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Limits on Efficient Computation in the Physical World

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Limits on Efficient Computation in the Physical World Limits on Efficient Computation in the Physical World by Scott Joel Aaronson Bachelor of Science (Cornell University) 2000 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Computer Science in the GRADUATE DIVISION of the UNIVERSITY of CALIFORNIA, BERKELEY Committee in charge: Professor Umesh Vazirani, Chair Professor Luca Trevisan Professor K. Birgitta Whaley Fall 2004 The dissertation of Scott Joel Aaronson is approved: Chair Date Date Date University of California, Berkeley Fall 2004 Limits on Efficient Computation in the Physical World Copyright 2004 by Scott Joel Aaronson 1 Abstract Limits on Efficient Computation in the Physical World by Scott Joel Aaronson Doctor of Philosophy in Computer Science University of California, Berkeley Professor Umesh Vazirani, Chair More than a speculative technology, quantum computing seems to challenge our most basic intuitions about how the physical world should behave. In this thesis I show that, while some intuitions from classical computer science must be jettisoned in the light of modern physics, many others emerge nearly unscathed; and I use powerful tools from computational complexity theory to help determine which are which. In the first part of the thesis, I attack the common belief that quantum computing resembles classical exponential parallelism, by showing that quantum computers would face serious limitations on a wider range of problems than was previously known. In partic- ular, any quantum algorithm that solves the collision problem—that of deciding whether a sequence of n integers is one-to-one or two-to-one—must query the sequence Ω n1/5 times. This resolves a question that was open for years; previously no lower bound better than constant was known. A corollary is that there is no “black-box” quantum algorithm to break cryptographic hash functions or solve the Graph Isomorphism problem in poly- nomial time. I also show that relative to an oracle, quantum computers could not solve NP-complete problems in polynomial time, even with the help of nonuniform “quantum advice states”; and that any quantum algorithm needs Ω 2n/4/n queries to find a local minimum of a black-box function on the n-dimensional hypercube. Surprisingly, the latter result also leads to new classical lower bounds for the local search problem. Finally, I give new lower bounds on quantum one-way communication complexity, and on the quantum query complexity of total Boolean functions and recursive Fourier sampling. The second part of the thesis studies the relationship of the quantum computing model to physical reality. I first examine the arguments of Leonid Levin, Stephen Wol- fram, and others who believe quantum computing to be fundamentally impossible. I find their arguments unconvincing without a “Sure/Shor separator”—a criterion that separates the already-verified quantum states from those that appear in Shor’s factoring algorithm. I argue that such a separator should be based on a complexity classification of quantum states, and go on to create such a classification. Next I ask what happens to the quantum computing model if we take into account that the speed of light is finite—and in particu- lar, whether Grover’s algorithm still yields a quadratic speedup for searching a database. Refuting a claim by Benioff, I show that the surprising answer is yes. Finally, I analyze hypothetical models of computation that go even beyond quantum computing. I show that 2 many such models would be as powerful as the complexity class PP, and use this fact to give a simple, quantum computing based proof that PP is closed under intersection. On the other hand, I also present one model—wherein we could sample the entire history of a hidden variable—that appears to be more powerful than standard quantum computing, but only slightly so. Professor Umesh Vazirani Dissertation Committee Chair iii Contents List of Figures vii List of Tables viii 1 “Aren’t You Worried That Quantum Computing Won’t Pan Out?” 1 2 Overview 6 2.1 LimitationsofQuantumComputers . ... 7 2.1.1 TheCollisionProblem.......................... 8 2.1.2 LocalSearch ............................... 9 2.1.3 Quantum Certificate Complexity . 10 2.1.4 TheNeedtoUncompute. .. .. .. .. .. .. .. 11 2.1.5 Limitations of Quantum Advice . 11 2.2 ModelsandReality................................ 13 2.2.1 Skepticism of Quantum Computing . 13 2.2.2 Complexity Theory of Quantum States . 13 2.2.3 Quantum Search of Spatial Regions . 14 2.2.4 Quantum Computing and Postselection . 15 2.2.5 ThePowerofHistory .......................... 16 3 Complexity Theory Cheat Sheet 18 3.1 TheComplexityZooJunior . 19 3.2 Notation...................................... 