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/1)) 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(m~(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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