Efficient Quantum Algorithm For Identifying Hidden Polynomials

We consider a natural generalization of an abelian Hidden Subgroup Problem where the subgroups and their cosets correspond to graphs of linear functions over a finite field F with d elements. The hidden functions of the generalized problem are not restricted to be linear but can also be m - variate polynomial functions of total degree n ≥ 2. The problem of identifying hidden m - variate polynomials of degree less or equal to n for fixed n and m is hard on a classical computer since ω(√d) black - b ox queries are required to guarantee a constant success probability. In contrast, we present a quantum algorithm that correctly identifies such hidden polynomials for all but a finite number of values of d with constant probability and that has a running time that is only polylogarithmic in d. © Rinton Press