For distinguishing conjugate hidden subgroups, the pretty good measurement is as good as it gets

Recently Bacon, Childs and van Dam showed that the “pretty good measurement” (PGM) is optimal for the Hidden Subgroup Problem on the dihedral group Dn 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. This generalizes the case of the dihedral group, and includes a number of other examples of interest. 1 The Hidden Conjugate Problem Consider the following special case of the Hidden Subgroup Problem, called the Hidden Conjugate Problem in [13]. Let G be a group, and H a non-normal subgroup of G; denote conjugates of H as H = gHg. Then we are promised that the hidden subgroup is H for some g, and our goal is to find out which one. The usual approach is to prepare a uniform superposition over the group, entangle the group element with a second register by calculating or querying the oracle function, and then measure the oracle function. This yields a uniform superposition over a random left coset of the hidden subgroup,