Efficient quantum algorithms for the hidden subgroup problem over semi-direct product groups

Authors:
Yoshifumi Inui;Francois Le Gall
Affiliations:
Department of Computer Science, The University of Tokyo, Bunkyo-ku, Tokyo, Japan and ERATO-SORST Quantum Computation and Information Project, JST, Bunkyo-ku, Tokyo, Japan;Department of Computer Science, The University of Tokyo, Bunkyo-ku, Tokyo, Japan and ERATO-SORST Quantum Computation and Information Project, JST, Bunkyo-ku, Tokyo, Japan
Venue:
Quantum Information & Computation
Year:
2007

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Abstract

In this paper, we consider the hidden subgroup problem (HSP) over the class of semidirect product groups Zpr ⋊ Zq, for p and q prime. We first present a classification of these groups in five classes. Then, we describe a polynomial-time quantum algorithm solving the HSP over all the groups of one of these classes: the groups of the form Zpr ⋊ Zp, where p is an odd prime. Our algorithm works even in the most general case where the group is presented as a black-box group with not necessarily unique encoding. Finally, we extend this result and present an efficient algorithm solving the HSP over the groups Zprm ⋊ Zp.