Quantum Wave Simulation with Sources and Loss Functions

• URL: http://arxiv.org/abs/2411.17630v3
• Date: Tue, 04 Feb 2025 19:08:41 GMT
• Title: Quantum Wave Simulation with Sources and Loss Functions
• Authors: Cyrill Bösch, Malte Schade, Giacomo Aloisi, Scott D. Keating, Andreas Fichtner,
• Abstract summary: We present a quantum algorithmic framework for simulating linear, anti-Hermitian (lossless) wave equations in heterogeneous media. Our framework is compatible with standard numerical discretization schemes. We show that subspace energies can be extracted and wave fields compared through an $l$ loss function.
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• 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.

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