An Efficient Quantum Algorithm for some Instances of the Group Isomorphism Problem

TL;DR

This paper introduces a quantum algorithm that significantly accelerates the process of testing isomorphism between certain nonabelian groups, outperforming classical methods by exponential factors.

Contribution

It presents the first quantum algorithm for specific nonabelian group isomorphism problems with exponential speedup over classical algorithms.

Findings

Quantum algorithm runs in polynomial time in the logarithm of group order.

First exponential speedup for nonabelian group isomorphism testing.

Applicable to groups formed as extensions of abelian groups by cyclic groups.

Abstract

In this paper we consider the problem of testing whether two finite groups are isomorphic. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian groups. Le Gall has constructed an efficient classical algorithm for a class of groups corresponding to one of the most natural ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group $A$ by a cyclic group $Z_m$ with the order of $A$ coprime with $m$. More precisely, the running time of that algorithm is almost linear in the order of the input groups. In this paper we present a quantum algorithm solving the same problem in time polynomial in the logarithm of the order of the input groups. This algorithm works in the black-box setting and is the first quantum algorithm solving instances of the nonabelian group isomorphism problem exponentially faster than the best known classical algorithms.