Quantum and Classical Query Complexities for Generalized Simon's Problem

Abstract: Simon's problem is an essential example demonstrating the faster speed of quantum computers than classical computers for solving some problems. The optimal separation between exact quantum and classical query complexities for Simon's problem has been proved by Cai $&$ Qiu. Generalized Simon's problem can be described as follows. Given a function $f:{{0, 1}}^n \to {{0, 1}}^m$, with the property that there is some unknown hidden subgroup $S$ such that $f(x)=f(y)$ iff $x \oplus y\in S$, for any $x, y\in {{0, 1}}^n$, where $|S|=2^k$ for some $0\leq k\leq n$. The goal is to find $S$. For the case of $k=1$, it is Simon's problem. In this paper, we propose an exact quantum algorithm with $O(n-k)$ queries and an non-adaptive deterministic classical algorithm with $O(k\sqrt{2^{n-k}})$ queries for solving the generalized Simon's problem. Also, we prove that their lower bounds are $\Omega(n-k)$ and $\Omega(\sqrt{k2^{n-k}})$, respectively. Therefore, we obtain a tight exact quantum query complexity $\Theta(n-k)$ and an almost tight non-adaptive classical deterministic query complexities $\Omega(\sqrt{k2^{n-k}}) \sim O(k\sqrt{2^{n-k}})$ for this problem.