Quantum algorithms for shifted subset problems

We consider a recently proposed generalisation of the abelian hidden subgroup problem: the shifted subset problem. The problem is to determine a subset S of some abelian group, given access to quantum states of the form |S+x>, for some unknown shift x. We give quantum algorithms to find Hamming spheres and other subsets of the boolean cube {0,1}^n. The algorithms have time complexity polynomial in n and give rise to exponential separations from classical computation.