An Improved Query for the Hidden Subgroup Problem

TL;DR

This paper introduces a new character query for the hidden subgroup problem that improves success probability over standard methods, especially for conjugate subgroups, by generalizing the phase kickback trick.

Contribution

The authors propose the character query, a novel approach that enhances subgroup identification success rates in abelian HSPs, extending the phase kickback technique.

Findings

Character query outperforms standard equal superposition query

Success probability increases for conjugate subgroups

Method generalizes phase kickback trick

Abstract

An equal superposition query with |0> in the response register is used in the "standard method" of single-query algorithms for the hidden subgroup problem (HSP). Here we introduce a different query, the character query, generalizing the well-known phase kickback trick. This query maximizes the success probability of subgroup identification under a uniform prior, for the HSP in which the oracle functions take values in a finite abelian group. We then apply our results to the case when the subgroups are drawn from a set of conjugate subgroups and obtain a success probability greater than that found by Moore and Russell.