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C Copyright 2013 Paul Pham c Copyright 2013 Paul Pham Low-depth quantum architectures for factoring Paul Pham A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2013 Reading Committee: Aram Harrow, Chair Paul Beame Mark Oskin Boris Blinov Program Authorized to Offer Degree: University of Washington Computer Science & Engineering University of Washington Abstract Low-depth quantum architectures for factoring Paul Pham Chair of the Supervisory Committee: Affiliate Professor Aram Harrow Computer Science & Engineering Quantum computing is a new field which combines computer science and quantum physics. Its most famous result, Shor’s factoring algorithm, would enable us to one day compromise the widely-used RSA cryptosystem if we are able to design efficient quan- tum architectures. Studying the depth of these architectures would allow us to solve this human problem within a human lifetime. Toward that end, we contribute two hy- brid factoring architectures in poly-logarithmic and sub-logarithmic depths, which are both exponential improvements over previous known works. We also present an im- proved procedure for generating quantum Fourier states useful for quantum compiling. Finally, we invent a new circuit resource called coherence which upper bounds the error- correcting effort needed in a future quantum computer. We use this to characterize a better time-space tradeoff for factoring as well as to provide configurable-depth factoring architectures. TABLE OF CONTENTS Page List of Figures . iii Chapter 0: Introduction . 1 0.1 Quantum Gates and Circuit Bases . 7 0.2 Quantum Architectures and Models . 14 0.3 Constant-depth Communication . 20 0.4 Modules and Long-Range Interactions . 25 0.5 Organization of Dissertation . 34 Chapter 1: Hybrid Factoring in Poly-logarithmic Depth . 36 1.1 Related Work . 37 1.2 The Constant-Depth Carry-Save Technique . 39 1.3 Quantum Modular Addition . 43 1.4 Quantum Modular Multiplication . 49 1.5 Quantum Modular Exponentiation . 56 1.6 Asymptotic Results . 60 1.7 Conclusion . 61 Chapter 2: Hybrid Factoring in Sub-logarithmic Depth . 64 2.1 Circuit Complexity Background . 65 2.2 Controlled Rotations for Factoring . 70 2.3 Quantum Majority Circuits for Modular Exponentiation . 73 2.4 Conclusion . 80 Chapter 3: Quantum Compiling . 82 3.1 Quantum Compiling Background . 83 3.2 The Solovay-Kitaev Algorithm . 89 3.3 Quantum Compiler Review . 91 3.4 Quantum Compiler Comparison . 98 i 3.5 Phase Kickback and Quantum Fourier States . 100 3.6 Circuit Resources for the KSV Quantum Compiler . 102 3.7 Single-Qubit Rotations for Quantum Majority Gate . 115 3.8 Conclusion . 117 Chapter 4: Quantum Circuit Coherence . 119 4.1 Definition of Circuit Coherence . 120 4.2 Measurement-Based Quantum Computing Background . 129 4.3 Circuit Coherence as a Time-Space Tradeoff . 135 4.4 Circuit Coherence of Factoring Architectures . 144 4.5 Conclusion . 146 Chapter 5: Conclusion . 148 Appendix A: KSV Error from Early Measurement . 151 A.1 Measurement Operators . 152 A.2 Notation . 153 A.3 Operators That Measure a Function . 154 A.4 Measurement Operators with Ancillae . 155 A.5 Error Calculations . 158 Bibliography . 166 ii LIST OF FIGURES Figure Number Page 1 A quantum circuit with single-qubit and two-qubit gates and with resources of depth, size, and width. 4 2 A circuit for measurement in the Bell state basis. 19 3 Constant-depth circuit based on [21, 22] for teleportation over n = 5 qubits [83]........................................... 21 4 Constant-depth circuits based on [21, 22] for fanout [45] of one qubit to n = 4 entangled copies. 22 5 A novel, constant-depth circuit for unbounded quantum unfanout on CCNTC 23 6 Three modules in the 2D CCNTCM model. 28 1.1 An example of carry-save encoding for the 5-bit conventional number 30.. 41 1.2 Carry-save adder circuit for a single bit position i: ai + bi + ci = ui + vi+1.. 41 1.3 The depth-efficient Toffoli gate decomposition from [4]. 42 1.4 The carry-save adder (CSA), or 3 2 adder, and carry-save 2 2 adder. 42 ! ! 1.5 A schematic proof of Gossett’s constant-depth modular reduction for n = 3. 45 1.6 Addition and three rounds of modular reduction for a 3-bit modulus. 47 1.7 Symbol for an n-bit 3 2 modular adder, also called a CSA tile. 48 ! 1.8 Modular multiple addition of quantum integers on a CSA tile architecture for t0 = 18 ...................................... 55 1.9 Parallel modular exponentiation . 57 2.1 One round of programmable ancillae rotation (PAR) to probabilistically achieve arbitrary single-qubit rotations [51]. 71 2.2 Decomposition of a controlled-Z (L(RZ(f))) rotation using three single- qubit rotations and CNOTs. 72 2.3 Decomposition of a controlled-Z (L(RZ(f))) rotation using one single- qubit rotation controlled-SWAPs, and an ancilla. 72 3.1 An arbitrary quantum circuit being compiled into single-qubit gates and CNOT. ........................................ 