Solutions to the Hidden Subgroup Problem on Some Metacyclic Groups

We describe a new polynomial-time quantum algorithm that solves the hidden subgroup problem (HSP) for a special class of metacyclic groups, namely $\mathbb{Z}_{p} \rtimes \mathbb{Z}_{q^s}$, with q |(p−1) and p/q=poly(log p), where p, q are any odd prime numbers and s is any positive integer. This solution generalizes previous algorithms presented in the literature. In a more general setting, without imposing a relation between p and q, we obtain a quantum algorithm with time and query complexity $2^{O(\sqrt{\log p})}$. In any case, those results improve the classical algorithm, which needs ${\Omega}(\sqrt{p})$ queries.