Permutation groups, minimal degrees and quantum computing

Abstract

We study permutation groups of given minimal degree without the classical primitivity assumption. We provide sharp upper bounds on the order of a permutation group of minimal degree m and on the number of its elements of any given support. These results contribute to the foundations of a non-commutative coding theory. A main application of our results concerns the Hidden Subgroup Problem for the symmetric group in Quantum Computing. We completely characterize the hidden subgroups of the symmetric group that can be distinguished from identity with weak Quantum Fourier Sampling, showing these are exactly the subgroups with bounded minimal degree. This implies that the weak standard method for the symmetric group has no advantage whatsoever over classical exhaustive search.