Quantum circuits of elastic wave equations via Schrödingerisation
Abstract
We construct explicit quantum circuits for the velocity-stress formulation of elastic wave equations in homogeneous media using Schrödingerisation. After rewriting the equations in symmetric matrix form(SMF), we use Fourier spectral discretization for source-driven periodic problems and staggered-grid finite differences for Dirichlet, Neumann, and impedance boundary conditions. The Schrödingerisation method transforms any linear PDEs into a system of Schrödinger-type PDEs -with unitary evolution- using the warped phase transformation that maps the equations in one higher dimension. For periodic problems, we develop circuits by combining controlled Fourier-mode rotations. For physical boundaries, interleaved encoding swaps the spatial and field-component registers, enabling a more efficient circuit construction using shift operators. We analyze pth-order Trotter-Suzuki formulas and derive splitting-error and CNOT-gate estimates, under the assumptions of smooth auxiliary initialization and solution-state recovery. Comparison with classical partitioned splitting methods identifies a polynomial improvement in spatial-resolution dependence for the quantum state-evolution subroutine. Noiseless circuit simulations of a forced periodic problem and an impedance problem illustrate the constructions and quantify their errors against exact solutions and classical reference computations.
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BibTeX
@misc{jin2026quantum,
title = {{Quantum circuits of elastic wave equations via Schrödingerisation}},
author = {Shi Jin and Chundan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37247},
eprint = {2610.37247},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37247},
}
RIS
TY - UNPB
TI - Quantum circuits of elastic wave equations via Schrödingerisation
AU - Shi Jin
AU - Chundan Zhang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37247
UR - https://arxiv.org/abs/2610.37247
AB - We construct explicit quantum circuits for the velocity-stress formulation of elastic wave equations in homogeneous media using Schrödingerisation. After rewriting the equations in symmetric matrix form(SMF), we use Fourier spectral discretization for source-driven periodic problems and staggered-grid finite differences for Dirichlet, Neumann, and impedance boundary conditions. The Schrödingerisation method transforms any linear PDEs into a system of Schrödinger-type PDEs -with unitary evolution- using the warped phase transformation that maps the equations in one higher dimension. For periodic problems, we develop circuits by combining controlled Fourier-mode rotations. For physical boundaries, interleaved encoding swaps the spatial and field-component registers, enabling a more efficient circuit construction using shift operators. We analyze pth-order Trotter-Suzuki formulas and derive splitting-error and CNOT-gate estimates, under the assumptions of smooth auxiliary initialization and solution-state recovery. Comparison with classical partitioned splitting methods identifies a polynomial improvement in spatial-resolution dependence for the quantum state-evolution subroutine. Noiseless circuit simulations of a forced periodic problem and an impedance problem illustrate the constructions and quantify their errors against exact solutions and classical reference computations.
ER -