The Symmetric Group Defies Strong Fourier Sampling: Part II

TL;DR

This paper proves that the hidden subgroup problem over the symmetric group remains unsolvable by any entangled measurement on coset states, extending prior results to more general measurement strategies.

Contribution

It demonstrates that even with entangled measurements on pairs of coset states, the hidden subgroup problem for the symmetric group cannot be efficiently solved.

Findings

Strong Fourier sampling fails even with entangled measurements

Hidden subgroups cannot be identified by polynomial experiments on coset states

Results apply to the symmetric group and related problems like Graph Isomorphism

Abstract

Part I of this paper showed that the hidden subgroup problem over the symmetric group--including the special case relevant to Graph Isomorphism--cannot be efficiently solved by strong Fourier sampling, even if one may perform an arbitrary POVM on the coset state. In this paper, we extend these results to entangled measurements. Specifically, we show that the hidden subgroup problem on the symmetric group cannot be solved by any POVM applied to pairs of coset states. In particular, these hidden subgroups cannot be determined by any polynomial number of one- or two-register experiments on coset states.