Efficient quantum algorithm for identifying hidden polynomials (pp0215-0230)
          Thomas Decker, Jan Draisma, and Pawel Wocjan
          doi: https://doi.org/10.26421/QIC9.3-4-3
Abstracts: 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(\sqrt d) black-box 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.
Key words: quantum algorithms, hidden subgroup problems, hidden polynomial problems