by Sean Hallgren, Alexander Russell, Amnon Ta-shma
SIAM Journal on Computing
http://www.cs.caltech.edu/~hallgren/reps1.ps
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Abstract:
The Hidden Subgroup Problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over Abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The non-Abelian case is open; an efficient solution would give rise to an efficient quantum algorithm for Graph Isomorphism. We fully analyze a natural generalization of the Abelian case algorithm to the non-Abelian case. We show that the algorithm finds the normal core of the hidden subgroup, and that, in particular, normal subgroups can be found. We show, however, that this immediate generalization of the Abelian algorithm does not efficiently solve Graph Isomorphism. 1
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