Solving nonlinear differential equations on Quantum Computers: A Fokker-Planck approach

TL;DR

This paper introduces quantum algorithms for solving nonlinear differential equations by transforming them into linear systems using the Fokker-Planck approach, demonstrating promising results on prototype systems.

Contribution

It proposes a novel method to transform nonlinear dynamical systems into linear forms suitable for quantum algorithms, using the Fokker-Planck equation and three integration strategies.

Findings

Quantum and classical solutions agree well on prototype systems

The proposed methods enable solving nonlinear equations on quantum hardware

Open opportunities for quantum advantage in nonlinear differential equations

Abstract

For quantum computers to become useful tools to physicists, engineers and computational scientists, quantum algorithms for solving nonlinear differential equations need to be developed. Despite recent advances, the quest for a solver that can integrate nonlinear dynamical systems with a quantum advantage, whilst being realisable on available (or near-term) quantum hardware, is an open challenge. In this paper, we propose to transform a nonlinear dynamical system into a linear system, which we integrate with quantum algorithms. Key to the method is the Fokker-Planck equation, which is a non-normal partial differential equation. Three integration strategies are proposed: (i) Forward-Euler stepping by unitary block encoding; (ii) Schroedingerisation, and (iii) Forward-Euler stepping by linear addition of unitaries. We emulate the integration of prototypical nonlinear systems with the proposed quantum solvers, and compare the output with the benchmark solutions of classical integrators. We find that classical and quantum outputs are in good agreement. This paper opens opportunities for solving nonlinear differential equations with quantum algorithms.