Optimal single-copy measurement for the hidden-subgroup problem

The optimization of measurements for the state distinction problem has recently been applied to the theory of quantum algorithms with considerable successes, including efficient alternative 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.