Quantum algorithm for identifying hidden polynomial function graphs
Thomas Decker, Jan Draisma, Paweł Wocjan · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · 2008
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 a finite field F with d elements. For the Hidden Polynomial Function Graph Problem the functions are not restricted to be linear but can also be m-variate polynomial functions of total degree n ≥ 2. For fixed m and bounded n the problem is hard on a classical computer as the black box query complexity is polynomial in d. In contrast, we reduce it to a quantum state identification problem so that its query complexity is n m +n m−1 +...+n, independent of d. We derive an efficient measurement for distinguishing the resulting quantum states provided that the characteristic of F is sufficiently large. Its success probability and implementation are closely related to a classical problem involving polynomial equations.