Modified group non-membership is in AWPP

TL;DR

This paper proves that the modified group non-membership problem, with group order as input, is in AWPP relative to any group oracle, using postselected quantum computing techniques.

Contribution

It introduces a new variant of the group non-membership problem and shows it belongs to AWPP, expanding understanding of its computational complexity.

Findings

The modified problem is in AWPP relative to any group oracle.

Uses postselected quantum computing to establish the complexity class.

Extends previous results on group non-membership problems.

Abstract

It is known that the group non-membership problem is in QMA relative to any group oracle and in ${\rm SPP}\cap{\rm BQP}$ relative to group oracles for solvable groups. We consider a modified version of the group non-membership problem where the order of the group is also given as an additional input. We show that the problem is in AWPP relative to any group oracle. To show the result, we use the idea of the postselected quantum computing.