Revisiting finite Abelian hidden subgroup problem and its distributed exact quantum algorithm

TL;DR

This paper introduces more efficient and exact quantum algorithms for the finite Abelian hidden subgroup problem, utilizing amplitude amplification and the Chinese Remainder Theorem, with extensions to non-Abelian groups and classical algorithms.

Contribution

It presents a concise, exact quantum algorithm for finite AHSP applicable to all finite Abelian groups and a distributed quantum approach requiring fewer resources, extending to some non-Abelian groups.

Findings

More concise exact quantum algorithm using amplitude amplification.

Distributed quantum algorithm with fewer qudits and no quantum communication.

Classical parallel algorithm with reduced query complexity.

Abstract

We revisit the finite Abelian hidden subgroup problem (AHSP) from a mathematical perspective and make the following contributions. First, by employing amplitude amplification, we present an exact quantum algorithm for the finite AHSP, our algorithm is more concise than the previous exact algorithm and applies to any finite Abelian group. Second, utilizing the Chinese Remainder Theorem, we propose a distributed exact quantum algorithm for finite AHSP, which requires fewer qudits, lower quantum query complexity, and no quantum communication. We further show that our distributed approach can be extended to certain classes of non-Abelian groups. Finally, we develop a parallel exact classical algorithm for finite AHSP with reduced query complexity; even without parallel execution, the total number of queries across all nodes does not exceed that of the original centralized algorithm under mild conditions.