Abstract
We introduce the Hidden Polynomial Function Graph Problem as a natural generalization of an abelian Hidden Subgroup Problem (HSP) where the subgroups and their cosets correspond to graphs of linear functions over the finite field F_p. For the Hidden Polynomial Function Graph Problem the functions are not restricted to be linear but can also be multivariate polynomial functions of higher degree. For a fixed number of indeterminates and bounded total degree the Hidden Polynomial Function Graph Problem is hard on a classical computer as its black box query complexity is polynomial in p. In contrast, this problem can be reduced to a quantum state identification problem so that the resulting quantum query complexity does not depend on p. For univariate polynomials we construct a von Neumann measurement for distinguishing the states. We relate the success probability and the implementation of this measurement to certain classical problems involving polynomial equations. We present an effic