20 3.3 Oracles ...................................... 21 4 Quantum Computing Cheat Sheet 23 4.1 Quantum Computers: N Qubits ........................ 24 4.2 FurtherConcepts................................. 27 I Limitations of Quantum Computers 29 5 Introduction 30 5.1 TheQuantumBlack-BoxModel. 31 5.2 OracleSeparations ............................... 32 iv 6 The Collision Problem 34 6.1 Motivation .................................... 36 6.1.1 OracleHardnessResults . 36 6.1.2 InformationErasure ........................... 36 6.2 Preliminaries ................................... 37 6.3 Reduction to Bivariate Polynomial . ..... 38 6.4 LowerBound ................................... 41 6.5 SetComparison.................................. 43 6.6 OpenProblems.................................. 46 7 Local Search 47 7.1 Motivation .................................... 49 7.2 Preliminaries ................................... 51 7.3 RelationalAdversaryMethod . 52 7.4 Snakes....................................... 57 7.5 SpecificGraphs.................................. 60 7.5.1 BooleanHypercube ........................... 60 7.5.2 Constant-Dimensional Grid Graph . 64 8 Quantum Certificate Complexity 67 8.1 SummaryofResults ............................... 68 8.2 RelatedWork................................... 70 8.3 Characterization of Quantum Certificate Complexity . .......... 70 8.4 Quantum Lower Bound for Total Functions . 72 8.5 AsymptoticGaps................................. 74 8.5.1 LocalSeparations............................. 76 8.5.2 Symmetric Partial Functions . 77 8.6 OpenProblems.................................. 78 9 The Need to Uncompute 79 9.1 Preliminaries ................................... 81 9.2 QuantumLowerBound ............................. 82 9.3 OpenProblems.................................. 87 10 Limitations of Quantum Advice 88 10.1Preliminaries .................................. 91 10.1.1 QuantumAdvice ............................. 92 10.1.2 TheAlmostAsGoodAsNewLemma . 93 10.2 Simulating Quantum Messages . 93 10.2.1 Simulating Quantum Advice . 96 10.3 A Direct Product Theorem for Quantum Search . ..... 99 10.4 TheTraceDistanceMethod . 103 10.4.1 Applications ............................... 106 10.5OpenProblems .................................. 110 v 11 Summary of Part I 112 II Models and Reality 114 12 Skepticism of Quantum Computing 116 12.1 Bell Inequalities and Long-Range Threads . ........ 119 13 Complexity Theory of Quantum States 126 13.1 Sure/ShorSeparators. 127 13.2 ClassifyingQuantumStates . 130 13.3BasicResults ................................... 135 13.4 Relations Among Quantum State Classes . 138 13.5LowerBounds................................... 141 13.5.1 SubgroupStates ............................. 142 13.5.2 ShorStates ................................ 146 13.5.3 Tree Size and Persistence of Entanglement . 148 13.6 Manifestly Orthogonal Tree Size . 149 13.7 ComputingWithTreeStates . 154 13.8 TheExperimentalSituation . 157 13.9 ConclusionandOpenProblems . 160 14 Quantum Search of Spatial Regions 162 14.1 SummaryofResults ............................... 162 14.2RelatedWork................................... 164 14.3 ThePhysicsofDatabases . 165 14.4TheModel .................................... 167 14.4.1 LocalityCriteria . 168 14.5GeneralBounds.................................. 169 14.6SearchonGrids.................................. 173 14.6.1 Amplitude Amplification . 174 14.6.2 DimensionAtLeast3 . .. .. .. .. .. .. .. 175 14.6.3 Dimension2................................ 180 14.6.4 MultipleMarkedItems. 181 14.6.5 Unknown Number of Marked Items . 184 14.7 SearchonIrregularGraphs . 185 14.7.1 Bits Scattered on a Graph . 189 14.8 Application to Disjointness . 190 14.9OpenProblems .................................. 191 15 Quantum Computing and Postselection 192 15.1 The Class PostBQP ................................ 193 15.2 Fantasy Quantum Mechanics . 196 15.3OpenProblems .................................. 198 vi 16 The Power of History 199 16.1 The Complexity of Sampling Histories . 200 16.2 OutlineofChapter ............................... 201 16.3 Hidden-VariableTheories . 203 16.3.1 ComparisonwithPreviousWork . 205 16.3.2 Objections ................................ 206 16.4 Axioms for Hidden-Variable Theories . ....... 206 16.4.1 ComparingTheories . 207 16.5 ImpossibilityResults . 208 16.6 SpecificTheories ................................ 211 16.6.1 FlowTheory ............................... 211 16.6.2 Schr¨odingerTheory . 215 16.7 TheComputationalModel. 218 16.7.1 BasicResults ............................... 219 16.8 TheJuggleSubroutine. 220 16.9 Simulating SZK .................................. 221 16.10Search in N 1/3 Queries.............................. 224 16.11ConclusionsandOpenProblems . 226 17 Summary of Part II 228 Bibliography 229 vii List of Figures
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