Permutation groups, minimal degrees and quantum computing

Julia Kempe, László Pyber, Aner Shalev
2007 Groups, Geometry, and Dynamics  
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 H Ä S n 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 S n in quantum computing. We completely characterize the hidden subgroups of S n that can be
more » ... inguished from identity with weak Quantum Fourier Sampling, showing that these are exactly the subgroups with bounded minimal degree. This implies that the weak standard method for S n has no advantage whatsoever over classical exhaustive search. Mathematics Subject Classification (2000) . 20B35, 20C15, 68Q25, 81P68.
doi:10.4171/ggd/24 fatcat:bhe56se7szbszbhjkgj6bbfhwy