Quantum simulation of elastic wave equations via Schr\"odingerisation

TL;DR

This paper develops quantum simulation algorithms for elastic wave equations using the Schr"odingerisation method, transforming PDEs into Schr"odinger-type equations suitable for quantum computers, demonstrating exponential space complexity advantage.

Contribution

It introduces a novel application of Schr"odingerisation to elastic wave equations, including velocity-stress and displacement forms, with complexity analysis showing quantum advantage.

Findings

Quantum algorithms for elastic wave equations are developed.

Exponential quantum advantage in space complexity is demonstrated.

Spectral and staggered grid methods are integrated with Schr"odingerisation.

Abstract

In this paper we study quantum simulation algorithms on the elastic wave equations using the Schr\"odingerisation method. The Schr\"odingerisation method transforms any linear PDEs into a system of Schr\"odinger-type PDEs -with unitary evolution-using the warped phase transformation that maps the equations in one higher dimension. This makes them suitable for quantum simulations. We expore the application in two forms of the elastic wave equations. For the velocity-stress equation in isotropic media, we explore the symmetric matrix form under the external forcing via Schr\"odingerisation combined with spectral method. For problems with variable medium parameters, we apply Schr\"odingerisation method based on the staggered grid method to simulate velocity and stress fields, and give the complexity estimates. For the wave displacement equation, we transform it into a hyperbolic system and apply the Schr\"odingerisation method, which is then discretized by the spectral method and central difference scheme. Details of the quantum algorithms will be provided, along with the complexity analysis which demontrate exponential quantum advantage in space dimensin over the classical algorithms.