Quantum Query Complexity and Distributed Computing ILLC Dissertation Series DS-2004-01 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-525 6051 fax: +31-20-525 5206 e-mail: [email protected] homepage: http://www.illc.uva.nl/ Quantum Query Complexity and Distributed Computing Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof.mr. P.F. van der Heijden ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Aula der Universiteit op dinsdag 27 januari 2004, te 12.00 uur door Hein Philipp R¨ohrig geboren te Frankfurt am Main, Duitsland. Promotores: Prof.dr. H.M. Buhrman Prof.dr.ir. P.M.B. Vit´anyi Overige leden: Prof.dr. R.H. Dijkgraaf Prof.dr. L. Fortnow Prof.dr. R.D. Gill Dr. S. Massar Dr. L. Torenvliet Dr. R.M. de Wolf Faculteit der Natuurwetenschappen, Wiskunde en Informatica The investigations were supported by the Netherlands Organization for Sci- entific Research (NWO) project “Quantum Computing” (project number 612.15.001), by the EU fifth framework projects QAIP, IST-1999-11234, and RESQ, IST-2001-37559, the NoE QUIPROCONE, IST-1999-29064, and the ESF QiT Programme. Copyright c 2003 by Hein P. R¨ohrig Revision 411 ISBN: 3–933966–04–3 v Contents Acknowledgments xi Publications xiii 1 Introduction 1 1.1 Computation is physical . 1 1.2 Quantum mechanics . 2 1.2.1 States . 2 1.2.2 Evolution . 5 1.2.3 Observables . 7 1.2.4 Entanglement . 12 1.2.5 Perspective . 16 1.3 Quantum computation and information . 17 1.3.1 Quantum circuits . 17 1.3.2 Quantum black-box algorithms . 19 1.3.3 Hallmark results . 23 I Quantum query complexity 25 2 Quantum search 27 2.1 Quantum amplitude amplification . 27 2.1.1 Grover’s algorithm . 27 2.1.2 Amplitude amplification . 33 2.2 Convolution products . 36 2.3 Search in the density-matrix formalism . 41 2.4 Energy levels of a Hamiltonian . 48 2.4.1 Sampling from the energy levels . 49 vii 2.4.2 Comparing degeneracy degrees . 52 2.4.3 Numerical simulations . 53 3 Property testing 57 3.1 Introduction . 57 3.2 Preliminaries . 58 3.3 Separating quantum and classical property testing . 59 3.4 An exponential separation . 62 3.5 Quantum lower bounds . 70 3.6 Further research . 73 4 Robustness 75 4.1 Introduction . 75 4.2 Robustly recovering all n bits . 78 4.3 The multiple-faulty-copies model . 83 4.4 Robust polynomials . 83 4.5 Discussion and open problems . 86 II Distributed quantum computing 87 5 Nonlocality 89 5.1 Introduction . 89 5.1.1 Bell inequalities . 91 5.1.2 Imperfections . 94 5.2 Definitions . 99 5.3 Bounds on multiparty nonlocality . 103 5.3.1 Combinatorial bounds . 103 5.3.2 Application to the GHZ correlations . 105 5.3.3 an addition theorem . 108 5.4 Reproducing quantum correlations . 110 5.5 Conclusions . 115 6 Quantum coin flipping 117 6.1 Introduction . 117 6.2 Two-party coin flipping with penalty for cheating . 120 6.3 The multiparty model . 122 6.3.1 Adversaries . 122 6.3.2 The broadcast channel . 123 6.4 Multiparty quantum coin-flipping protocols . 125 6.5 Lower bound . 128 6.5.1 The two-party bound . 128 viii 6.5.2 More than two parties . 132 6.6 Summary . 134 Bibliography 135 Index 147 Samenvatting 151 Abstract 155 ix Acknowledgments I am indebted to Harry Buhrman for his guidance, advice, and comradeship. Most problems addressed in this thesis were raised by him; his ideas also contributed to many of the solutions. I am also very grateful to Paul Vit´anyi, who offered me the PhD position and whose pragmatic yet highly competent management style I admire. Special thanks go to Ronald de Wolf. He was the first real quantum- computing researcher I ever talked to, the night before AQIP 98. His close reading of the thesis improved it a lot; all remaining errors are of course mine. He, John Tromp, and I shared an office—the drie ‘heren’ frequently were the only people at the institute at night and during the weekend. I also thank my other coauthors of the papers that are the foundation of this thesis: Andris Ambainis, Yevgeniy Dodis, Lance Fortnow, Peter Høyer, Serge Massar, and Ilan Newman. During my studies, I spent a substantial amount of time as guest of Lov Grover at Bell Labs. What I know about quantum search, I learned from him. Two long-time mentors deserve special mention here: Prof. Dr. Karl Hensen, my “Vertrauensdozent” in the Studienstiftung, was a constant point of reference outside my specialty. From Prof. Dr. Dr. h.c. mult. G¨unter Hotz I learned to critically review objectives of research in a wider context. For many pleasant, instructive, and fruitful scientific discussions, I thank Scott Aaronson, Luis Antunes, Eldar Fischer, P´eter G´acs, Mart de Graaf, Pe- ter Gr¨unwald, Jaap-Henk Hoepman, Troy Lee, Ashwin Nayak, Daniel Preda, Yaoyun Shi, Robert