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arXiv quant-ph/0412033quant-phcs.DS
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一类半直积群上隐子群问题的有效量子算法

Efficient Quantum Algorithms for the Hidden Subgroup Problem over a Class of Semi-direct Product Groups

Yoshifumi Inui, Francois Le Gall

备注：
10 pages, final version. Algorithms modified to work with black-box groups too
发表信息：
Quantum Information and Computation, Vol. 7, No. 5&6 (2007), 559-570
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摘要：

本文研究了半直积群类$\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_q$上的隐子群问题（HSP），其中p和q是素数.我们首先提出了一个分类，这些群体在五个类。然后，我们描述了一个多项式时间量子算法求解HSP的所有这些类之一的组：组的形式$\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_p$，其中p是一个奇素数。我们的算法工作，即使在最一般的情况下，该组作为一个黑盒组，不一定是唯一的编码。最后，我们推广了这个结果，给出了一个求解群$\mathbb{Z}^m_{p^r}\rtimes\mathbb{Z}_p$上HSP的有效算法。

英文摘要：

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 $\mathbb{Z}^m_{p^r}\rtimes\mathbb{Z}_p$.

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