Alternative Models for Quantum Computation by Cedric Yen-Yu Lin MASSACHUSETTS INSTITUTE OF TECHNOLOLGY Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of JUN 3 0 2015 Doctor of Philosophy in Physics LIBRAR IES at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2015 @ Massachusetts Institute of Technology 2015. All rights reserved. Signature redacted A uthor .................. .... (V 4/ Department of Physics May 22, 2015 Signature redacted Certified by............ Elward H. Farhi Professor of Physics; Director, Center for Theoretical Physics Thesis Supervisor Signature redacted Accepted by............. Nergis Mavalvala Curtis and Kathleen Marble Professor of Astrophysics Associate Head for Education, Physics 2 Alternative Models for Quantum Computation by Cedric Yen-Yu Lin Submitted to the Department of Physics on May 22, 2015, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Abstract We propose and study two new computational models for quantum computation, and in- fer new insights about the circumstances that give quantum computers an advantage over classical ones. The bomb query complexity model is a variation on the query complexity model, inspired by the Elitzur-Vaidman bomb tester. In this model after each query to the black box the result is measured, and the algorithm fails if the measurement gives a 1. We show that the bomb query complexity is asymptotically the square of the usual quantum query complexity. We then show a general method of converting certain classical algorithms to bomb query algorithms, which then give improved quantum algorithms. We apply this general method to graph problems, giving improved quantum query algorithms for single-source shortest paths and maximum bipartite matching. Normalizer circuits are a class of restricted quantum circuits defined on Hilbert spaces associated with Abelian groups. These circuits generalize the Clifford group, and are com- posed of gates implementing quantum Fourier transforms, automorphisms, and quadratic phases. We show that these circuits can be simulated efficiently on a classical computer even on infinite Abelian groups (the finite case is known [1, 21), as long as the group is decomposed into primitve subgroups. This result gives a generalization of the Gottesman-Knill theorem to infinite groups. However, if the underlying group is not decomposed (the group is a black box group) then normalizer circuits include many well known quantum algorithms, includ- ing Shor's factoring algorithm. There is therefore a large difference in computational power between normalizer circuits over explicitly decomposed versus black box groups. In fact, we show that a version of the problem of decomposing Abelian groups is complete for the com- plexity class associated with normalizer circuits over black box groups: any such normalizer circuit can be simulated classically given the ability to decompose Abelian groups. Thesis Supervisor: Edward H. Farhi Title: Professor of Physics; Director, Center for Theoretical Physics 3 4 Acknowledgments I have many people to thank, without whom completing this thesis would be impossible. First and foremost, I thank Eddie Farhi for being a wonderful advisor and collaborator; for giving me guidance on research yet allowing me plenty of freedom; and for teaching me how to be inquisitive, how to think about quantum computation, and how to do science. I thank Peter Shor for being like a second advisor to me, for his frequent help and guidance in my research and being a frequent collaborator. I thank Aram Harrow and Scott Aaroson for frequent discussions and answering many questions, and also for inspiring me how to think about quantum (and classical) computation. I thank Eddie Farhi, Aram Harrow, and David Kaiser for the time and effort they have spent on reviewing my thesis and thesis defense. I also thank Aram, Peter, and Ike Chuang for serving on my general exam committee. I thank Robert Rautendorf and Ian Affleck for their support and collaboration at the beginning of my research career as an undergraduate, and also for introducing me to the world of quantum computation. I thank Han-Hsuan Lin, Juan Bermejo-Vega, Maarten Van den Nest, Elizabeth Crosson, and Shelby Kimmel for being great collaborators and friends. I especially thank Han-Hsuan and Juan for years of working together, and Shebly for her invaluable help when I was applying for a postdoctoral position. I thank Ike Chuang, Scott Aaronson, and Jon Kelner for their phenomenal classes on quantum and classical computation. In particular I thank Ike for the term project in his class, which eventually lead to Chapter 5 of this thesis. I thank David Gosset and Andy Lutomirski for helping me get used to graduate life at MIT. I