Efficient Quantum Algorithm for Identifying Hidden Polynomials

TL;DR

This paper introduces a quantum algorithm capable of efficiently identifying hidden multivariate polynomials over finite fields, significantly outperforming classical methods in terms of query complexity.

Contribution

The paper presents a novel quantum algorithm for identifying hidden multivariate polynomials of degree n over finite fields, with polylogarithmic query complexity.

Findings

Quantum algorithm correctly identifies hidden polynomials with high probability

Classical approach requires Omega(sqrt{d}) queries, quantum method is more efficient

Algorithm works for all but finitely many field sizes d

Abstract

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 Omega(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.