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Quantum Separation of Local Search and Fixed Point Computation

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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.

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Authors and Affiliations

  1. Institute for Advanced Study, Princeton, USA

    Xi Chen

  2. Tsinghua University, Beijing, China

    Xiaoming Sun

  3. Boston University, Boston, USA

    Shang-Hua Teng

Authors
  1. Xi Chen
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  2. Xiaoming Sun
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  3. Shang-Hua Teng
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Corresponding author

Correspondence to Xi Chen.

Additional information

X. Chen is supported by NSF Grant DMS-0635607.

X. Sun is supported by the National Natural Science Foundation of China Grant (60553001, 60603005 and 60621062), and the National Basic Research Program of China Grant 2007CB807900, 2007CB807901.

S.-H. Teng is supported by NSF grants CCR-0635102 and ITR CCR-0325630.

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Chen, X., Sun, X. & Teng, SH. Quantum Separation of Local Search and Fixed Point Computation. Algorithmica 56, 364–382 (2010). https://doi.org/10.1007/s00453-009-9289-0

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