Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations

TL;DR

This paper develops quantum algorithms for solving nonlinear differential equations by transforming them into linear PDEs, comparing different quantum methods and proposing a Schr"odinger framework for improved efficiency.

Contribution

It introduces a novel approach using linear representations to analyze nonlinear ODEs and HJE with quantum algorithms, including a Schr"odinger framework for the Liouville equation.

Findings

Quantum simulation methods outperform linear systems algorithms in time complexity.

The Schr"odinger framework offers a promising approach for solving the Liouville equation.

Comparison shows different discretisations impact quantum algorithm efficiency.

Abstract

We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schr\"odinger framework to solve the Liouville equation for the HJE with the Hamiltonian formulation of classical mechanics, since it can be recast as the semiclassical limit of the Wigner transform of the Schr\"odinger equation. Comparsion between the Schr\"odinger and the Liouville framework will also be made.