For Distinguishing Conjugate Hidden Subgroups, the Pretty Good Measurement is as Good as it Gets

TL;DR

This paper proves that the pretty good measurement (PGM) is optimal for distinguishing conjugate hidden subgroups across various groups, extending previous results and providing bounds on success probabilities.

Contribution

It generalizes the optimality of PGM for conjugate hidden subgroups to all groups and subgroups, especially when forming a Gel'fand pair, and offers success probability bounds.

Findings

PGM is optimal for conjugate hidden subgroups in all groups.

When H forms a Gel'fand pair, PGM is optimal for any number of registers.

Provides bounds on the success probability of the optimal measurement.

Abstract

Recently Bacon, Childs and van Dam showed that the ``pretty good measurement'' (PGM) is optimal for the Hidden Subgroup Problem on the dihedral group D_n in the case where the hidden subgroup is chosen uniformly from the n involutions. We show that, for any group and any subgroup H, the PGM is the optimal one-register experiment in the case where the hidden subgroup is a uniformly random conjugate of H. We go on to show that when H forms a Gel'fand pair with its parent group, the PGM is the optimal measurement for any number of registers. In both cases we bound the probability that the optimal measurement succeeds. This generalizes the case of the dihedral group, and includes a number of other examples of interest.