Efficient quantum algorithms for the hidden subgroup problem over
semi-direct product groups
(pp559-570)
Yoshifumi
Inui and Francois Le Gall
doi:
https://doi.org/10.26421/QIC7.5-6-9
Abstracts:
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 $\mathbb{Z}^m_{p^r}\rtimes\mathbb{Z}_p$.
Key words: Quantum
Algorithms, Hidden Subgroup Problem, Black-box Groups |