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Control, Gates, and Error Suppression with Hamiltonians in Quantum Computation by MASSACHUSETTS INSTITUTE OF TECHNOLOGY Adam Darryl Bookatz JUN 09 2016 Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of LIBRARIES Doctor of Philosophy in Physics ARCHIVES at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2016 @ Massachusetts Institute of Technology 2016. All rights reserved. Signature redacted Author ............ Department of Physics May 12, 2016 Signature redacted Certified by........ ........................................................ Edward Farhi Cecil and Ida Green Professor of Physics; Director, Center for Theoretical Physics Thesis Supervisor Accepted by ............. Signature redacted Nergis Mavalvala Curtis and Kathleen Marble Professor of Astrophysics Associate Department Head for Education, Physics 2 Control, Gates, and Error Suppression with Hamiltonians in Quantum Computation by Adam Darryl Bookatz Submitted to the Department of Physics on May 12, 2016, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Abstract In this thesis we are primarily interested in studying how to suppress errors, perform sim- ulation, and implement logic gates in quantum computation within the context of using Hamiltonian controls. We also study the complexity class QMA-complete. We first investigate a method (introduced by Jordan, Farhi, and Shor) for suppress- ing environmentally induced errors in Hamiltonian-based quantum computation, involving encoding the system with a quantum error-detecting code and enforcing energy penalties against leaving the codespace. We prove that this method does work in principle: in the limit of infinitely large penalties, local errors are completely suppressed. We further derive bounds for the finite-penalty case and present numerical simulations suggesting thatthe method achieves even greater protection than these bounds indicate. We next consider the task of Hamiltonian simulation, i.e. effectively changing a sys- tem Hamiltonian to some other desired Hamiltonian by applying external time-dependent controls. We propose protocols for this task that rely solely on realistic bounded-strength control Hamiltonians. For systems coupled to an uncontrollable environment, our approach may be used to perform simulation while simultaneously suppressing unwanted decoherence. We also consider the scenario of removing unwanted couplings in many-body quantum systems obeying local system Hamiltonians and local environmental interactions. We present protocols for efficiently switching off the Hamiltonian of a system, i.e. simulating thezero Hamiltonian, using bounded-strength controls. To this end, we introduce the combinatorial concept of balanced-cycle orthogonal arrays, show how to construct them from classical error-correcting codes, and show how to use them to decouple n-qudit `-local Hamiltonians ` 1 using protocols of length at most O(n − log n). We then present a scheme for implementing high-fidelity quantum gates using afew interacting bosons obeying a Bose-Hubbard Hamiltonian on a line. We find high-fidelity logic operations for a gate set (including the cnot gate) that is universal for quantum information processing. Lastly, we discuss the quantum complexity class QMA-complete, surveying all known such problems, and we introduce the “quantum non-expander” problem, proving that it is QMA-complete. A quantum expander is a type of rapidly-mixing quantum channel; we show that estimating its mixing time is a co-QMA-complete problem. Thesis Supervisor: Edward Farhi Title: Cecil and Ida Green Professor of Physics; Director, Center for Theoretical Physics 3 4 Acknowledgements First and foremost, I would like to thank my wife, Josepha, for her steadfast support and encouragement and for cheerfully putting up with having a graduate student for a husband. I am especially grateful to my advisor, Eddie Farhi, for being a great thesis supervisor, giving me both direction and freedom, as well as exemplifying how to write academic papers with precision and clarity. I also especially thank Pawel Wocjan, for being like a second advisor to me, sharing much wisdom – in science, career, and life – and working on many projects with me. In addition, I greatly appreciate the help of all of my collaborators, Yoav Lahini, Martin Roetteler, Leo Zhou, Greg Steinbrecher, Stephen Jordan, Yi-Kai Liu, Lorenza Viola, and Dirk Englund. Thank you to my thesis committee members, Edward Farhi, Aram Harrow, and Boleslaw Wyslouch for carefully reading and checking my thesis, thereby comforting me that at least someone has read it. I feel privileged to have