Hamiltonian simulation for 3D elastic wave equations in homogeneous elastic media

2026-04-22

AI Summary

This work proposes an explicit quantum algorithm for simulating the three-dimensional elastic wave equation in homogeneous isotropic media without relying on black-box oracles. Starting from the first-order velocity–stress formulation, the equation is discretized via finite differences and recast into a Schrödinger-type system. By exploiting a register structure that separates spatial and component degrees of freedom, the Hamiltonian is decomposed into tensor product terms, enabling the explicit construction of first- and second-order Trotter–Suzuki evolution circuits. The approach yields precise estimates of qubit count and CNOT gate complexity, with resource bounds explicitly dependent on discretization parameters, simulation time, target accuracy, and material properties. Numerical experiments confirm the accuracy of both time evolution and physical field reconstruction.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

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Abstract

We present an explicit quantum circuit construction for Hamiltonian simulation of a first-order velocity--stress formulation of the three-dimensional elastic wave equation in homogeneous isotropic media. Previous studies have shown how elastic wave equations can be cast into forms amenable to Hamiltonian simulation, but they typically rely on black box Hamiltonian access assumptions, making gate complexity estimation difficult. Starting from the first-order velocity--stress formulation, we discretize the system by finite differences, transform it into Schrödinger form, and exploit the separation between the component register and the spatial register to decompose the Hamiltonian into structured tensor product terms. This yields explicit implementations of first-order and second-order Trotter formulas for the resulting time evolution operator. We derive corresponding error bounds and constant sensitive qubit and CNOT complexity estimates in terms of the discretization parameter, simulation time, target accuracy, and material parameters. Numerical experiments validate the proposed framework through comparisons with the exact time evolution and reconstructed physical fields.

Problem

• Hamiltonian simulation
• elastic wave equation
• quantum circuit
• Trotter formula
• finite difference

Innovation

• Hamiltonian simulation
• elastic wave equation
• quantum circuit construction
• Trotterization
• finite difference discretization