Quantum Computation Beyond the Circuit Model by Stephen Paul Jordan Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2008 c Stephen Paul Jordan, MMVIII. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author............................................. ............................... Department of Physics May 2008 Certified by.......................................... .............................. Edward H. Farhi Professor of Physics Thesis Supervisor arXiv:0809.2307v1 [quant-ph] 13 Sep 2008 Accepted by......................................... .............................. Thomas J. Greytak Professor of Physics 2 Quantum Computation Beyond the Circuit Model by Stephen Paul Jordan Submitted to the Department of Physics on May 2008, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Abstract The quantum circuit model is the most widely used model of quantum computation. It provides both a framework for formulating quantum algorithms and an architecture for the physical construction of quantum computers. However, several other models of quantum computation exist which provide useful alternative frameworks for both discovering new quantum algorithms and devising new physical implementations of quantum computers. In this thesis, I first present necessary background material for a general physics audience and discuss existing models of quantum computation. Then, I present three new results relating to various models of quantum computation: a scheme for improving the intrinsic fault tolerance of adiabatic quantum computers using quantum error detecting codes, a proof that a certain problem of estimating Jones polynomials is complete for the one clean qubit complexity class, and a generalization of perturbative gadgets which allows k-body interactions to be directly simulated using 2-body interactions. Lastly, I discuss general principles regarding quantum computation that I learned in the course of my research, and using these principles I propose directions for future research. Thesis Supervisor: Edward H. Farhi Title: Professor of Physics 3 4 Acknowledgments I first wish to thank the people who have participated most directly in my formation as a quantum information scientist. At the top of this list is Eddie Farhi, who I thank for offering me invaluable help on matters both scientific and logistical, and for being a pleasure to work with. Without him, my graduate school experience would not have been as rich and rewarding as it was. I thank Peter Shor for being always willing to chat about any scientific topic, being a frequent collaborator, serving on my general exam and thesis committees, and teaching quantum computation courses that I greatly enjoyed. In many ways, I feel that he is like a second advisor to me. I thank Isaac Chuang for serving on my general exam and thesis committees, teaching quantum computation courses which I greatly benefitted from, and for interesting conversations. I thank Seth Lloyd for interesting conversations and for serving on my general exam committee. I thank Al´an Aspuru-Guzik at Harvard for bringing me in on his chemical dynamics project, and for spreading his enthusiasm for research. I also thank each of these people for aiding me in obtaining a postdoctoral position. I wish to thank those who have collaborated with me directly on papers, some of which form much of the content of this thesis: Eddie Farhi, Peter Shor, Al´an Aspuru-Guzik, Ivan Kassal, Peter Love, Masoud Mohseni, David Yeung, and Richard Cleve. I am also grateful to Richard Cleve for many long and interesting conversations about quantum com- putation. I thank John Preskill, Howard Barnum, Lov Grover, Daniel Lidar, Joe Traub, and the people of Perimeter Institute for inviting me to visit them, and having interesting conversations when I did. I am grateful to many people for having interesting conversa- tions that shaped my thoughts on quantum computing, and for being helpful in various ways. These include Scott Aaronson, Dave Bacon, Jacob Biamonte, Andrew Childs, Wim van Dam, David DiVincenzo, Pavel Etingof, Steve Flammia, Andrew Fletcher, Joe Giraci, Jeffrey Goldstone, Daniel Gottesman, Sam Gutmann, Aram Harrow, Elham Kashefi, Jor- dan Kerenidis, Ray Laflamme, Mike Mosca, Andrew Landahl, Debbie Leung, Michael Levin, William Lopes, Carlos Mochon, Shay Mozes, Markus Mueller, Daniel Nagaj, Ashwin Nayak, Robert Raussendorf, Mark Rudner, Rolando Somma, Madhu Sudan, Jake Taylor, David Vogan, and Pawel Wocjan. I wish to thank the people and organizations who have supported me and my research. I thank the society of presidential fellows for supporting me during my first year at