Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

TL;DR

This paper introduces a quantum algorithm based on homotopy perturbation for efficiently solving nonlinear dissipative ordinary differential equations, achieving exponential speedup over classical methods.

Contribution

It develops a novel quantum algorithm that transforms nonlinear ODEs into linear ones and solves them with exponential efficiency gains.

Findings

Achieves exponential improvement in solving nonlinear ODEs over classical algorithms.

Successfully embeds nonlinear ODEs into linear ODEs for quantum solution.

Provides a quantum method with complexity polylogarithmic in problem parameters.

Abstract

While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation method, we propose a quantum algorithm for solving $n$-dimensional nonlinear dissipative ordinary differential equations (ODEs). Our algorithm first converts the original nonlinear ODEs into the other nonlinear ODEs which can be embedded into finite-dimensional linear ODEs. Then we solve the embedded linear ODEs with quantum linear ODEs algorithm and obtain a state $\epsilon$-close to the normalized exact solution of the original nonlinear ODEs with success probability $\Omega(1)$. The complexity of our algorithm is $O(g\eta T{\rm poly}(\log(nT/\epsilon)))$, where $\eta$, $g$ measure the decay of the solution. Our algorithm provides exponential improvement over the best classical algorithms or previous quantum algorithms in $n$ or $\epsilon$.