ar X iv : 0 70 7 . 12 60 v 1 [ qu an t - ph ] 9 J ul 2 00 7 An efficient quantum algorithm for the hidden subgroup problem in nil - 2 groups ∗
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چکیده
In this paper we extend the algorithm for extraspecial groups in [12], and show that the hidden subgroup problem in nil-2 groups, that is in groups of nilpotency class at most 2, can be solved efficiently by a quantum procedure. The algorithm presented here has several additional features. It contains a powerful classical reduction for the hidden subgroup problem in nilpotent groups of constant nilpotency class to the specific case where the group is a p-group of exponent p and the subgroup is either trivial or cyclic. This reduction might also be useful for dealing with groups of higher nilpotency class. The quantum part of the algorithm uses well chosen group actions based on some automorphisms of nil-2 groups. The right choice of the actions requires the solution of a system of quadratic and linear equations. The existence of a solution is guaranteed by the Chevalley-Warning theorem, and we prove that it can also be found efficiently. Research supported by the European Commission IST Integrated Project Qubit Applications (QAP) 015848, the OTKA grants T42559 and T46234, the NWO visitor’s grant Algebraic Aspects of Quantum Computing, and by the ANR Blanc AlgoQP grant of the French Research Ministry. SZTAKI, Hungarian Academy of Sciences, H-1111 Budapest, Hungary. [email protected] UMR 8623 Université Paris–Sud 91405 Orsay, France. [email protected] CNRS–LRI, UMR 8623 Université Paris–Sud 91405 Orsay, France. [email protected]
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تاریخ انتشار 2008