84 3.2 Pseudo-code for the Solovay-Kitaev algorithm [28]. 90 iii ACKNOWLEDGMENTS This dissertation is a product made by me, but I am a product made by many others. I’ve benefitted greatly from having the following people in my life. My family, especially my parents Minh and Phiet, my sister Tracie and her husband Jeremy, my uncle Le and aunt Thuy, my sisters Mai and Ann, my brother-in-law Nash, and my nieces Nadia and Maria. I also wish to thank the following people and things in my life in no particular order. You all supported me, gave me ideas, and helped me grow: My coworking group, fellow tEps, and fastidious last-minute proofreaders: Blake Stacey and Eric Downes. My brother from another mother and graduation buddy: Dr. Alex B. Jaffe, esq. The quantum theory group at UW, and then MIT: Kamil Michnicki, Lukas Svec, Kevin Zatloukal, Isaac Crosson, David Rosenbaum. My housemates at 419 Boylston: Mike Toomim, Nate Morris, and Marius Nita. The grad students I started out with: Nodira Khoussainova, Kayur Patel, Liz Tseng, Travis Kriplean, Yaw Anokwa, Suporn Pongnumkul, Anna Gringauze, and Jonah Cohen. My friends and fellow UW grad students: Eva Ringstrom, Ian Simon, Widad Machmouchi, Adrian Sampson, Mark Yatskar, Summer Thyme, Cyrus Rashtchian, Thach Nguyen, and Nguyen Nguyen. My workout buddies Ray Cheng, Dimi Gklezakos, and Yun-En Liu. These fine UW profes- sors: Gaetano Borriello, David Notkin, Oren Etzioni, Steve Tanimoto. Mike Ernst, who has been my mentor and role model at both MIT and UW. The Evergreen State College, and everyone else who has ever given me a job. My officemates: Jinna Lei, Ezgi Mercan, Kivanc Muslu, Gabe Schubiner, Nicholas Fitzgerald, Bryan Ferris, Stanley Kok, Masaharu Kobashi, Punya Biswal. My number one cheerleader and daughter of the Pacific Northwest: Maggie Stringfel- low Herring. My friends at MIT for both the first and second times: Adrian Bischoff, iv Jason Rolfe, Malima Wolf, Brian Neltner, Keith Durand, Mark Tobenkin, Dan Taub, and Josh Gordonson. My Seattle friends: Abbie Smith-Howe, Justin Nafziger, Nick Braddy, Corinne Pascale, and Kate Bowlus. My students at UW, especially Noah Siegel, Andrea McCool, Harshad Petwe, and Jeffrey Booth. My design classmates at UW, especially Katherine Wong and Claire Simon. My Burning Man campmates: Eric Miles, Lee Granas, David Morris, Ivan, Julianna, and Igor Mordatch. My extended Burning Man family: Kether Axelrod, Tibet Boyer, Robert Jaffe, Taliesin Jaffe, and Anika Levy. My life coach, Bi Lai. These fine entertainments: Futurama, the tenth season; Star Treks, both The Next Generation and Deep Space Nine; Baz Luhrmann’s Great Gatsby 3D movie and soundtrack; Stephen Colbert and Jon Stewart; Florence + the Machine; The National; Ellie Goulding; Imagine Dragons. My thesis committee: Paul Beame, Mark Oskin, and Boris Blinov. My Microsoft Research mentor Krysta. And finally I would like to thank my advisor Aram, who is always willing to listen and to teach. This work was supported by the IARPA QCS program (D11PC20167). The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DoI/NBC, or the U.S. Government. The work in Chapter 1 was partially supported by Microsoft Research. v DEDICATION To my family, teachers, friends, and muses. This is for everyone who believed that I could do this. vi 1 Chapter 0 INTRODUCTION The combination of existing scientific disciplines has often been the cradle and the playground for entirely new discoveries. Quantum computing is one such unification of two seemingly disparate fields. Computer science and quantum physics originated as contemporaries in the early part of the 20th century, both culminating in technology that decisively ended World War II. They have remained largely independent until recently, but the common thread connecting them has always existed: engineering. From the perspective of computer science, engineering is problem-driven. We humans wish to solve a problem, like calculating ballistic weapon trajectories, and we work back- wards to electromechanical relays or punched cards. To calculate, we need to perform fast arithmetic on stored numbers, for which we need to design devices in a certain configu- ration, for which we need certain materials from nature. At heart, computer science is the study of structured problem-solving which historically has had a strong mechanical bias. This perspective asks the questions: what do we want to do, what is desirable, and how can we build it? It is like a child going into a LEGO1 store with a pre-existing plan: the child only buys the bricks needed, or orders custom bricks. It is an efficient plan for brick procurement, but not necessarily creative in exploring the space of all possible designs. From the perspective of quantum physics, engineering is phenomena-based.
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