Quantum Wave Simulation with Sources and Loss Functions

TL;DR

This paper introduces a quantum algorithmic framework for simulating a broad class of wave equations, achieving significant speed-ups over classical methods while handling sources, boundary conditions, and energy measurements.

Contribution

The paper presents a versatile quantum simulation framework for wave equations that includes source handling, boundary conditions, and optimal measurement strategies, with proven quartic speed-up in 3D.

Findings

Achieves quartic speed-up over classical solvers in 3D simulations.

Compatible with standard numerical discretizations and multiple sources.

Provides methods for energy extraction and boundary condition incorporation.

Abstract

We present a quantum algorithmic framework for simulating linear, anti-Hermitian (lossless) wave equations in heterogeneous, anisotropic, and time-independent media. This framework encompasses a broad class of wave equations, including the acoustic wave equation, Maxwell$'$s equations and the elastic wave equation. Our formulation is compatible with standard numerical discretization schemes and allows for the efficient implementation of multiple practically relevant time- and space-dependent sources. Furthermore, we demonstrate that subspace energies can be extracted and wave fields compared through an $l_2$ loss function, achieving optimal precision scaling with the number of samples taken. Additionally, we introduce techniques for incorporating boundary conditions and linear constraints that preserve the anti-Hermitian nature of the equations. Leveraging the Hamiltonian simulation algorithm, our framework achieves a quartic speed-up over classical solvers in 3D simulations, under conditions of sufficiently global measurements and compactly supported sources and initial conditions. This quartic speed-up is optimal for time-domain solutions, as the Hamiltonian of the discretized wave equations has local couplings. In summary, our framework provides a versatile approach for simulating wave equations on quantum computers, offering substantial speed-ups over state-of-the-art classical methods.