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The hidden subgroup problem (HSP) plays an important role in quantumcomputation, because many quantum algorithms that are exponentially faster thanclassical algorithms can be casted in the HSP structure. In this paper, wepresent a new polynomial-time quantum algorithm that solves the HSP over thegroup $\Z_{p^r} \rtimes \Z_{q^s}$, when $p^r/q= \up{poly}(\log p^r)$, where$p$, $q$ are any odd prime numbers and $r, s$ are any positive integers. Tofind the hidden subgroup, our algorithm uses the abelian quantum Fouriertransform and a reduction procedure that simplifies the problem to find cyclicsubgroups.

Solution to the Hidden Subgroup Problem for a Class of Noncommutative Groups