Quantum algorithms for solvable groups

TL;DR

This paper presents a polynomial-time quantum algorithm for solving key problems in solvable groups, including order computation and subgroup testing, outperforming classical methods in black-box group settings.

Contribution

The paper introduces the first polynomial-time quantum algorithms for several fundamental problems in solvable groups, including order computation and subgroup membership testing.

Findings

Quantum algorithms solve problems in solvable groups efficiently.

Classical algorithms cannot solve these problems in polynomial time.

Quantum states can be used to represent subgroups for further analysis.

Abstract

In this paper we give a polynomial-time quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, reduce to computing orders of solvable groups and therefore admit polynomial-time quantum algorithms as well. Our algorithm works in the setting of black-box groups, wherein none of these problems can be computed classically in polynomial time. As an important byproduct, our algorithm is able to produce a pure quantum state that is uniform over the elements in any chosen subgroup of a solvable group, which yields a natural way to apply existing quantum algorithms to factor groups of solvable groups.