Polynomial time quantum algorithms for certain bivariate hidden polynomial problems

TL;DR

This paper introduces a new quantum algorithm for solving certain bivariate hidden polynomial problems efficiently, even with complex input structures, advancing quantum capabilities in algebraic problem-solving.

Contribution

The paper develops a novel quantum approach for the bivariate hidden polynomial graph problem, handling complex superpositions and extending to elliptic curves.

Findings

Efficient quantum algorithm for bivariate HPGP with superpositions.

Polylogarithmic complexity for constant degree problems.

Applicable to elliptic curves and quadratic forms.

Abstract

We present a new method for solving the hidden polynomial graph problem (HPGP) which is a special case of the hidden polynomial problem (HPP). The new approach yields an efficient quantum algorithm for the bivariate HPGP even when the input consists of several level set superpositions, a more difficult version of the problem than the one where the input is given by an oracle. For constant degree, the algorithm is polylogarithmic in the size of the base field. We also apply the results to give an efficient quantum algorithm for the oracle version of the HPP for an interesting family of bivariate hidden functions. This family includes diagonal quadratic forms and elliptic curves.