Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem

Schur duality decomposes many copies of a quantum state into subspaces
labeled by partitions, a decomposition with applications throughout quantum
information theory. Here we consider applying Schur duality to the problem of
distinguishing coset states in the standard approach to the hidden subgroup
problem. We observe that simply measuring the partition (a procedure we call
weak Schur sampling) provides very little information about the hidden
subgroup. Furthermore, we show that under quite general assumptions, even a
combination of weak Fourier sampling and weak Schur sampling fails to identify
the hidden subgroup. We also prove tight bounds on how many coset states are
required to solve the hidden subgroup problem by weak Schur sampling, and we
relate this question to a quantum version of the collision problem.