Three Complexity Classification Questions at the ITUTE Quantum/Classical Boundary OFTENO by OCT 0 3 2019 Daniel Grier LIBRARIES ARCHIVES Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2019 @ Massachusetts Institute of Technology 2019. All rights reserved. Signature redacted A u th o r ................................................................ Department of Electrical Engineering and Computer Science August 30, 2019 redacted Certified by.. Signature Scott Aaronson Professor of Computer Science, The University of Texas at Austin Thesis Supervisor Signature redacted A ccepted by ............... ............... Ac b /Leslie A. Kolodziejski Professor of Electrical Engineering and Computer Science Chair, Department Committee on Graduate Students 2 Three Complexity Classification Questions at the Quantum/Classical Boundary by Daniel Grier Submitted to the Department of Electrical Engineering and Computer Science on August 30, 2019, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract A central promise of quantum computers is their ability to solve some problems dramatically more efficiently than their classical counterparts. Thus, to understand feasible computation in our physical world, we must turn to quantum rather than classical complexity theory. That said, classical complexity theory has a long and successful history of developing tools and techniques to analyze the power of various computing models. Can we use classical complexity theory to aid our understanding of the quantum world? As it turns out, the answer is yes. There is actually a very fruitful connection between quantum and classical complexity theory, each field informing the other. We will add to this perspective through the lens of classification-attempts to categorize all variations of the object of study as thoroughly and completely as possible. First, we will show that every regular language has quantum query complexity E(1), 5(Vi), or O(n). Combining quantum query complexity with these fundamental classical languages not only reveals new structure in these languages, but also leads to a generalization of Grover's famous quantum search algorithm. Second, we will discuss the complexity of computing the permanent over various matrix groups. In particular, this will show that computing the permanent of a unitary matrix is #P-hard. The theorem statement is classical, and yet, the proof is almost entirely the result of exploiting well-known theorems in quantum linear optics. Finally, we give a complete classification of Clifford operations over qubits. Although the Clifford operations are classically simulable, they also exhibit distinct quantum behavior, making them a particularly interesting gate set at the quantum/classical boundary. Thesis Supervisor: Scott Aaronson Title: Professor of Computer Science, The University of Texas at Austin 3 4 Acknowledgments I should probably start my acknowledgments by stating how inadequate they will be in expressing my gratitude for all the people who have contributed to my graduate school experience. My advisor Scott Aaronson might refer to that sentiment as "forehead- bangingly obvious," but sometimes you just have to set the record straight. Of course, Scott is on the top of my list when it comes to people to thank. For all my years, Scott has been an academic heavyweight role model. He bleeds interesting research ideas, maintains a highly-successful blog, has two rambunctious children, all while being one of the most supportive advisors I've seen in the business. Even when forces drew him towards UT Austin during my third year, he fought to keep me financially supported at MIT as long as possible. I will always be grateful to have had Scott as an advisor, and I hope to never lose his infectious passion for science. On the topic of advisors, I would also like to thank my undergraduate advisor Stephen Fenner at the University of South Carolina for introducing me to computa- tional complexity theory, and more generally to the world of research in theoretical computer science. It seems rather unlikely that I'd be sitting here typing these ac- knowledgments in this thesis were it not for him giving me some problem about the complexity of a strange combinatorial game many years ago. I will always appreciate his patience and encouragement during these early research years, and he remains a friend and collaborator to this day. There are many faculty members at MIT who helped me somewhere along the way. First, I would like to thank both Aram Harrow and Ryan Williams for serving on my thesis committee. I would also like to thank those professors who helped to fund my studies through a TA position after Scott moved. Amongst them, I would especially like to mention Ronitt Rubinfeld for her support during this time. Finally, I am incredibly grateful for the kindness of Srini Devadas, who single-handedly made it possible for me to do research full-time for my final semester. I hope the work I did