Controlling Quantum Information Thesis by Andrew J. Landahl In Partial Ful¯llment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2002 (Defended May 21, 2002) ii c 2002 ° Andrew J. Landahl All rights Reserved iii Dedicated to my parents Nancy and John Landahl and to the memory of my grandparents Barbara and Aldo Martin iv Acknowledgements I am pleased to acknowledge all of the people who have contributed to my per- sonal and professional development during my graduate career. My most heartfelt gratitude goes to my friends and family, especially to my parents Nancy and John, to my sister Heidi, and to my aunt Carol for their collective unwavering love and support. I am also especially grateful to my mentor, colleague, and friend John Preskill. John provided me with research suggestions when I ¯rst began my studies, the opportunity to travel and establish scienti¯c collaborations, the freedom to pursue my own intellectual interests, and the steadfast encouragement to help me see my research through. John's organization of thought and clarity of expression continue to be deep personal inspirations. I will be fortunate indeed if even a fraction of those qualities have rubbed o® on me through my years of study in his company at Caltech. This thesis would not have been possible were it not from the long list of collaborators from whom I have learned so much: Charlene Ahn, Andrew Childs, Eric Dennis, Enrico Deotto, Andrew Doherty, Eddie Farhi, Sam Gutmann, Je®rey Goldstone, Alexei Kitaev, and John Preskill. Their insights have been invaluable. I also thank Je® Kimble, Hideo Mabuchi, and Kip Thorne who graciously gave of their time to sit on both my candidacy and thesis defense committees. I consider them additional mentors, from whom I have learned greatly, both through formal instruction and informal discussions. A special thanks goes to Ike Chuang and his quanta research group in the Media Lab at MIT for their hospitality during a fruitful visit which both initiated the research described in Chapter 5 and which has generated an ongoing research collaboration. One of the wonderful things about my Caltech research experience has been the vast number of quantum information scientists I have met and learned so much from, many of whom who have graciously come and visited Caltech, bringing with v them new ideas and fresh perspectives. I would like to thank the following people for lively and engaging scienti¯c interactions: Scott Aaronsen, Dorit Aharonov, Srinivas Aji, Paul Alsing, Dave Bacon, Howard Barnum, Dave Beckman, Charlie Bennett, Herb Bernstein, Ken Brown, Carl Caves, Nicholas Cerf, Richard Cleve, John Cortese, Sumit Daftuar, Win van Dam, Keshav Dani, Eric Dennis, Ivan Deutsch, David DiVincenzo, Steven van Enk, Chris Fuchs, Daniel Gottesman, Bob Gingrich, Salman Habib, Sean Hallgren, Jim Harrington, Aram Harrow, Patrick Hayden, Peter H¿yer, Lawrence Ip, Kurt Jacobs, Daniel Jonathan, Richard Jozsa, Julia Kempe, Rowan Killip, Manny Knill, Inna Kozinsky, Greg Kuperberg, Debbie Leung, Seth Lloyd, Hoi-Kwong Lo, Gerard Milburn, Paul McFadden, Carlos Mon- chon, Mike Mosca, Ashwin Nayak, Michael Nielsen, Tobias Osbourne, Ben Rahn, Eric Rains, Ben Recht, Mary-Beth Ruskai, RudigerÄ Schack, Leonard Schulman, Yaoyun Shi, Ben Schumacher, Peter Shor, John Smolin, Federico Spedialari, Dan Stamper-Kurn, Andrew Steane, Dan Steck, Barbara Terhal, Ashish Thalpiyal, Ben Toner, Umesh Vazarani, Guifre Vidal, Lorenza Viola, Chenyang Wang, Mike Westmoreland, Birgitta Whaley, Clint White, Howard Wiseman, Dave Wineland, Ronald de Wolf, Bill Wootters, and Nathan Wozny. I would also like to thank my many friends and colleagues in experimental quantum optics at Caltech who have taught me about the importance of keeping close to experiment in theoretical research. Never again will I begin a sentence with \Suppose you have three spins in a box." Thanks to: Mike Armen, John Au, Andy Berglund, Kevin Birnbaum, Andreea Boca, Glov Boi, Dave Boozer, Joe Buck, James Chou, Akira Furusawa, JM Geremia, Win Goh, Christina Hood, Alex Kuzmich, Ron Legere, Ben Lev, Peter Lodahl, Theresa Lynn, Jason McK- eever, Christof Naegerl, Dominik Schraeder, John Stockton, David Vernooy, Jon Williams, and Jun Ye. The administrative assistance I received from Beth Adams, Donna Driscoll, Ann Harvey, Sheri Stoll, and Helen Tuck has been phenomenal; I thank them for going the extra mile so many times. Without them I would have surely been lost in so many ways. vi As this thesis signals the end of my formal education, but certainly not the end of a lifetime of learning, I would like to take this opportunity to thank some of the teachers I have had over the years who had major impacts on my intellectual development. In particular, I would