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