Architecture of a Quantum Multicomputer Optimized for Shor’s Factoring Algorithm A dissertation submitted to the Department of computer science of Keio University in partial fulfillment of the requirements arXiv:quant-ph/0607065 v1 11 Jul 2006 for the degree of doctor of philosophy Rodney Doyle Van Meter III July 2006 c Copyright by Rodney Doyle Van Meter III 2006 All Rights Reserved ii Abstract Quantum computers exist, and offer tantalizing possibilities of dramatic increases in computational power, but scaling them up to solve problems that are classically in- tractable offers enormous technical challenges. Distributed quantum computation of- fers a way to surpass the limitations of an individual quantum computer. I propose a quantum multicomputer as a form of distributed quantum computer. The quantum multicomputer consists of a large number of small nodes and a qubus interconnect for creating entangled state between the nodes. The primary metric chosen is the perfor- mance of such a system on Shor’s algorithm for factoring large numbers: specifically, the quantum modular exponentiation step that is the computational bottleneck. This dissertation introduces a number of optimizations for the modular exponen- tiation, including quantum versions of the classical carry-select and conditional-sum adders, improvements in the modular arithmetic, and a means for reducing the amount of expensive, error-prone quantum computation by increasing the amount of cheaper, more reliable classical computation. Parallel implementations of these circuits are eval- uated in detail for two abstract architectural models, one (called AC) which supports long-distance communication between quantum bits, or qubits, and one which allows only communication between nearest neighbors in a linear layout (called NTC). My algorithms reduce the latency, or circuit depth, to complete the modular exponentiation of an n-bit number from O(n3) to O(n log2 n) for AC and O(n2 log n) for NTC. In- cluding improvements in the constant factors, calculations show that these algorithms are one million times and thirteen thousand times faster on AC and NTC, respectively, when factoring a 6,000-bit number. These circuits also reduce the demands on quantum error correction from 210n4 to 12n3 log n for AC and 3n4 for NTC, potentially ∼ ∼ 2 ∼ reducing the number of levels of error-correction encoding or allowing execution on more error-prone hardware. Extending to the quantum multicomputer, I calculate the performance of several types of adder circuits for several different hardware configurations. Five different iii qubus interconnect topologies and two different node sizes are considered, and two forms of carry-ripple adder are found to be the fastest for a wide range of performance parameters. Small nodes (up to five logical qubits) and a linear interconnection network provide adequate performance; more complex networks are unnecessary until n reaches several hundred bits. As node size grows, it is important that the I/O bandwidth of a node grow, as well, or performance can actually decline despite the overall decrease in network activity. The links in the quantum multicomputer are serial; parallel links would provide only very modest improvements in system reliability and performance. Two levels of the Steane [[23,1,7]] error correction code will adequately protect our data for factoring a 1,024-bit number even when the qubit teleportation failure rate is one percent. iv Acknowledgements I had the good fortune to become acquainted very early with some characters of very high standing, and to feel the incessant wish that I could even become what they were. Thomas Jefferson, Autobiography Many, many people have demonstrated a faith in me that can never be repaid, start- ing with my parents, who never even suggested, so far as I recall, that there were any limits to what I could accomplish (despite sometimes overwhelming evidence to the contrary). (Though, at the same time, I have no recollection that they ever suggested I had a future in the NBA.) Most especially, I must thank my daughters Sophia and Esther and my wife Mayumi, who put up with many hours of Daddy being physically present but mentally elsewhere. My sisters Sheila and Lera for nearly forty years have suffered the indignities and lo- gistical difficulties of a weird, sartorially challenged older brother who lives thousands of miles away. To my grandparents, aunts, uncles, cousins and brother-in-law in a large and close family I also owe an apology for living so far away. Every life has its cusps, its turning points that forever change you. The biggest was Caltech, but joining ISI was an unanticipated stroke of fortune. The friends and men- tors I made in the Caltech and ISI days still carry me forward. Ross Berteig dragged me to the three most life-changing classes I took at Caltech (Feynman, Ayres, and Scudder), including the