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 $p$th-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.

The paper addresses explicit quantum-circuit implementation of homogeneous-medium elastic wave propagation with time-independent forcing and physical boundary conditions. It symmetrizes the velocity–stress equations, uses Fourier spectral discretization for periodic problems and staggered-grid finite differences for Dirichlet, Neumann, and impedance boundaries, and applies Schrödingerisation to embed non-unitary dynamics into unitary evolution with one additional coordinate. Periodic circuits use Fourier-mode-controlled rotations, with explicit field-component circuits in two and three dimensions. For physical boundaries, interleaving spatial and component registers produces shift-operator circuits; Neumann evolution follows by conjugating the Dirichlet construction, while impedance terms use exact rotations on boundary subspaces. Nested $p$th-order Trotter–Suzuki formulas provide commutator-based error and CNOT estimates. Assuming smooth auxiliary initialization and the stated recovery conditions, the one-dimensional periodic cost is $\widetilde O_p(s_p^2T^{1+1/p}R_T^{1+1/p}\varepsilon^{-1/p}h^{-1})$, where $R_T=\max\{1,\|u(0)\|/\|u(T)\|\}$ and $s_p$ is the product-formula stage count. Physical-boundary costs scale as $\widetilde O_p(s_p^4T^{1+1/p}R_T^{1+1/p}\varepsilon^{-1/p}h^{-1-1/p})$ in one dimension. Comparison with classical partitioned splitting identifies a polynomial spatial-resolution advantage for state evolution only. Noiseless simulations use 19 qubits for a forced two-dimensional periodic example and 14 for a one-dimensional impedance example. Periodic circuit errors approximately halve when the timestep halves; the impedance circuit error is about $3.90\times10^{-2}$. The impedance pulse remains away from the boundaries throughout the test. Input preparation, observable estimation, finite-gate-set implementation, complex media, and more general boundary geometries remain outside the demonstrated scope.