The Optimal Single Copy Measurement for the Hidden Subgroup Problem

TL;DR

This paper determines the optimal single-copy measurement for the hidden subgroup problem when all subgroups are equally likely, generalizing previous results and combining earlier measurement strategies.

Contribution

It introduces a new optimal measurement strategy for the hidden subgroup problem with uniformly distributed subgroups, unifying previous approaches.

Findings

Identifies the optimal measurement for uniformly distributed subgroups

Shows the optimal measurement is a hybrid of previous strategies

Enhances understanding of quantum measurements for subgroup problems

Abstract

The optimization of measurements for the state distinction problem has recently been applied to the theory of quantum algorithms with considerable successes, including efficient new quantum algorithms for the non-abelian hidden subgroup problem. Previous work has identified the optimal single copy measurement for the hidden subgroup problem over abelian groups as well as for the non-abelian problem in the setting where the subgroups are restricted to be all conjugate to each other. Here we describe the optimal single copy measurement for the hidden subgroup problem when all of the subgroups of the group are given with equal a priori probability. The optimal measurement is seen to be a hybrid of the two previously discovered single copy optimal measurements for the hidden subgroup problem.