian States ......... 43 4.1 Introduction................................. 43 4.2 Background................................. 44 4.2.1 Second quantization and the canonical anticommutation relations 44 4.2.2 Mapping fermions to qubits.................... 45 4.3 Algorithms for state preparation...................... 45 4.3.1 Preparing Slater determinants................... 45 4.3.2 Preparing fermionic Gaussian states................ 48 4.4 Conclusion................................. 55 5 Hartree-Fock on a Superconducting Qubit Processor .............. 57 5.1 Introduction................................. 57 5.2 Background................................. 59 5.2.1 The electronic structure problem.................. 59 5.2.2 The Pauli exclusion principle................... 60 5.2.3 Spatial orbitals and spin orbitals.................. 60 5.2.4 Slater determinants......................... 61 5.2.5 Second quantization........................ 62 5.2.6 The Hartree-Fock method..................... 63 5.3 Methods................................... 65 5.3.1 Representation........................... 65 5.3.2 Circuit ansatz............................ 65 5.3.3 Energy measurement........................ 65 5.3.4 Error mitigation.......................... 66 5.3.5 Optimization............................ 66 v 5.4 Results.................................... 68 5.5 Conclusion................................. 69 6 Generating Certified Random Numbers on a Superconducting Qubit Processor 73 6.1 Introduction................................. 73 6.2 Preliminaries................................ 75 6.2.1 Random quantum circuit sampling................. 75 6.2.2 Cross-entropy benchmarking (XEB)................ 76 6.2.3 Randomness extractors....................... 77 6.3 The computational hardness assumption.................. 77 6.4 The protocol................................. 78 6.5 Entropy estimation............................. 80 6.5.1 The case of an honest server.................... 80 6.5.2 Adjustments to the min-entropy bound.............. 81 6.6 Security and verification time........................ 82 6.7 Experimental implementation........................ 83 6.8 Conclusion................................. 83 Appendices ....................................... 86 Bibliography ...................................... 127 vi List of Figures 2.1 Optimization progress of SPSA in simulated experiments on a Sherrington- Kirkpatrick model Hamiltonian using two different hyperparameter settings: the ones used by default in the implementation from the software package Qiskit (Unoptimized), and ones that were found by searching for good settings (Optimized). The solid line represents the mean energy over 50 runs with dif- ferent PRNG seeds, and the shaded region represents a width of one standard deviation of the mean. The dotted lines are 10 example trajectories. The dot- ted gray line corresponds to the ansatz optimum. SPSA fails to converge with the unoptimized hyperparameters......................... 20 2.2 Wall clock time for optimization to achieve precision 1e-3 for the Sherrington- Kirkpatrick model at p = 1. Times are averaged over 50 experiments with different PRNG seeds. The black lines at the tips of the bars represent a width of one standard deviation. The best choice of optimizer can depend on the wall clock time model, with MGD, MPG, and SGD benefiting greatly from the ability to request execution of a batch of circuits............... 22 2.3 Success probability and time to solution for varying levels of required pre- cision at p = 5. Top: The probability of converging (out of 50 trials) to the optimal value of the ansatz at the given precision. Bottom: The average wall clock time the optimizer took to reach the given precision. Error bars represent 1 standard deviation. Time to solution is only reported if the prob- ability of convergence was at least 75% (dotted horizontal gray line). We see that Nelder-Mead and BOBYQA are the least likely to converge and often the slowest to converge when they do succeed. Meanwhile, MGD and MPG have the highest probability of converging as well as usually the fastest convergence times........................................ 24 2.4 Probability of convergence as a function of gate error level under a model of rotation error for the 3-regular graph model. Shown is the probability, over 50 trials with different PRNG seeds, of converging to within a precision of 5e-3, as a function of gate error level. Error bars represent one standard deviation. In this scenario, Nelder-Mead is the least resilient to this noise, while MPG is the most, and MGD follows............................ 25 vii 3.1 We studied three families of optimization problems: a. Hardware Grid prob- lems with a graph matching the hardware connectivity of the 23 qubits used in this experiment. b. MaxCut on random 3-regular graphs, with the largest instance depicted (22 qubits). c. The fully-connected Sherrington-Kirkpatrick (SK) model shown at the largest size (17 qubits). d. QAOA uses p applications of problem and driver unitaries to approximate solutions to optimization prob- lems. The parameters γ and β are shared among qubits in a layer but different for each of the p layers.............................. 29 3.2 a. The linear swap network can route a 17-qubit SK model problem unitary to n layers of nearest-neighbor two qubit interactions. b. The e−iγwZZ SWAP · interaction is a composite phasing and SWAP operation which can be synthe- sized from three applications of our hardware native entangling SYC and γw- dependent single-qubit rotations (yellow boxes). c. The definition of the SYC gate........................................ 33 3.3 Comparison of simulated (left) and experimental (right) p = 1 landscapes, with a clear correspondence of landscape features. An overlaid optimization trace
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