DETERMINISTIC ALGORITHMS FOR THE HIDDEN SUBGROUP PROBLEM

We present deterministic algorithms for the Hidden Subgroup Problem. The first algorithm, for abelian groups, achieves the same asymptotic worst-case query complexity as the optimal randomized algorithm, namely O(root n), where n is the order of the group. The analogous algorithm for non-abelian groups comes within a root log n factor of the optimal randomized query complexity. The best known randomized algorithm for the Hidden Subgroup Problem has expected query complexity that is sensitive to the input, namely O(root n/m), where m is the order of the hidden subgroup. In the first version of this article [1, Sec. 5], we asked if there is a deterministic algorithm whose query complexity has a similar dependence on the order of the hidden subgroup. Prompted by this question, Ye and Li [2] present deterministic algorithms for abelian groups which solve the problemwith O (root n/m) queries, and find the hidden subgroup with O(root n (logm)/m + logm) queries. Moreover, they exhibit instances which show that in general, the deterministic query complexity of the problem may be o(p n/m), and that of finding the entire subgroup may also be o(root n/m) or even omega(root n/m). We present a different deterministic algorithm for the Hidden Subgroup Problem that also has query complexity O(root n/m) for abelian groups. The algorithm is arguably simpler. Moreover, it works for non-abelian groups, and has query complexity O(p (n/ m) log(n/m)) for a large class of instances, such as those over supersolvable groups. We build on this to design deterministic algorithms to find the hidden subgroup for all abelian and some nonabelian instances, at the cost of a logm multiplicative factor increase in the query complexity.