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Quantum Separation of Local Search and Fixed Point Computation

  • Xi Chen
  • Xiaoming Sun
  • Shang-Hua Teng
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5092)

Abstract

We give a lower bound of Ω (n (d − 1)/2) on the quantum query complexity for finding a fixed point of a discrete Brouwer function over grid [n] d . Our lower bound is nearly tight, as Grover Search can be used to find a fixed point with O(n d/2) quantum queries. Our result establishes a nearly tight bound for the computation of d-dimensional approximate Brouwer fixed points defined by Scarf and by Hirsch, Papadimitriou, and Vavasis. It can be extended to the quantum model for Sperner’s Lemma in any dimensions: The quantum query complexity of finding a panchromatic cell in a Sperner coloring of a triangulation of a d-dimensional simplex with n d cells is Ω(n (d − 1)/2). For d = 2, this result improves the bound of Ω(n 1/4) of Friedl, Ivanyos, Santha, and Verhoeven.

More significantly, our result provides a quantum separation of local search and fixed point computation over [n] d , for d ≥ 4. Aaronson’s local search algorithm for grid [n] d , using Aldous Sampling and Grover Search, makes O (n d/3) quantum queries. Thus, the quantum query model over [n] d for d ≥ 4 strictly separates these two fundamental search problems.

Keywords

Local Search Point Computation Query Complexity Query Model Quantum Query 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Xi Chen
    • 1
  • Xiaoming Sun
    • 2
  • Shang-Hua Teng
    • 3
  1. 1.Institute for Advanced Study 
  2. 2.Tsinghua University 
  3. 3.Boston University 

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