An efficient quantum algorithm for the hidden subgroup problem in nil-2 groups

TL;DR

This paper presents a quantum algorithm that efficiently solves the hidden subgroup problem in nil-2 groups, extending previous work and introducing a classical reduction and automorphism-based quantum techniques.

Contribution

It extends quantum algorithms for extraspecial groups to all nil-2 groups and introduces a classical reduction to p-groups with trivial or cyclic subgroups.

Findings

Efficient quantum algorithm for nil-2 groups

Classical reduction to p-groups of exponent p

Solution of quadratic and linear equations via Chevalley-Warning theorem

Abstract

In this paper we extend the algorithm for extraspecial groups in \cite{iss07}, and show that the hidden subgroup problem in nil-2 groups, that is in groups of nilpotency class at most 2, can be solved efficiently by a quantum procedure. The algorithm presented here has several additional features. It contains a powerful classical reduction for the hidden subgroup problem in nilpotent groups of constant nilpotency class to the specific case where the group is a $p$-group of exponent $p$ and the subgroup is either trivial or cyclic. This reduction might also be useful for dealing with groups of higher nilpotency class. The quantum part of the algorithm uses well chosen group actions based on some automorphisms of nil-2 groups. The right choice of the actions requires the solution of a system of quadratic and linear equations. The existence of a solution is guaranteed by the Chevalley-Warning theorem, and we prove that it can also be found efficiently.