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arXiv:2609.01904

Quantum Circuit and Tensor Network Implementation of the 2D Acoustic Wave Equation

Tamas Nemeth, Gabor Vattay

quant-phcond-mat.dis-nncs.MSSubmitted Sep 1, 2026• Announced Sep 3, 2026• 0 citations
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Abstract

We present a cohesive framework for simulating seismic wave propagation utilizing quantum computing paradigms and their classical tensor network equivalents. We detail a quantum circuit-based formulation for the explicit finite-difference time-domain (FDTD) solution of the two-dimensional acoustic wave equation and map this quantum architecture onto a tensor train representation, namely for Matrix Product State (MPS). The MPS solver enables deterministic simulation of large-scale wavefield dynamics on classical high-performance computing systems. We demonstrate the MPS representation by computing 2D seismic wavefields on the Marmousi model. Our results indicate that the MPS representation is a viable direction for computing and scaling wavefield propagation.

Comments
7 pages, 1 figure

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We present a cohesive framework for simulating seismic wave propagation utilizing quantum computing paradigms and their classical tensor network equivalents.
We detail a quantum circuit-based formulation for the explicit finite-difference time-domain (FDTD) solution of the two-dimensional acoustic wave equation and
map this quantum architecture onto a tensor train representation, namely for Matrix Product State (MPS).
The MPS solver enables deterministic simulation of large-scale wavefield dynamics on classical high-performance computing systems.
We demonstrate the MPS representation by computing 2D seismic wavefields on the Marmousi model.
Our results indicate that the MPS representation is a viable direction for computing and scaling wavefield propagation.
Numerical models are essential for investigating and understanding the physics and sound propagation and scattering, given the ocean’s complexity and the limitations of real-world field experiments. High-performance computing has become a primary tool for researchers to simulate long-range sound propagation. Quantum computing is an emerging technique that offers potential speedups over conventional binary computers. However, classical algorithms cannot be directly implemented in quantum systems due to fundamental differences in computing principles. This necessitates the development of quantum ocean acoustic models that remain interpretable within the same context as classical algorithms. This paper presents a quantum formulation for a finite-difference time-domain (FDTD) solver and a parabolic equation (PE) method. The results demonstrate that these quantum algorithms implemented on a simulator reproduce results identical to their classical counterparts, providing a foundation for further developments in quantum-centric ocean acoustics.
In this work, we implement a tensor network-based quantum algorithm to solve unsteady, nonlinear partial differential equations (PDEs). The challenge lies in how to effectively represent, encode, process, and evolve the nonlinear system of PDEs on quantum computers. We will discuss the new techniques using the compressible 1-dimensional (1D) Burgers' equation as an example, because it represents the fundamental nonlinear feature and yet removes certain complexity in physics, allowing us to focus on the design of quantum algorithms. Previous attempts to solve nonlinear PDEs in quantum computation have often involved storing multiple copies of solutions or employing linearizations. Neither is practical due to exponential scaling with evolution time or insufficient solution accuracy. Our framework is based on matrix product states (MPSs) and matrix product operators (MPOs). For example, the velocity field is represented by MPS, whereas the linear and nonlinear spatial differential terms of the velocity field are processed by MPOs. Our primary focus herein is to verify and validate the various tensor network components of the algorithm using solutions obtained by the classical algorithms on high performance computing (HPC) architectures. We use a classical time marching method to demonstrate the functionality of the tensor network operations to model the PDE and their robustness with the time evolution of the system. Our classical simulation results demonstrate the utility of tensor network-based operations in modeling nonlinear PDEs and highlight the necessity as well as potential advantages of using quantum simulations for these techniques.
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