Deterministic Algorithms for the Hidden Subgroup Problem

Abstract

We consider deterministic algorithms for the well-known hidden subgroup problem ($\mathsf{HSP}$): for a finite group $G$ and a finite set $X$, given a function $f:G \to X$ and the promise that for any $g_1, g_2 \in G, f(g_1) = f(g_2)$ iff $g_1H=g_2H$ for a subgroup $H \le G$, the goal of the decision version is to determine whether $H$ is trivial or not, and the goal of the identification version is to identify $H$. An algorithm for the problem should query $f(g)$ for $g\in G$ at least as possible. Nayak asked whether there exist deterministic algorithms with $O(\sqrt{\frac{|G|}{|H|}})$ query complexity for $\mathsf{HSP}$. We answer this problem by proving the following results, which also extend the main results of Ref. [30], since here the algorithms do not rely on any prior knowledge of $H$. (i)When $G$ is a general finite Abelian group, there exist an algorithm with $O(\sqrt{\frac{|G|}{|H|}})$ queries to decide the triviality of $H$ and an algorithm to identify $H$ with $O(\sqrt{\frac{|G|}{|H|}\log |H|}+\log |H|)$ queries. (ii)In general there is no deterministic algorithm for the identification version of $\mathsf{HSP}$ with query complexity of $O(\sqrt{\frac{|G|}{|H|}})$, since there exists an instance of $\mathsf{HSP}$ that needs $\omega(\sqrt{\frac{|G|}{|H|}})$ queries to identify $H$. $f(x)$ is said to be $\omega(g(x))$ if for every positive constant $C$, there exists a positive constant $N$ such that for $x>N$, $f(x)\ge C\cdot g(x)$, which means $g$ is a strict lower bound for $f$. On the other hand, there exist instances of $\mathsf{HSP}$ with query complexity far smaller than $O(\sqrt{\frac{|G|}{|H|}})$, whose query complexity is $O(\log \frac{|G|}{|H|})$ and even $O(1)$.