Quantum advantage for numerical wave simulation in heterogeneous media with sources, boundary conditions and loss function estimation

We propose a quantum algorithmic framework for the simulation of linear, anti-Hermitian (lossless) wave equations within media that can be heterogeneous and anisotropic, but with time-independent properties. Starting from the acoustic wave equation, our approach extends to a wide range of wave systems, including Maxwell’s equations and elastic wave propagation.

Our method is compatible with common numerical discretization techniques and handles various source terms with both spatial and temporal dependencies, requiring minimal assumptions about the medium’s regularity. In addition, we demonstrate that energy in specific subspaces can be efficiently extracted, and wavefields can be analyzed using an optimally estimated l₂-norm. We also outline methods for implementing boundary conditions and enforcing linear constraints while preserving the anti-Hermitian characteristics of the wave equations. Leveraging the Hamiltonian simulation algorithm, our framework achieves a quartic speedup over classical methods for 3D wave simulations under certain global measurement conditions and with localized sources or initial states. In conclusion, this approach offers a versatile and highly efficient method for wave equation simulation, providing significant performance gains compared to the best classical algorithms.