An exact quantum hidden subgroup algorithm and applications to solvable groups

TL;DR

This paper introduces a polynomial-time exact quantum algorithm for the hidden subgroup problem in certain abelian groups, with applications to group structure determination, without requiring factorization of the group order.

Contribution

It presents a novel exact quantum algorithm for the hidden subgroup problem in groups like Z_{m^k}^n, applicable to smooth and general m, and extends to group structure analysis.

Findings

Algorithm works in polynomial time for specific groups

Exact quantum Fourier transform can be used without factorization

Applications include determining structure of abelian and solvable groups

Abstract

We present a polynomial time exact quantum algorithm for the hidden subgroup problem in $Z_{m^k}^n$. The algorithm uses the quantum Fourier transform modulo m and does not require factorization of m. For smooth m, i.e., when the prime factors of m are of size poly(log m), the quantum Fourier transform can be exactly computed using the method discovered independently by Cleve and Coppersmith, while for general m, the algorithm of Mosca and Zalka is available. Even for m=3 and k=1 our result appears to be new. We also present applications to compute the structure of abelian and solvable groups whose order has the same (but possibly unknown) prime factors as m. The applications for solvable groups also rely on an exact version of a technique proposed by Watrous for computing the uniform superposition of elements of subgroups.