Quantum Algorithm for Solving a Quadratic Nonlinear System of Equations

TL;DR

This paper presents a quantum algorithm that exponentially accelerates solving quadratic nonlinear systems of equations, embedding the problem into linear systems and using quantum linear solvers for efficient solutions.

Contribution

The paper introduces a novel quantum algorithm that solves quadratic nonlinear systems of equations with exponential speedup over classical methods.

Findings

Complexity is $O({\rm polylog}(n/\epsilon))$, exponentially faster than classical algorithms.

The algorithm achieves an $\e$-close solution with high success probability.

It has wide applications in nonlinear science and related fields.

Abstract

Solving a quadratic nonlinear system of equations (QNSE) is a fundamental, but important, task in nonlinear science. We propose an efficient quantum algorithm for solving $n$-dimensional QNSE. Our algorithm embeds QNSE into a finite-dimensional system of linear equations using the homotopy perturbation method and a linearization technique; then we solve the linear equations with a quantum linear system solver and obtain a state which is $\epsilon$-close to the normalized exact solution of the QNSE with success probability $\Omega(1)$. The complexity of our algorithm is $O({\rm polylog}(n/\epsilon))$, which provides an exponential improvement over the optimal classical algorithm in dimension $n$, and the dependence on $\epsilon$ is almost optimal. Therefore, our algorithm exponentially accelerates the solution of QNSE and has wide applications in all kinds of nonlinear problems, contributing to the research progress of nonlinear science.