On solving systems of random linear disequations

An important subcase of the hidden subgroup problem is equivalent to the shift problem over abelian groups. An efficient solution to the latter problem would serve as a building block of quantum hidden subgroup algorithms over solvable groups. The main idea of a promising approach to the shift problem is reduction to solving systems of certain random disequations in finite abelian groups. The random disequations are actually generalizations of linear functions distributed nearly uniformly over those not containing a specific group element in the kernel. In this paper we give an algorithm which finds the solutions of a system of N random linear disequations in an abelian p-group A in time polynomial in N, where N=(log|A|)^{O(q)}, and q is the exponent of A.Comment: 13 page

The Optimal Single Copy Measurement for the Hidden Subgroup Problem

The optimization of measurements for the state distinction problem has recently been applied to the theory of quantum algorithms with considerable successes, including efficient new quantum algorithms for the non-abelian hidden subgroup problem. Previous work has identified the optimal single copy measurement for the hidden subgroup problem over abelian groups as well as for the non-abelian problem in the setting where the subgroups are restricted to be all conjugate to each other. Here we describe the optimal single copy measurement for the hidden subgroup problem when all of the subgroups of the group are given with equal a priori probability. The optimal measurement is seen to be a hybrid of the two previously discovered single copy optimal measurements for the hidden subgroup problem.Comment: 8 pages. Error in main proof fixe