like to thank Mrs. Elaine Romanias, my ¯fth- and sixth-grade teacher who encouraged me to explore problems from creative points-of-view, Mr. Michael Stueben, my geometry teacher who challenged me with the puzzles he developed for his Discover Magazine Brain Bogglers column, Mr. Don Hyatt, my computer science teacher who mentored me as I undertook a Westinghouse Science Talent Search project and opened my eyes to the exciting ¯eld of research science, Dr. John Dell, my AP physics teacher who gave so gen- erously of his time, lent me many of his physics books, answered my seemingly endless list of physics and math questions, and by whose example I strengthened my resolve to become a physicist, Prof. Yoshi Oono, my Research Experience for Undergraduates mentor who taught me that fundamental physics isn't de¯ned by energy scales, Prof. Lay-Nam Chang, my honors thesis advisor who encouraged me to pursue my research interests in quantum information science, Prof. Kip Thorne, who taught me how to appreciate physics independently from mathemat- ics, and lastly Prof. John Preskill, my Ph.D. advisor who taught me by example that excellence in research and excellence in teaching are not mutually exclusive alternatives. Since the dawn of science, generous patrons have enabled scienti¯c progress. In my case, I would like to thank Caltech, DARPA, the NSF, and IBM for their ¯nancial support during my graduate career. Without their assistance, this thesis would surely not have been possible. Finally, I would like to thank the following diversions for preserving my sanity during my graduate career: table tennis, fantasy baseball, swing dancing, the a- hats softball team, and Carl's Jr. hamburgers. vii Controlling Quantum Information by Andrew J. Landahl In Partial Ful¯llment of the Requirements for the Degree of Doctor of Philosophy Abstract Quantum information science explores ways in which quantum physical laws can be harnessed to control the acquisition, transmission, protection, and processing of information. This ¯eld has seen explosive growth in the past several years from progress on both theoretical and experimental fronts. Essential to this endeavor are methods for controlling quantum information. In this thesis, I present three new approaches for controlling quantum informa- tion. First, I present a new protocol for continuously protecting unknown quantum states from noise. This protocol combines and expands ideas from the theories of quantum error correction and quantum feedback control. The result can outper- form either approach by itself. I generalize this protocol to all known quantum stabilizer codes, and study its application to the three-qubit repetition code in detail via Monte Carlo simulations. Next, I present several new protocols for controlling quantum information that are fault-tolerant. These protocols require only local quantum processing due to the topological properties of the quantum error correcting codes upon which they are built. I show that each protocol's fault-dependence behavior exhibits an order- disorder phase transition when mapped onto an associated statistical-mechanical model. I review the critical error rates of these protocols found by numerical study of the associated models, and I present new analytic bounds for them using a self- avoiding random walk argument. Moreover, I discuss fault-tolerant procedures for viii encoding, error-correction, computing, and decoding quantum information using these protocols, and calculate the accuracy threshold of fault-tolerant quantum memory for protocols using them. I end by presenting a new class of quantum algorithms that solve combinatorial optimization problems solely by measurement. I compute the running times of these algorithms by establishing an explicit dynamical model for the measurement process. This model, the digitized version of von Neumann's measurement model, is recognized as Kitaev's phase estimation algorithm. I show that the running times of these algorithms are closely related to the running times of adiabatic quantum algorithms. Finally, I present a two-measurement algorithm that achieves a quadratic speedup for Grover's unstructured search problem. ix Preface In the spring of 1996, I visited Caltech as a prospective graduate student eager to pursue research in the nascent ¯eld of quantum computing. I was drawn to Caltech by the recent experimental demonstration of quantum logic by Je® Kimble's group. Here was a place where ground-breaking research was being done! Although I leaned more towards theory, I was willing to convert to an experimentalist if it meant being involved in this exciting new ¯eld. What a surprise it was to meet John Preskill that fateful week|a theoretical physicist at Caltech