An exact quantum hidden subgroup algorithm and applications to solvable groups

We present a polynomial time exact quantum algorithm for the hidden subgroup problem in $\Z_{m^k}^n$. The algorithm uses the quantum Fourier transform modulo $m$ and does not require factorization of $m$. For smooth $m$, i.e., when the prime factors of $m$ are of size $(\log m)^{O(1)}$, the quantum Fourier transform can be exactly computed using the method discovered independently by Cleve and Coppersmith, while for general $m$, the algorithm of Mosca and Zalka is available. Even for $m=3$ and $k=1$ our result appears to be new. We also present applications to compute the structure of abelian and solvable groups whose order has the same (but possibly unknown) prime factors as $m$. The applications for solvable groups also rely on an exact version of a technique proposed by Watrous for computing the uniform superposition of elements of subgroups.