Efficient quantum algorithms for the hidden subgroup problem over semi-direct product groups
Keyword(s):
In this paper, we consider the hidden subgroup problem (HSP) over the class of semi-direct product groups $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_q$, 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 $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_p$, 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 $\mathbb{Z}^m_{p^r}\rtimes\mathbb{Z}_p$.
2006 ◽
Vol 359
(2)
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pp. 114-116
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2003 ◽
Vol 14
(05)
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pp. 723-739
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2005 ◽
Vol 35
(1)
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pp. 170-188
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2008 ◽
Vol 105
(48)
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pp. 18681-18686
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2016 ◽
Vol 120
(32)
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pp. 6459-6466
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