Deterministic Algorithms for the Hidden Subgroup Problem

TL;DR

This paper investigates deterministic algorithms for the hidden subgroup problem, establishing bounds on query complexity for different group types and showing limitations of deterministic approaches.

Contribution

It provides new deterministic algorithms with optimal query bounds for Abelian groups and proves the impossibility of such bounds in general cases.

Findings

Deterministic algorithms exist with $O(\sqrt{|G|/|H|})$ queries for Abelian groups.

No universal deterministic algorithm with $O(\sqrt{|G|/|H|})$ queries for all groups.

Some instances require more than $O(\sqrt{|G|/|H|})$ queries, while others need fewer, like $O(1)$ or $O(\log |G|/|H|)$.

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 $O(1)$.