Quantum Separation of Local Search and Fixed Point Computation

In this paper, we give a lower bound of (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(nd/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 quantum 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 nd cells is (n(d-1)/2 ). For d = 2, this result improves the bound of (n1/4 ) 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 sea...

• Type: Journal
• Year: 2010
• Where: ALGORITHMICA
• Authors: Xi Chen, Xiaoming Sun, Shang-Hua Teng
• Keywords: ALGORITHMICA 2010, Fixed Point, Quantum, Quantum Query Complexity