On the Quantum Query Complexity of Local Search in Two and Three Dimensions

The quantum query complexity of searching for local optima has been a subject of much interest in the recent literature. For the d-dimensional grid graphs, the complexity has been determined asymptotically for all fixed d \geqslant 5, but the lower dimensional cases present special difficulties, and considerable gaps exist in our knowledge. In the present paper we present near-optimal lower bounds, showing that the quantum query complexity for the 2-dimensional grid [n]^2 is \Omega \left( {n^{1/2 - \delta } } \right), and that for the 3-dimensional grid [n]^3 is \Omega (n^{1- \delta } ), for any fixed \delta \ge 0. A general lower bound approach for this problem, initiated by Aaronson [1](based on Ambainis? adversary method [3] for quantum lower bounds), uses random walks with low collision probabilities. This approach encounters obstacles in deriving tight lower bounds in low dimensions due to the lack of degrees of freedom in such spaces. We solve this problem by the novel construction and analysis of random walks with non-uniform step lengths. The proof employs in a nontrivial way sophisticated results of Sarkozy and Szemeredi [14], Bose and Chowla [5], and Halasz [9] from combinatorial number theory, as well as less familiar probability tools like Esseen?s Inequality.