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

摘要

In this paper, we consider the hidden subgroup problem (HSP) over the class of semi-direct product groups Z(p)(r) X Z(q), 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 Z(p)(r) X Z(p), 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 Z(pr)(m) x Z(p)