Spalek,ˇ and Arjen de Vries. For introducing me to new worlds, for advice and friendship, and generally a good time I thank Torben Hagerup, Ute R¨ohrig, Richard Hahnloser, Sebastian Seung, Tarmo Johannes, Dasha Beltsiukova, Liddy Shriver, Markus Jakobsson, and Susanne Wetzel. Thanks to Rudi Cilibrasi and Rudolf Janz for proofreading drafts of this the- sis and instructing me to exorcize parentheses; Stefan Manegold and Kolja Sulimma provided valuable advice about printing this thesis. I am grateful to my parents for their support throughout my studies. And Tzveta, thanks for always stimulating me to finish this thesis and for providing distraction from work! Hein R¨ohrig Amsterdam, December 2003 xi Publications The following publications are the base for Chapters 2–6. • H. Buhrman, L. Fortnow, I. Newman, and H. R¨ohrig. Quantum prop- erty testing. In Proceedings of 14th SODA, pages 480–488, 2003, quant- ph/0201117. • H. Buhrman, I. Newman, H. R¨ohrig, and R. de Wolf. Robust quantum algorithms and polynomials. Submitted, quant-ph/0309220. • H. Buhrman, P. Høyer, S. Massar, and H. R¨ohrig. Combinatorics and quantum nonlocality. Physical Review Letters, 91(4):047903, 2003, quant-ph/0209052. • H. Buhrman and H. R¨ohrig. Distributed quantum computing. In B. Rovan and P. Vojtas, editors, Mathematical Foundations of Com- puter Science 2003, volume 2747 of Lecture Notes in Computer Science, pages 1–20. Springer, 2003. • A. Ambainis, H. Buhrman, Y. Dodis, and H. R¨ohrig. Multiparty quan- tum coin flipping. Submitted, 2003, quant-ph/0304112. xiii Chapter 1 Introduction 1.1 Computation Is Physical The last two decades have seen a renewed interest in the relation between computation and physics. In particular, in the early 1980s Feynman [54, 55] pointed out that simulating quantum mechanics appears to be difficult in models of computation that were then thought to represent the strongest feasible form of computation. Moreover, he raised the question whether a quantum-mechanical computer could be more powerful than a “classical” computer, e.g., in analyzing other quantum-mechanical systems. Most physi- cists believe that the world is quantum-mechanical, or at least is more ac- curately described by quantum mechanics than by classical theories. In this case, it should in principle be possible to actually build such quantum com- puters. Conversely, quantum computers can also be regarded as experiments for verifying the predictions of quantum mechanics, and principal obstacles may very well indicate limitations of quantum mechanics. This would be of great interest since to date there is no experimental data contradicting quantum mechanics. Initially, interest among computer scientists was limited. In part this was due to the similarity of Feynman’s proposal to conventional “analog” com- puters. Deutsch [47] laid the theoretical groundwork for a “digital” variant of quantum computing, and in the 1980s and in the early 1990s a sequence of quantum algorithms appeared [48, 22, 109] that showed that quantum com- puters are in certain aspects significantly more powerful than classical com- puters. However, the area really became popular only after Shor presented an efficient quantum algorithm for factoring integers [108], a problem considered so difficult classically that the most important cryptographic systems both in theory and practice rely on its hardness. Further theoretical discoveries included the feasibility of correcting errors in quantum computation. This 1 2 Chapter 1. Introduction answered affirmatively the question whether an imperfect real-world quan- tum mechanical device could benefit from these theoretical advantages; thus, one could say that quantum error correction is an example of a contribution to physics by computer scientists. 1.2 Quantum Mechanics An atom is not a soccer ball: the measures and laws of classical physics, which describe the motion of a soccer ball in space, fail to explain physical phenomena at the atomic scale. For example, the laws of classical physics do not adequately describe the structure of the atomic nucleus, the wave-particle character of light, and discrete absorption spectra. These phenomena can be explained by the physical theory of quantum mechanics. This theory has been developed from 1925 on chiefly by Heisenberg and Schr¨odinger. Despite some seemingly “unnatural” model assumptions, quantum mechanics is today accepted by most physicists as the best tool to describe nature on the very small-scale level or where tiny differences of energy are involved. In the limit of many particles and great energy differences, quantum mechanical laws converge to their classical counterparts.