thank Alex Arkhipov, Mohammad Bavarian, Adam Bookatz, Adam Bouland, An- drew Childs, Richard Cleve, Matt Coudron, Lior Eldar, David Gosset, Daniel Gottesman, Jeongwan Haah, Stephen Jordan, Robin Kothari, Chris Laumann, Andy Lutomirski, Ashley Montanaro, Daniel Nagaj, Anand Natarajan, Cyril Stark, Pawel Wocjan, and Henry Yuen for many useful discussions. I thank the staff of the CTP, LNS, and the Department of Physics for their help over the years, and for making our lives so much easier. Special thanks to Scott Morley, Joyce Berggren, and Charles Suggs for keeping the CTP as nice as it is. I thank NSERC and the ARO for supporting me financially so that I could concentrate on research. I thank my friends for their support and friendship, and for all the good times we've had together. Lastly, I thank my family for always being there when I need them. I thank my par- ents Carol and Wen-Chien, my brother Terence, and my grandparents for their unwavering support and encouragement throughout the years. I am lucky to have them in my life. 5 6 Contents 1 Introduction 11 1.1 Classical Computation ...... ...... ....... 1.1.1 Turing and the theory of computability ..... ...... 13 1.1.2 Complexity theory .......... ....... ...... 15 1.2 Quantum Computation ..... .......... .... ...... 18 1.3 Shor's Algorithm for Factoring ........ ...... .... .. 18 1.3.1 The Quantum Fourier Transform .... ..... ...... 20 1.3.2 Shor's Algorithm for Order Finding and Factoring .... .. 218 1.3.3 Hidden Subgroup Problem ... ...... .... ..... 22 1.4 Query Complexity and Grover's Algorithm . ....... ...... 22 1.4.1 Black Boxes and Grover's Problem ...... .. 1.4.2 Grover's algorithm ........ ......... ...... 23 1.4.3 Query Complexity .... ......... .... ...... 24 1.5 Organization of this thesis ..... ....... ..... ...... 26 2 Upper Bounds for Quantum Query Complexity Base I on the Elitzur- Vaidman Bomb Tester 29 2.1 Introduction ......... .......... ...... .. ... 29 2.2 The Elitzur-Vaidman bomb testing problem ....... .. .. 31 2.3 Bomb query complexity .. ....... ...... ... .. ..... 32 2.4 M ain result ..... ....... ...... ....... ..... 34 2.4.1 Upper bound ...... ...... ...... .. .. ..... 34 2.4.2 Lower bound .. ......... ......... .. ..... 37 2.5 Generalizations and Applications ..... ....... ... .. 37 2.5.1 Generalizing the bomb query model .... ... .. .. ..... 38 2.5.2 Using classical algorithms to design bomb query algorithms . ... 39 2.5.3 Explicit quantum algorithm for Theorem 24 . ....... .. ..... 40 2.6 Improved upper bounds on quantum query complexity . ..... ..... 42 2.6.1 Single source shortest paths for unweighted graphs .... ..... 42 2.6.2 Maximum bipartite matching .. ....... ....... .. .. 44 2.7 Projective query complexity . ........ ....... ..... ..... 46 3 Normalizer circuits over infinite-dimensional systems: an introduction 49 3.1 Introduction ..... ..... ...... ..... ..... ..... ..... 49 3.2 Outline of this chapter ... ...... ....... ...... ....... 52 3.3 Summary of concepts . ..... ...... ..... ..... ..... .... 52 3.3.1 T he setting . ..... ..... ..... ..... .... ..... .. 52 7 3.3.2 Examples .................................. 52 3.4 Abelian groups ............. ...................... 55 3.4.1 Z: the group of integers . ............. ............ 55 3.4.2 1T: the circle group ............................. 55 3.4.3 Finite Abelian groups ........................... 55 3.4.4 Black box groups ................... ........... 56 3.5 The Hilbert space of a group ......... ............. ..... 56 3.5.1 Finite Abelian groups ..................... ...... 56 3.5.2 The integers Z ............................... 57 3.5.3 Total Hilbert space ............................. 58 3.6 Normalizer circuits (without black boxes) ....... ............. 59 3.6.1 Normalizer gates .............................. 59 3.6.2 Normalizer circuits ............................. 61 3.6.3 Classical encodings of normalizer gates ....... ........... 62 3.7 Normalizer circuits over black box groups ............. ....... 63 3.8 Group and Character Theory .............. ............. 66 3.8.1 Elementary Abelian groups ........................ 66 3.8.2 Characters ........ ............. ............ 67 3.8.3 Simplifying characters via the bullet group ............ .... 69 3.8.4 Annihilators ........... ............. ......... 70 3.9 Homomorphisms and matrix representations ... .............. .. 70 3.9.1 Homomorphisms ......... ..................... 70 3.9.2 Matrix representations . ......................... 71 3.10 Quadratic functions ... ............. .............. .. 73 3.10.1 Definitions ............. ............. ....... 74 3.10.2 Normal form of bicharacters .................. ...... 74 3.10.3 Normal form of quadratic functions