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. © 2008 Springer-Verlag Berlin Heidelberg.