Strong Fourier Sampling Fails over $G^n$

TL;DR

This paper proves that strong Fourier sampling cannot distinguish certain subgroups in powers of a fixed group G, including all nonabelian simple groups, indicating limitations of this approach for solving the hidden subgroup problem.

Contribution

It establishes a negative result showing the failure of strong Fourier sampling for specific groups, highlighting fundamental limitations in quantum algorithms for these cases.

Findings

Strong Fourier sampling fails for some subgroups of G^n.

Applicable to all nonabelian simple groups.

Shows limitations for quantum algorithms on certain nilpotent groups.

Abstract

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.