Quantum Separation of Local Search and Fixed Point Computation

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.