Optimal Separation in Exact Query Complexities for Simon's Problem

Simon’s problem is one of the most important problems demonstrating the power of quantum computers, which achieves a large separation between quantum and classical query complexities. However, Simon’s discussion on his problem was limited to bounded-error setting, which means his algorithm can not always get the correct answer. Exact quantum algorithms for Simon’s problem have also been proposed, which deterministically solve the problem with O(n) queries. Although these algorithms are either complicated or specialized, their results give a O(n) versus Ω( √ 2n) separation in exact query complexities for Simon’s problem. Later results indicate the the quantum lower bound for Simon’s problem, but it has not been proved whether this separation is optimal. In this paper, we propose another exact quantum algorithm for solving Simon’s problem with O(n) queries, which is simple, concrete and does not rely on some special query oracles. Our algorithm combines Simon’s algorithm with the quantum amplitude amplification technique to ensure its determinism. In particular, we show that Simon’s problem can be solved by a classically deterministic algorithm with O( √ 2n) queries (as we are aware, there were no classically deterministic algorithms for solving Simon’s problem with O( √ 2n) queries). Combining some previous results, we prove the optimal separation in exact query complexities for Simon’s problem: Θ(n) versus Θ( √ 2n).