The Symmetric Group Defies Strong Fourier Sampling: Part I
  We resolve the question of whether Fourier sampling can efficiently solve the hidden subgroup problem. Specifically, we show that the hidden subgroup problem over the symmetric group cannot be efficiently solved by strong Fourier sampling, even if one may perform an arbitrary POVM on the coset state. Our results apply to the special case relevant to the Graph Isomorphism problem. 
Cristopher Moore, Alexander Russell, and Leonard J. Schulman