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UC Berkeley UC Berkeley Electronic Theses and Dissertations UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Applications of Near-Term Quantum Computers Permalink https://escholarship.org/uc/item/2fg3x6h7 Author Huggins, William James Publication Date 2020 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Applications of Near-Term Quantum Computers by William Huggins A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Chemistry in the Graduate Division of the University of California, Berkeley Committee in charge: Professor K. Birgitta Whaley, Chair Professor Martin Head-Gordon Professor Joel Moore Summer 2020 Applications of Near-Term Quantum Computers Copyright 2020 by William Huggins 1 Abstract Applications of Near-Term Quantum Computers by William Huggins Doctor of Philosophy in Chemistry University of California, Berkeley Professor K. Birgitta Whaley, Chair Quantum computers exist today that are capable of performing calculations that challenge the largest classical supercomputers. Now the great challenge is to make use of these devices, or their successors, to perform calculations of independent interest. This thesis focuses on a family of algorithms that aim to accomplish this goal, known as variational quantum algorithms, and the challenges that they face. We frame these challenges in terms of two kinds of resources, the number of operations that we can afford to perform in a single quantum circuit, and the overall number of circuit repetitions required. Before turning to our specific contributions, we provide a review of the electronic structure problem, which serves as a central example for our work. We also provide a self-contained introduction to the field of quantum computing and the notion of a variational quantum algorithm. We begin the main body of the thesis by studying a generalization of the standard unitary coupled cluster ansatz. We present a sparse version of unitary coupled cluster and show that it can accurately represent the ground and excited states of small molecular systems using fewer quantum gates than the full unitary coupled cluster. We then introduce a strategy for representing molecular ground states as linear combinations of parameterized wavefunctions, allowing for a tradeoff between the number of operations required for each circuit and the number of circuit repetitions. We provide circuit primitives that allow for the efficient measurement of the required matrix elements between these wavefunctions. Subsequently, we show how the cost of estimating the energy of a quantum chemical wavefunction on a near-term quantum computer can be dramatically reduced by using a factorization of the two-electron integral tensor. Furthermore, we explain how this measurement strategy helps mitigate against errors during state preparation and measurement. We then present a Monte Carlo version of a classical algorithm for calculating the partition function of two-dimensional lattice models that shows a similar kind of tradeoff between two resources, albeit in a different context. Finally, we show how ideas based on tensor networks can inform the design of quantum circuits for machine learning tasks. We argue that this application is especially 2 tolerant to noise and present numerical data substantiating this argument. We conclude by explaining explicitly how our work addresses the problems of the limited resources available to near-term quantum computers and offering some optimism about the future of the field. i Dedicated to my parents, John and Diane, my brother Bobby, and my partner Hilary Your support and encouragement has made this dissertation, and so much else in my life, possible. ii Contents Contents ii List of Figures vi List of Tables xiv 1 Introduction1 1.1 Noisy Intermediate-Scale Quantum Computing................. 2 1.2 Outline....................................... 4 2 The Electronic Structure Problem6 2.1 Useful Approximations.............................. 7 2.2 Fermionic Wavefunctions............................. 9 2.2.1 First Quantization ............................ 9 2.2.1.1 Second Quantization...................... 10 2.2.1.2 Properties of Quantum Chemical Hamiltonians and Wave- functions ............................ 12 2.3 Classical Techniques and Their Limitations................... 13 2.3.0.1 Hartree-Fock and Beyond ................... 14 2.3.0.2 Configuration Interaction and Exact Diagonalization . 16 2.3.0.3 Coupled Cluster ........................ 17 2.4 The Promise of Quantum Computing for Quantum Chemistry . 18 3 Noisy Intermediate-Scale Quantum Computing 22 3.1 The Formalism of Quantum Computing..................... 24 3.1.1 Quantum Gates in the Circuit Model.................. 24 3.1.2 Measurement ............................... 