Solving Nonlinear Partial Differential Equations via a Hybrid Newton Method Using Quantum Linear System Solver

TL;DR

This paper introduces a hybrid quantum-classical method using a novel quantum linear system solver to efficiently approximate solutions to complex nonlinear partial differential equations, potentially enhancing computational capabilities in fluid dynamics.

Contribution

A new quantum linear system solver integrated with Newton's method for solving nonlinear PDEs, requiring less eigenvalue information and enabling hybrid quantum-classical computation.

Findings

Demonstrates how quantum computation can improve PDE solving efficiency.

Provides a resource estimation for practical quantum PDE applications.

Shows potential for handling nonlinearities with quantum systems.

Abstract

To approximate solutions of complex nonlinear partial differential equations remains a computational challenge, especially for sets of equations relevant in industry, such as Euler or Navier-Stokes equations. Even the most sophisticated computational fluid dynamic algorithms coupled with powerful supercomputers can not find approximate solutions for several design challenges in both adequate time and scale-resolving accuracy. One difficulty arises from solving high dimensional, strongly nonlinear partial differential equations, such as the Navier-Stokes equations, which capture the underlying physics. For nearly all classical algorithms, methods closely related to Newton's method are used to approximate a solution to the problem. Approximately solving the large-scale linear systems of equations occurring in this iterative scheme is generally a main contributor to the total computational complexity. In this paper a new quantum linear system solver supporting Newton's classical method to solve nonlinear partial differential equations is introduced. We present a new variant of the HHL algorithm, requiring less apriori information regarding the eigenvalues of the corresponding matrix. We apply this quantum linear system solver in a hybrid quantum-classical fashion to solve nonlinear partial differential equations. Moreover, a resource estimation for advanced use-cases of practical relevance is provided. Our results demonstrate how quantum computation may improve existing classical methodologies for solving nonlinear partial differential equations. This approach provides another promising application of quantum computers and presents a possible way forward for handling nonlinearities on inherently linear quantum systems.