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International Journal for Numerical Methods in Engineering

Simple Harmonic Motion Analysis of Elastic Structures Using Hamiltonian Simulation via Quantum Singular Value Transformation

Koya Wagatsuma, Katsuhiro Endo, Kenjiro Terada

Published Aug 21, 2026• 0 citations
Publisher

Abstract

ABSTRACT This study proposes a quantum computing algorithm for dynamic structural analysis, assuming the use of quantum computers. Specifically, we apply a quantum algorithm known as Hamiltonian Simulation (HS), which calculates the time evolution of the Schrödinger equation with time‐independent coefficients, to the equations of motion governing the simple harmonic motion problem of a two‐dimensional elastic structure. This is implemented by applying the Quantum Singular Value Transformation (QSVT) method to the finite‐element‐discretized equations. The performance of the proposed method is evaluated through several numerical experiments and compared with theoretical computational complexity and error estimates. Practical challenges are also discussed.

DOI
10.1002/nme.70407

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ABSTRACT This study proposes a quantum computing algorithm for dynamic structural analysis, assuming the use of quantum computers. Specifically, we apply a quantum algorithm known as Hamiltonian Simulation (HS), which calculates the time evolution of the Schrödinger equation with time‐independent coefficients, to the equations of motion governing the simple harmonic motion problem of a two‐dimensional elastic structure. This is implemented by applying the Quantum Singular Value Transformation (QSVT) method to the finite‐element‐discretized equations. The performance of the proposed method is evaluated through several numerical experiments and compared with theoretical computational complexity and error estimates. Practical challenges are also discussed.
Quantum computing can be used to speed up the simulation time (more precisely, the number of queries of the algorithm) for physical systems; one such promising approach is the Hamiltonian simulation (HS) algorithm. Recently, it was proven that the quantum singular value transformation (QSVT) achieves the minimum simulation time for HS. An important subroutine of the QSVT-based HS algorithm is the amplitude amplification operation, which can be realized via the oblivious amplitude amplification or the fixed-point amplitude amplification in the QSVT framework. In this work, we execute a detailed analysis of the error and number of queries of the QSVT-based HS and show that the oblivious method is better than the fixed-point one in the sense of simulation time. Based on this finding, we apply the QSVT-based HS to the one-dimensional linearized Vlasov-Poisson equation and demonstrate that the linear Landau damping can be successfully simulated.
The most widely used approach for simulating the dynamics of time-dependent Hamiltonians via quantum computation depends on the quantum-classical hybrid variational quantum time evolution algorithm, in which ordinary differential equations of the variational coefficients for determining time evolution are solved via classical simulations with a time discretization method. We here present a full-quantum approach, in which ordinary differential equations of the variational coefficients are transformed into static linear equations via the Chebyshev spectral discretization method and then solved via the quantum singular value transformation algorithm. Our full quantum algorithm avoids classical feedback, achieves exponential convergence for smooth Hamiltonians, and yields a quantum circuit depth that is independent of the number of time steps. We demonstrate two implementation strategies, with a global formulation designed for fault-tolerant architectures and a sequential formulation tailored to near-term devices, and validate the approach through numerical simulations of proton-hydrogen charge-transfer dynamics, a prototypical time-dependent quantum chemistry problem. This work establishes a systematic pathway from quantum-classical hybrid variational quantum algorithms to full-quantum solvers for general time-dependent Hamiltonians, particularly those whose dynamics admit compact variational descriptions, opening a route toward full quantum computational advantages in time-dependent simulations.
Quantum computing can be used to speed up the simulation time (more precisely, the number of queries of the algorithm) for physical systems; one such promising approach is the Hamiltonian simulation (HS) algorithm. Recently, it was proven that the quantum singular-value transformation (QSVT) achieves the minimum simulation time for HS. An important subroutine of the QSVT-based HS algorithm is the amplitude amplification operation, which can be realized via the oblivious amplitude amplification or the fixed-point amplitude amplification in the QSVT framework. In this work we execute a detailed analysis of the error and number of queries of the QSVT-based HS and show that the oblivious method is better than the fixed-point one in the sense of simulation time. Based on this finding, we apply the QSVT-based HS to the one-dimensional linearized Vlasov-Poisson equation and demonstrate that the linear Landau damping can be successfully simulated.
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.
SUMMARY Quantum computing has attracted considerable attention in recent years because it promises speedups that conventional supercomputers cannot offer, at least for some applications. Though existing quantum computers (QCs) are, in most cases, still too small to solve significant problems, their future impact on domain sciences is already being explored now. Within this context, we present a quantum computing concept for 1-D elastic wave propagation in heterogeneous media with two components: a theoretical formulation and an implementation on a real QC. The method rests on a finite-difference approximation, followed by a sparsity-preserving transformation of the discrete elastic wave equation to a Schrödinger equation, which can be simulated directly on a gate-based QC. An implementation on an error-free quantum simulator verifies our approach and forms the basis of numerical experiments with small problems on the real QC IBM Brisbane. The latter produce simulation results that qualitatively agree with the error-free version but are contaminated by quantum decoherence and noise effects. Complementing the discrete transformation to the Schrödinger equation by a continuous version allows the replacement of finite differences by other spatial discretization schemes, such as the spectral-element method. Anticipating the emergence of error-corrected quantum chips, we analyse the computational complexity of the best quantum simulation algorithms for future QCs. This analysis suggests that our quantum computing approach may lead to wavefield simulations that run exponentially faster than simulations on classical computers.
In this paper we study quantum simulation algorithms on the elastic wave equations using the Schrödingerisation method. 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. This makes them suitable for quantum simulations. We explore the application in two forms of the elastic wave equations. For the velocity-stress equation in isotropic media, we explore the symmetric matrix form under the external forcing via Schrödingerisation combined with spectral method. For problems with variable medium parameters, we apply Schrödingerisation method based on the staggered grid method to simulate velocity and stress fields, and give the complexity estimates. For the wave displacement equation, we transform it into a hyperbolic system and apply the Schrödingerisation method, which is then discretized by the spectral method and central difference scheme. Details of the quantum algorithms will be provided, along with the complexity analysis which demonstrates an exponential quantum advantage in space dimension over the classical algorithms if only considering the quantum subroutine which evolves the initial quantum state to the final quantum state.