26 3.1.3 The Quantum Phase Estimation Algorithm .............. 27 3.2 Near-Term Hardware............................... 29 3.3 The Noise in \Noisy"............................... 32 3.4 Variational Quantum Algorithms ........................ 35 3.5 The Variational Quantum Eigensolver...................... 37 iii 3.5.1 Fermionic Wavefunctions on Qubits................... 38 3.5.2 Ansatz Design............................... 41 3.5.3 Repeated State Preparation and Measurement............. 42 3.5.4 Optimization ............................... 44 3.6 Some Challenges for Variational Quantum Algorithms............. 46 4 Generalized Unitary Coupled Cluster Wavefunctions for Quantum Chem- istry on a Quantum Computer 49 4.1 Preface....................................... 49 4.2 Introduction.................................... 49 4.3 Theory....................................... 51 4.3.1 Coupled-Cluster Theory ......................... 51 4.3.1.1 Traditional Coupled Cluster.................. 51 4.3.1.2 Unitary CC........................... 52 4.3.1.3 Generalized CC......................... 53 4.3.2 Generalized Unitary CC......................... 54 4.3.2.1 Unitary Pair CC with Generalized Singles and Doubles Prod- uct Wavefunctions ....................... 54 4.3.3 Excited State Algorithms......................... 55 4.3.3.1 Previous Work ......................... 55 4.3.3.2 Orthogonally Constrained VQE................ 56 4.3.3.3 Energy Error Analysis of OC-VQE.............. 57 4.4 Quantum Resource Requirements........................ 58 4.4.1 Quantum implementation of Overlap Measurements.......... 59 4.5 Benchmark implementations on a Classical Computer............. 60 4.5.1 Computational Details.......................... 60 4.5.2 Applications to Chemical Systems.................... 61 4.5.2.1 H4(in D4h and D2h symmetry)................. 61 4.5.2.2 Double Dissociation of H2O (C2v)............... 65 4.5.2.3 Dissociation of N2 ....................... 66 4.5.2.4 Discussion of Excited State Energies............. 68 4.5.2.5 Summary of Chemical Applications.............. 69 4.6 Summary and Outlook.............................. 70 4.7 Additional Computational Details........................ 71 5 A Non-Orthogonal Variational Quantum Eigensolver 76 5.1 Preface....................................... 76 5.2 Introduction.................................... 76 5.3 Theory....................................... 79 5.3.1 Matrix Element Measurement...................... 79 5.3.2 Diagonalization With Uncertainty.................... 82 5.3.3 Experiment Design Heuristic....................... 84 iv 5.3.4 Implementation.............................. 84 5.3.4.1 The k-UpCCGSD Ansatz ................... 85 5.3.4.2 Computational Details..................... 86 5.4 Results....................................... 86 5.4.1 NOVQE Ground State Energies..................... 87 5.4.1.1 A Hydrogen Complex, H4 ................... 87 5.4.1.2 Hexatriene ........................... 88 5.4.2 NOVQE Matrix Element Measurements ................ 90 5.4.2.1 A Hydrogen Complex, H4 ................... 91 5.4.2.2 Hexatriene ........................... 92 5.5 Discussion and Outlook ............................. 94 5.6 Additional Computational Details........................ 96 5.7 Hexatriene Geometries.............................. 98 6 Efficient and Noise Resilient Measurement for Quantum Chemistry on a Quantum Computer 100 6.1 Preface....................................... 100 6.2 Introduction.................................... 100 6.3 Results....................................... 103 6.3.1 Using Hamiltonian Factorization for Measurements . 103 6.3.2 Circuit Repetitions Required for Energy Measurement . 106 6.3.3 Error Mitigation ............................. 109 6.4 Discussion..................................... 115 6.5 Variance Bounds ................................. 116 6.6 Applying the fermionic RDM Constraints to the Qubit Hamiltonian . 122 6.7 Low Rank Decomposition ............................ 123 6.8 Additional Computational Details........................ 124 7 Monte Carlo Approaches to the Tensor Renormalization Group 127 7.1 Preface....................................... 127 7.2 Tensor Network Background Material...................... 127 7.3 Introduction.................................... 131 7.4 Numerical Results................................. 136 7.5 Discussion..................................... 138 7.6 Additional Computational Details........................ 140 8 Towards Quantum Machine Learning with Tensor Networks 142 8.1 Preface....................................... 142 8.2 Machine Learning Background Material..................... 142 8.3 Introduction ................................... 144 8.4 Learning
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