Notes on the hidden subgroup problem on some semi-direct product groups

TL;DR

This paper investigates the hidden subgroup problem on certain semi-direct product groups, providing a classification of subgroup structures and demonstrating an efficient quantum algorithm for solving the problem in this class.

Contribution

The paper introduces a class of semi-direct product groups with easily classifiable subgroups and presents an efficient quantum algorithm for the hidden subgroup problem on these groups.

Findings

Classified subgroup structures in the specified semi-direct product groups

Developed an efficient quantum algorithm for the hidden subgroup problem in this class

Identified conditions on N and p for the algorithm's applicability

Abstract

We consider the hidden subgroup problem on the semi-direct product of cyclic groups $\Z_{N}\rtimes\Z_{p}$ with some restriction on $N$ and $p$. By using the homomorphic properties, we present a class of semi-direct product groups in which the structures of subgroups can be easily classified. Furthermore, we show that there exists an efficient quantum algorithm for the hidden subgroup problem on the class.