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Electronic Colloquium on Computational Complexity

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Scott Aaronson

Limits on Efficient Computation in the Physical World


UC Berkeley, filed in Fall 2004


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 particular, 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 Omega(n^{1/5}) 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 polynomial 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 Omega(2^{n/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 Wolfram, 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 particular, 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 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.

Table of Contents

    1 "Aren't You Worried That Quantum Computing Won't Pan Out?"
    2 Overview
    3 Complexity Theory Cheat Sheet
    4 Quantum Computing Cheat Sheet
    PART I: Limitations of Quantum Computers
      5 Introduction
      6 The Collision Problem
      7 Local Search
      8 Quantum Certificate Complexity
      9 The Need to Uncompute
      10 Limitations of Quantum Advice
      11 Summary of Part I
    PART II: Models and Reality
      12 Skepticism of Quantum Computing
      13 Complexity Theory of Quantum States
      14 Quantum Search of Spatial Regions
      15 Quantum Computing and Postselection
      16 The Power of History
      17 Summary of Part II

Number of pages: 258

ISSN 1433-8092 | Imprint