been able to attend MIT, with its strong series of quantum information courses, and I thank my teachers, Professors Isaac Chuang, Scott Aaronson, and Peter Shor for their classes from which I benefited greatly. The help, advice, and discussions with MIT postdocs and students has also been invaluable, and I want to thank Shelby Kimmel, Cedric Lin, Han-Hsuan Lin, David Gosset, Lior Eldar, Kristen Temme, Iman Marvian, as well as professors Barton Zwiebach, Aram Harrow, Seth Lloyd, and Sam Gutmann. I thank the many people of the faculty and staff of the CTP, the Department of Physics, and MIT who have helped me and taught me, and I also thank the MIT Rowing Club for helping me ward off physical atrophy long enough to write this thesis. Lastly, I thank my parents, Brian and Sandra, for their continuous support throughout my entire life, and my brothers, David and Gidon, for being great brothers and keeping me suitably distracted when appropriate; I am truly fortunate to have such a family. תושלב00ע 5 6 Contents 0 Thesis introduction 11 0.1 Outline . 13 1 Background material 17 1.1 Quantum mechanics . 17 1.1.1 Quantum states . 17 1.1.2 Measurement . 18 1.1.3 Hamiltonians and evolution . 18 1.1.4 Composite systems . 21 1.1.5 Mixed states . 24 1.1.6 Bosonic systems . 26 1.2 Hamiltonian-based quantum computing . 27 1.2.1 Adiabatic quantum computation . 28 1.2.2 Feynman’s model . 29 1.2.3 Continuous-time quantum walks . 30 1.2.4 Analogue Hamiltonian simulation and algorithms . 30 1.3 Circuit model . 32 1.3.1 Classical circuits . 32 1.3.2 Quantum circuits . 33 1.4 Complexity theory . 37 1.4.1 Brief mathematical background . 37 1.4.2 Complexity classes . 38 1.4.3 Hamiltonian complexity and QMA-completeness . 40 1.5 Error-correcting codes . 41 1.5.1 Classical linear error-correcting codes . 41 1.5.2 Quantum error-detecting/correcting codes . 43 Chapter bibliography . 47 2 Error suppression in Hamiltonian-based quantum computation using energy penalties 49 2.1 Introduction . 49 2.2 Quantum error-detecting codes . 51 2.3 The Hamiltonian model and energy penalties . 52 2.4 Error suppression through energy penalties . 54 2.4.1 The infinite EP case . 54 2.4.2 The finite EP case . 58 2.5 Numerical simulation for one logical qubit . 63 7 2.6 Outlook . 73 2.7 Conclusion . 75 2.8 Afterword . 75 Chapter appendices . 77 2.A Beyond 1-local errors . 77 Chapter bibliography . 79 3 Hamiltonian quantum simulation with bounded-strength controls 81 3.1 Introduction . 82 3.2 Principles of Hamiltonian simulation . 83 3.2.1 Control-theoretic framework . 83 3.2.2 Hamiltonian simulation with bang-bang controls . 87 3.3 Hamiltonian simulation with bounded controls . 88 3.3.1 Eulerian simulation of the trivial Hamiltonian . 88 3.3.2 Eulerian simulation protocols beyond no-op ............... 90 3.3.3 Simple two-qubit example . 93 3.3.4 Eulerian simulation while decoupling from an environment . 94 3.3.5 Eulerian simulation protocol requirements . 96 3.4 Illustrative applications . 99 3.4.1 Eulerian simulation in closed Heisenberg-coupled qubit networks . 99 3.4.2 Error-corrected Eulerian simulation in open Heisenberg-coupled qubit networks . 100 3.4.3 Eulerian simulation of Kitaev’s honeycomb lattice Hamiltonian . 102 3.5 Conclusion and outlook . 105 Chapter bibliography . 107 4 Improved bounded-strength decoupling schemes for local Hamiltonians 113 4.1 Introduction . 114 4.2 Description of the control-theoretic model . 115 4.3 Balanced cycles . 117 4.4 Balanced-cycle Orthogonal Arrays . 121 4.5 Construction of balanced-cycle orthogonal arrays . 124 4.6 BOA decoupling schemes from BCH codes . 127 4.7 Tables of best known BOA schemes for small systems . 129 4.8 Examples . 131 4.9 Conclusion . 135 Chapter bibliography . 137 5 Quantum logic with interacting bosons in 1D 139 5.1 Introduction . 139 5.2 Defining qubits on a lattice . 141 5.3 Implementing quantum gates . 141 5.3.1 Single-qubit gates . 142 5.3.2 cnot gate . 143 5.4 Compiling a three-qubit primitive . 145 5.5 Computational methods . 146 5.6 Conclusions . 147 Chapter appendices . 149 8 5.A Noise analysis . 149 5.B Quantum process tomography for cnot ..................... 151 Chapter bibliography . 153 6 Testing quantum expanders is co-QMA-complete 157 6.1 Introduction . 157 6.2 Preliminaries . 158 6.2.1 The quantum non-expander problem . 158 6.2.2 Thermalization of open quantum systems . 159 6.2.3 Quantum Merlin-Arthur . 161 6.3 Quantum non-expander is in QMA . 162 6.4 Some technical tools . 163 6.4.1 The Frobenius norm . 163 6.4.2 Controlled expanders . 164 6.5 Quantum non-expander is QMA-hard . 166 6.5.1 Outline of the proof . ..

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