Variational Quantum Evolution Equation Solver

TL;DR

This paper introduces a variational quantum algorithm for solving evolution equations, demonstrating faster convergence and scalability on near-term quantum computers, with extensions to complex systems like Navier-Stokes.

Contribution

It proposes a novel variational quantum algorithm for evolution equations, including implicit time-stepping and semi-implicit schemes for non-linear systems, extending quantum PDE solving capabilities.

Findings

Statevector simulations show favorable scaling with ansatz volume.

Time-to-solution scales with the diffusion parameter.

Semi-implicit scheme valid for non-linear evolution equations.

Abstract

Variational quantum algorithms offer a promising new paradigm for solving partial differential equations on near-term quantum computers. Here, we propose a variational quantum algorithm for solving a general evolution equation through implicit time-stepping of the Laplacian operator. The use of encoded source states informed by preceding solution vectors results in faster convergence compared to random re-initialization. Through statevector simulations of the heat equation, we demonstrate how the time complexity of our algorithm scales with the ansatz volume for gradient estimation and how the time-to-solution scales with the diffusion parameter. Our proposed algorithm extends economically to higher-order time-stepping schemes, such as the Crank-Nicolson method. We present a semi-implicit scheme for solving systems of evolution equations with non-linear terms, such as the reaction-diffusion and the incompressible Navier-Stokes equations, and demonstrate its validity by proof-of-concept results.