during that time would make him proud. Compared to many of my peers, I have had relatively few distinct collaborators 5 during my time at MIT, which, I suppose, means that I am all the more grateful for the time I've worked with them: Scott Aaronson, Adam Bouland, Matt Coudron, Stephen Fenner, Ramis Movassagh, Luke Schaeffer, Aarthi Sundaram, and John Watrous. By far, my closest collaborator was the inimitable Luke Schaeffer. Almost no day went by during my time at MIT that didn't have some sort of research check-in with Luke, often lasting two, three, four hours. He is on all but a few of my published papers, including all three I will discuss in this thesis. I can't really imagine what grad school would have been like without Luke, but I'm guessing it would have been less fun and less successful. I'd also like to thank the entire theory group at MIT. It was an intensely social group of people that knew how to play a mean game of soccer, sing their hearts out during karaoke, and were even pretty good at math if you gave them the chance. I have no doubt that many of them will go on (or have already gone on) to be world class researchers, and I feel privileged to have been part of the group. Purely out of fear of leaving anybody out, I will not try to enumerate all of my theory friends, but needless to say... I will miss you guys. I'd also like to thank the various organizations I was part of during my time at MIT that made it that much more enjoyable. First, to the cycling club team at MIT, consider this my formal apology for not training harder. I didn't deserve to be part of such a fun and successful team, but I sure did enjoy being along for the ride. I'd also like to thank my hockey team, who took a chance on me when I messaged them out of the blue asking to join. Sports have always been a large part of my life, and I'm really lucky to have have been part of two great teams. Finally, I would like to thank the EECS Communications Lab for hiring me on as a writing advisor. I would especially like to thank my manager Alison Takemura, who not only helped me with my own communication tasks, but also helped me think about helping others as well. Finally, I would like to thank my entire family-Jon, Marion, and Ben Grier- and my girlfriend Amy for the incredible support and good times over the years. To say that they've helped me become a better and happier person would just be forehead-bangingly obvious. 6 Previously Published Material The three main results of this thesis are based on previous papers. Chapter 3 is from a conference paper in joint work with Scott Aaronson and Luke Schaeffer [7]. Chapter 4 and Chapter 5 are based on a conference and workshop paper, respectively, with Luke Schaeffer [49, 50]. 7 8 Contents 1 Introduction 13 1.1 R esults .................................. 15 1.1.1 Quantum query complexity of regular languages ....... 15 1.1.2 Hardness results for matrix permanents ............ 16 1.1.3 Classification of Clifford operations .............. 17 1.2 Related Work .............................. 17 2 Background 21 2.1 Quantum Computers .......................... 21 2.1.1 States .............................. 22 2.1.2 Operations. ........................... 23 2.1.3 Measurement .......................... 24 2.2 Query Complexity ........................... 24 2.2.1 Relationships ................ .......... 26 2.3 Complexity Classes ........................... 27 2.3.1 Decision classes ......................... 27 2.3.2 Function classes ............... .......... 28 3 A Quantum Query Complexity 'richotomy for Regular Languages 33 3.1 Results Overview ............................ 36 3.1.1 Proof Techniques ..... ................... 38 3.1.2 Related Work .......................... 38 3.2 Background ..... 40 9 3.2.1 Regular languages ... ............ .......... 40 3.2.2 Query complexity with non-binary alphabets ... ....... 43 3.3 Applications of Star-free Algorithm .... ...... ...... ... 44 3.3.1 Dynamic AND-OR .... ........ ........ .... 44 3.3.2 Bounded Dyck language .. ........... ........ 45 3.3.3 Addition ...... ......... ........ ....... 45 3.3.4 Length-2 Word Break ...... ............ ..... 46 3.3.5 Grid problems .............. ............. 48 3.4 Formal Statement of Trichotomy Theorem ............... 49 3.4.1 Flattening .................... ......... 50 3.4.2 Trichotomy Theorem ................... .... 53 3.4.3 Equivalence of algebraic and regular expression definitions .. 54 3.4.4 Monotonic query complexity .. ............ ..... 56 3.4.5 Structure of the proof ....... ............ ... 57 3.5 Upper Bounds . ......................... ..... 57 3.5.1 Proof techniques .. ....................... 58 3.5.2 O(Vi) algorithm for star-free languages ............ 64 3.6 Dichotomy Theorems ............ ............... 67 3.7 Lower Bounds ........ ............ ........... 71 3.8 Context-Free Languages ..... .................... 74 3.8.1 Context-free languages do not obey the trichotomy .....