MIT. I thank Isaac Chuang and Ulrich Becker for giving me the opportunity to TA for their junior lab course during my second year. I thank Mark Heiligman, T.R. Govindan, Mel Currie, and all the other people of ARO and DTO for supporting me for the rest of my time at graduate school as a QuaCGR fellow. I thank Senthil Todadri and Alan Guth for serving as academic advisors. I thank the US Department of Energy for funding MIT’s center for theoretical physics, where I work, and I thank the administrative staff of CTP, Scott Morley, Charles Suggs, and Joyce Berggren for being so helpful and making CTP a functional and pleasant place to work. Lastly, but by no means least importantly I wish to thank those people who have con- tributed to this thesis indirectly by contributing to my development in general: my parents Eric and Janet, my wife Sara, my sisters Katherine and Elizabeth, my undergraduate re- search advisors, Moses Chan, Rafael Garcia, and Vincent Crespi, my high school physics and chemistry teacher Kevin McLaughlin, and all of the friends and teachers who I have been lucky enough to have. 5 6 Contents 1 Introduction 9 1.1 Classical Computation Preliminaries . ....... 9 1.2 Quantum Computation Preliminaries . ..... 17 1.3 QuantumAlgorithms............................... 24 1.3.1 Introduction ............................... 24 1.3.2 Algebraic and Number Theoretic Problems . 27 1.3.3 OracularProblems ............................ 31 1.3.4 Approximation and BQP-complete Problems . 35 1.3.5 Commentary ............................... 37 1.4 What makes quantum computers powerful? . 38 1.5 FaultTolerance.................................. 40 1.6 ModelsofQuantumComputation. 42 1.6.1 Adiabatic ................................. 42 1.6.2 Topological ................................ 44 1.6.3 QuantumWalks ............................. 45 1.6.4 OneCleanQubit............................. 45 1.6.5 Measurement-based ........................... 46 1.6.6 QuantumTuringMachines . 47 1.7 OutlineofNewResults............................. 47 2 Fault Tolerance of Adiabatic Quantum Computers 49 2.1 Introduction.................................... 49 2.2 ErrorDetectingCode .............................. 50 2.3 NoiseModel.................................... 52 2.4 HigherWeightErrors.............................. 54 3 DQC1-completeness of Jones Polynomials 57 3.1 Introduction.................................... 57 3.2 OneCleanQubit ................................. 57 3.3 JonesPolynomials ................................ 60 3.4 Fibonacci Representation . 63 3.5 ComputingtheJonesPolynomialinDQC1 . 65 3.6 DQC1-hardness of Jones Polynomials . ..... 68 3.7 Conclusion .................................... 72 3.8 Jones Polynomials by Fibonacci Representation . ......... 73 3.9 Density of the Fibonacci representation . ....... 77 3.10 Fibonacci and Path Model Representations . ....... 80 7 3.11 Unitaries on Logarithmically Many Strands . ........ 81 3.12 Zeckendorf Representation . ..... 88 4 Perturbative Gadgets 91 4.1 Introduction.................................... 91 4.2 PerturbationTheory .............................. 94 4.3 Analysis of the Gadget Hamiltonian . 95 4.4 NumericalExamples ............................... 99 4.5 Derivation of Perturbative Formulas . 100 4.6 Convergence of Perturbation Series . 103 5 Multiplicity and Unity 107 5.1 Multiplicity .................................... 107 5.2 Unity ....................................... 112 A Classical Circuit Universality 115 B Optical Computing 117 C Phase Estimation 121 D Minimizing Quadratic Forms 125 E Principle of Deferred Measurement 127 F Adiabatic Theorem 129 F.1 MainProof .................................... 129 F.2 SupplementaryCalculation . 130 8 Chapter 1 Introduction 1.1 Classical Computation Preliminaries This thesis is about quantum algorithms, complexity, and models of quantum computation. In order to discuss these topics it is necessary to use notations and concepts from classical computer science, which I define in this section. The “big-O” family of notations greatly aids in analyzing both classical and quantum algorithms without getting mired in minor details. Although it may seem like a trivial notation, it is the first step in a chain of increasing abstraction which allows computer scientists to analyze the general laws of computation which apply whether the computer is using base 2 or base 20, and whether it is made of transistors or tinker toys. By following this chain we will reach the major open questions about complexity classes and their relations to one another. The big-O notation is defined as follows. Definition 1. Given