one that led me to ISI. At ISI, I met Wook, and my life would never be the same in ways beyond enumerating; I owe no one a greater debt. Dale Chase taught me how to be a good employee and person (and how to play team vol- leyball). Andi, Bobo, Brenda, Daryll, Dave, Dennis, Edie, Gabrielle, Grace, Greg, Harold, Hugo, Irene, Jessica, John, John, Kevin, Kyu, Liralen, Mimi, Michelle, Min, Myles, Pam, Rick, Ryuji, Sandy, Steve, Suz, Tiger, Yosufi and the entire CINC-PAC, WoW, volleyball, Half Moon Bay, Quantum, Nokia, NII, Keio, and Network Alchemy v crowds, just for being there (wherever “there” happens to be). Thanks to Jim Hughes for help on classical cryptography, Reagan Moore for su- percomputing advice, and the rest of the MSSTC EC for years of companionship and learning. Without the support and encouragement of Takashi and Nobunori Shigezaki, Mark Holzbach and the folks at Asaca and ShibaSoku, and the teaching of Yuko Yamaguchi at Kichijoji Language School and Misaki-sensei at Keio, Japan would have remained a remote, foreign land rather than the second home it has become. When I began working on quantum computing three years ago, I received important early encouragement from Prof. Kohei Itoh and Eisuke Abe of Keio University, Prof. Kunihiro of the University of Electro-Communications, Drs. Kawano and Takahashi of NTT CRL, Prof. Iwama of Kyoto University, Prof. Yamashita of NAIST, Dave Bacon of Washington, Mark Oskin of Washington, and Dr. Yamaguchi of Stanford. Professors Yoshi Yamamoto, Seth Lloyd, Isaac Chuang, Andrew Steane, Mio Murao, Hiroshi Imai, Seigo Tarucha and Akira Furusawa, and Yasunobu Nakamura and J.W. Tsai of NEC, and many researchers at NTT provided access to their labs and students, without which I would never understand how to actually build a quantum computer. Prof. Yamamoto and the others who created and staffed the summer schools in Okinawa and Kochi not only taught me but brought me into their community. I look forward to deepening collaborations with all of you over the coming years. Although my name goes on the thesis, my coauthors on the half-dozen papers that are incorporated deserve much of the credit: Kohei Itoh, Mark Oskin, Thaddeus Ladd, Kae Nemoto, and Bill Munro. Thaddeus gets a special call out for writing advice as well as teaching me physics. Joe Touch, Ted Faber, Bill Manning and Nick Burke all read (sometimes awful) drafts of various papers and provided other important support. Kevin Binkley’s stochastic engine, especially the genetic algorithms, provided insight into optimization problems that remain open. Suzanne and Bob Diller get credit for cogent advice on the title and abstract of my dissertation, as well as providing years of friendship. Michael Cohen and Prof. Sagawa of Aizu-Wakamatsu and Prof. Jun Murai, Ryuji Wakikawa and Shoko Mikawa at the School of Internet ASIA Project provided teaching opportunities which turned into wonderful learning experiences for me. Chip Elliott and others at BBN have provided important encouragement, as well. The music of Tatopani, Billy Higgins, John Coltrane, Louis Armstrong, King Crim- son, Kodo, and a plethora of others kept me sane. Surprisingly, we have no favorite chef vi in this country (our favorite anywhere is Jose Luis Ugalde of Cafe Gibraltar), but ev- eryone who has fed me startling and wonderful meals – you keep me going. Without the cooking and baby-sitting of the indomitable Kazuko Arai, this thesis would have taken a decade to complete, if it ever got done at all. This work was supported in part by Ken Adelman and Dave Kashtan under the Network Alchemy basic research funding plan, and by CREST-JST. Kae Nemoto also provided funding for travel and a desk at NII. Karl and Pattie and Danner and Jenny put me up in Cambridge. This thesis was created using 100% free software. Thanks to the creators of Linux, X, TEXand LATEX, GNU Emacs, xfig, dia, POVray, maxima/MACSYMA, octave, gnu- plot, bison, flex, gcc, and more — and, in some cases, to the researchers on whose work these tools are founded, though the code base has changed. And thanks to thecreators of the arXiv, scholar.google.com, researchindex.org, and citebase, without which I would have missed much important research, and probably been forced to recreate it poorly and tediously on my own. I thank Y. Nakamura, T. Yamamoto, D. Wineland, and K. M. Itoh for the figures. Thanks to all of the patient physicists who have put up with my slowness, including Viv Kendon for help with Shor’s algorithm. I have had, in effect, four advisers: Fumio Teraoka in computer networking, Kohei Itoh in experimental physics, and Kae Nemoto and Bill Munro in theoretical physics. All four have been oustanding. Kae worked harder on reviewing this dissertation than anyone else; it would not be as clear and correct without her. Bill is the one person not on my committee without whom this technical work could not have been done.