  We present a negative result regarding the hidden subgroup problem on the
powers $G^n$ of a fixed group $G$. Under a condition on the base group $G$, we
prove that strong Fourier sampling cannot distinguish some subgroups of $G^n$.
Since strong sampling is in fact the optimal measurement on a coset state, this
shows that we have no hope of efficiently solving the hidden subgroup problem
over these groups with separable measurements on coset states (that is, using
any polynomial number of single-register coset state experiments). Base groups
satisfying our condition include all nonabelian simple groups. We apply our
results to show that there exist uniform families of nilpotent groups whose
normal series factors have constant size and yet are immune to strong Fourier
sampling.
