UvA-DARE (Digital Academic Repository) Quantum query complexity and distributed computing Röhrig, H.P. Publication date 2004 Document Version Final published version Link to publication Citation for published version (APA): Röhrig, H. P. (2004). Quantum query complexity and distributed computing. Institute for Logic, Language and Computation. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:28 Sep 2021 Heinn P. Rohrig Quantumm Query Complexityy and Distributedd Computing ILLCC Dissertation Series DS-2004-01 INSTITUTEE FOR LOGIC, LANGUAGE AND COMPUTATION Forr further information about ILLC-publications, please contact Institutee for Logic, Language and Computation Universiteitt van Amsterdam Plantagee Muidergracht 24 10188 TV Amsterdam phone:: +31-20-525 6051 fax:: +31-20-525 5206 e-mail:: illc@wins. uva. nl homepage:: http://www.illc.uva.nl/ Quantumm Query Complexityy and Distributedd Computing ACADEMISCHH PROEFSCHRIFT terr verkrijging van de graad van doctor aan de Universiteitt van Amsterdam opp gezag van de Rector Magnificus prof.mr.. P.F. van der Heijden tenn overstaan van een door het college voor promotiess ingestelde commissie, in het openbaar tee verdedigen in de Aula der Universiteit opp dinsdag 27 januari 2004, te 12.00 uur door r Heinn Philipp Röhrig geborenn te Frankfurt am Main, Duitsland. Promotores:: Prof.dr. H.M. Buhrman Prof.dr.ir.. P.M.B. Vitanyi Overigee 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 Faculteitt der Natuurwetenschappen, Wiskunde en Informatica Thee investigations were supported by the Netherlands Organization for Sci- entificc Research (NWO) project "Quantum Computing" (project number 612.15.001),, by the EU fifth framework projects QAIP, 1ST-1999-11234, and RESQ,, IST-2001-37559, the NoE QUIPROCONE, 1ST-1999-29064, and the ESFF QiT Programme. Copyrightt © 2003 by Hein P. Röhrig Revisionn 411 ISBN:: 3-933966-04-3 Contents s Acknowledgmentss xi Publicationss xiii 11 Introduction 1 1.11 Computation is physical 1 1.22 Quantum mechanics 2 1.2.11 States . 2 1.2.22 Evolution 5 1.2.33 Observables 7 1.2.44 Entanglement 12 1.2.55 Perspective 16 1.33 Quantum computation and information 17 1.3.11 Quantum circuits 17 1.3.22 Quantum black-box algorithms 19 1.3.33 Hallmark results 23 11 Quantum query complexity 25 22 Quantum search 27 2.11 Quantum amplitude amplification 27 2.1.11 Grover's algorithm 27 2.1.22 Amplitude amplification 33 2.22 Convolution products 36 2.33 Search in the density-matrix formalism 41 2.44 Energy levels of a Hamiltonian 48 2.4.11 Sampling from the energy levels 49 vii i 2.4.22 Comparing degeneracy degrees 52 2.4.33 Numerical simulations 53 33 Property testing 57 3.11 Introduction 57 3.22 Preliminaries 58 3.33 Separating quantum and classical property testing 59 3.44 An exponential separation 62 3.55 Quantum lower bounds 70 3.66 Further research 73 44 Robustness 75 4.11 Introduction 75 4.22 Robustly recovering all n bits 78 4.33 The multiple-faulty-copies model 83 4.44 Robust polynomials 83 4.55 Discussion and open problems 86 III Distributed quantum computing 87 55 Nonlocality 89 5.11 Introduction 89 5.1.11 Bell inequalities 91 5.1.22 Imperfections . 94 5.22 Definitions 99 5.33 Bounds on multiparty nonlocality 103 5.3.11 Combinatorial bounds 103 5.3.22 Application to the GHZ correlations 105 5.3.33 an addition theorem 108 5.44 Reproducing quantum correlations 110 5.55 Conclusions 115 66 Quantum coin flipping 117 6.11 Introduction 117 6.22 Two-party coin flipping with penalty for cheating 120 6.33 The multiparty model 122 6.3.11 Adversaries 122 6.3.22 The broadcast channel 123 6.44 Multiparty quantum coin-flipping protocols 125 6.55 Lower bound 128 6.5.11 The two-party bound 128 viii i 6.5.22 More than two parties 132 6.66 Summary 134 Bibliographyy 135 Indexx 147 Samenvattingg 151 Abstractt 155 be e Acknowledgments s II am indebted to Harry Buhrman for his guidance, advice, and comradeship. Mostt problems addressed in this thesis were raised by him; his ideas also contributedd to many of the solutions. I am also very grateful to Paul Vitanyi, whoo offered me the PhD position and whose pragmatic yet highly competent managementt style I admire. Speciall thanks go to Ronald de Wolf. He was the first real quantum- computingg researcher I ever talked to, the night