Quantum Separation of Local Search and Fixed Point Computation

Xi Chen, Xiaoming Sun, Shang-Hua Teng
<span title="2009-02-25">2009</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qhi3z76be5c5xeihac5cyiid3m" style="color: black;">Algorithmica</a> </i> &nbsp;
In this paper, 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 [1 : n] d . Our bound is nearly tight, as the Grover search algorithm 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 as defined by Scarf and by Hirsch, Papadimitriou, and Vavasis. It can also be extended to the
more &raquo; ... ntum model for Sperner's Lemma in any dimensions: The quantum query complexity of finding a panchromatic cell in a Sperner coloring of a uniform 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 ) obtained by Friedl, Ivanyos, Santha, and Verhoeven. More significantly, our result provides a quantum separation of local search and fixed point computation over grid [1 : n] d , for d ≥ 4. Combining Aldous sampling with Grover search, Aaronson gave an algorithm for local search over [1 : n] d that makes O(n d/3 ) quantum queries. Thus, the quantum query model over [1 : n] d for d ≥ 4 strictly separates these two fundamental search problems.
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