Quantum Property Testing of Group Solvability

TL;DR

This paper introduces a quantum algorithm that efficiently tests whether a finite set with a binary operation forms a solvable group or is far from any solvable group, extending previous work on Abelian groups.

Contribution

It presents the first quantum property testing algorithm for solvable groups, improving the scope of group property testing beyond Abelian groups.

Findings

Quantum algorithm uses polylogarithmic queries

Efficiently distinguishes solvable groups from non-solvable ones

Extends property testing to a broader class of groups

Abstract

Testing efficiently whether a finite set with a binary operation over it, given as an oracle, is a group is a well-known open problem in the field of property testing. Recently, Friedl, Ivanyos and Santha have made a significant step in the direction of solving this problem by showing that it it possible to test efficiently whether the input is an Abelian group or is far, with respect to some distance, from any Abelian group. In this paper, we make a step further and construct an efficient quantum algorithm that tests whether the input is a solvable group, or is far from any solvable group. More precisely, the number of queries used by our algorithm is polylogarithmic in the size of the set.