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.

翻译：我们提出了一个显式量子电路构造，用于均匀各向同性介质中三维弹性波动方程的一阶速度-应力形式的哈密顿模拟。先前的研究已展示如何将弹性波动方程转化为适合哈密顿模拟的形式，但这些研究通常依赖于黑箱哈密顿访问假设，使得门复杂度估算困难。从一阶速度-应力公式出发，我们通过有限差分对系统进行离散化，将其转化为薛定谔形式，并利用分量寄存器和空间寄存器之间的分离性，将哈密顿分解为结构化的张量积项。由此，针对生成的时间演化算子，实现了显式的一阶和二阶特罗特公式。我们推导了相应的误差界，并基于离散化参数、模拟时间、目标精度和材料参数，给出了门复杂度与CNOT数量的常数敏感估算。数值实验通过与精确时间演化及重构物理场的比较，验证了所提框架的有效性。