before AQIP 98. His close readingg 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 onlyy people at the institute at night and during the weekend. II also thank my other coauthors of the papers that are the foundation of thiss thesis: Andris Ambainis, Yevgeniy Dodis, Lance Fortnow, Peter H0yer, Sergee Massar, and Ilan Newman. During my studies, I spent a substantial amountt of time as guest of Lov Grover at Bell Labs. What I know about quantumm search, I learned from him. Twoo long-time mentors deserve special mention here: Prof. Dr. Karl Hensen,, my "Vertrauensdozent" in the Studienstiftung, was a constant point off reference outside my specialty. FVom Prof. Dr. Dr. h.c. mult. Günter Hotz II learned to critically review objectives of research in a wider context. Forr many pleasant, instructive, and fruitful scientific discussions, I thank Scottt Aaronson, Luis Antunes, Eldar Fischer, Peter Gacs, Mart de Graaf, Pe- terr Grünwald, Jaap-Henk Hoepman, Troy Lee, Ashwin Nayak, Daniel Preda, Yaoyunn Shi, Robert Spalek, and Arjen de Vries. For introducing me to new worlds,, for advice and friendship, and generally a good time I thank Tor ben Hagerup,, Ute Röhrig, Richard Hahnloser, Sebastian Seung, Tarmo Johannes, Dashaa Beltsiukova, Liddy Shriver, Markus Jakobsson, and Susanne Wetzel. Thankss to Rudi Cilibrasi and Rudolf Janz for proofreading drafts of this the- siss and instructing me to exorcize parentheses; Stefan Manegold and Kolja Sulimmaa provided valuable advice about printing this thesis. II am grateful to my parents for their support throughout my studies. Andd Tzveta, thanks for always stimulating me to finish this thesis and for providingg distraction from work! Heinn Röhrig Amsterdam,, December 2003 xi i Publications s Thee following publications are the base for Chapters 2-6. H. Buhrman, L. Fortnow, I. Newman, and H. Röhrig. Quantum prop- ertyy testing. In Proceedings of 14th SODA, pages 480-488, 2003, quant- ph/0201117. H. Buhrman, I. Newman, H. Röhrig, and R. de Wolf. Robust quantum algorithmss and polynomials. Submitted, quant-ph/0309220. H. Buhrman, P. H0yer, S. Massar, and H. Röhrig. Combinatorics andd quantum nonlocality. Physical Review Letters, 91(4):047903, 2003, quant-ph/0209052. H. Buhrman and H. Röhrig. Distributed quantum computing. In B.. Rovan and P. Vojtas, editors, Mathematical Foundations of Com- puterputer Science 2003, volume 2747 of Lecture Notes in Computer Science, pagess 1-20. Springer, 2003. A. Ambainis, H. Buhrman, Y. Dodis, and H. Röhrig. Multiparty quan- tumm coin flipping. Submitted, 2003, quant-ph/0304112. xüi i Chapterr 1 Introduction n 1.11 Computation Is Physical Thee last two decades have seen a renewed interest in the relation between computationn and physics. In particular, in the early 1980s Feynman [54, 55] pointedd out that simulating quantum mechanics appears to be difficult in modelss of computation that were then thought to represent the strongest feasiblee form of computation. Moreover, he raised the question whether a quantum-mechanicall computer could be more powerful than a "classical" computer,, e.g., in analyzing other quantum-mechanical systems. Most physi- cistss believe that the world is quantum-mechanical, or at least is more ac- curatelyy 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 forr verifying the predictions of quantum mechanics, and principal obstacles mayy very well indicate limitations of quantum mechanics. This would be off great interest since to date there is no experimental data contradicting quantumm mechanics. Initially,, interest among computer scientists was limited. In part this was duee to the similarity of Feynman's proposal to conventional "analog" com- puters.. Deutsch [47] laid the theoretical groundwork for a "digital" variant off quantum computing, and in the 1980s and in the early 1990s a sequence of quantumm algorithms appeared [48, 22, 109] that showed that quantum com- puterss are in certain aspects significantly more powerful than classical com- puters.. However, the area really became popular only after Shor presented an efficientt quantum algorithm for factoring integers [108], a problem considered soo difficult classically that the most important cryptographic systems both inn theory and practice rely on its hardness.