A quantum algorithm to solve nonlinear differential equations

TL;DR

This paper presents a quantum algorithm that efficiently solves sparse nonlinear differential equations with polynomial nonlinearities, offering an exponential speedup over classical methods.

Contribution

It introduces a novel quantum algorithm combining nonlinear amplitude transformations and Euler's method for solving differential equations.

Findings

Expected resource requirements are polylogarithmic in variables

Achieves exponential speedup over classical algorithms

Applicable to sparse systems with polynomial nonlinearities

Abstract

In this paper we describe a quantum algorithm to solve sparse systems of nonlinear differential equations whose nonlinear terms are polynomials. The algorithm is nondeterministic and its expected resource requirements are polylogarithmic in the number of variables and exponential in the integration time. The best classical algorithm runs in a time scaling linearly with the number of variables, so this provides an exponential improvement. The algorithm is built on two subroutines: (i) a quantum algorithm to implement a nonlinear transformation of the probability amplitudes of an unknown quantum state; and (ii) a quantum implementation of Euler's method.