Quantum computational algorithm for hidden symmetry subgroup problems on semi-direct product of cyclic groups

TL;DR

This paper develops a quantum algorithm to efficiently solve hidden symmetry subgroup problems for a specific class of semi-direct product groups, expanding quantum capabilities in group theory problems.

Contribution

It introduces a polynomial-time quantum algorithm for hidden subgroup problems on semi-direct products of cyclic groups with particular algebraic properties.

Findings

Algorithm successfully solves the problem in polynomial time

Extends quantum solutions to new classes of semi-direct product groups

Provides algebraic characterization of the groups involved

Abstract

We characterize the algebraic structure of semi-direct product of cyclic groups, $\Z_{N}\rtimes\Z_{p}$, where $p$ is an odd prime number which does not divide $q-1$ for any prime factor $q$ of $N$, and provide a polynomial-time quantum computational algorithm solving hidden symmetry subgroup problem of the groups.