Consider the following generalized hidden shift problem:
given a function f on {0,...,M − 1} × ZN promised to be
injective for fixed b and satisfying f(b, x) = f(b + 1, x + s)
for b = 0, 1,...,M − 2, find the unknown shift s ∈ ZN.
For M = N, this problem is an instance of the abelian
hidden subgroup problem, which can be solved efficiently on
a quantum computer, whereas for M = 2, it is equivalent
to the dihedral hidden subgroup problem, for which no
efficient algorithm is known. For any fixed positive �, we give
an efficient (i.e., poly(logN)) quantum algorithm for this
problem provided M ≥ N^∈. The algorithm is based on the
“pretty good measurement” and uses H. Lenstra’s (classical)
algorithm for integer programming as a subroutine
