Decomposition-Free Variational Quantum Eigensolvers for Natural Frequency Analysis in Structural Engineering

Abstract

The analysis of natural frequency is an important problem of structural engineering, where complex structures often lead to expensive eigenvalue problems. Variational quantum eigensolvers provide a potentially efficient solution by using quantum computing, but their application is limited by the costly need to decompose system matrices into a linear combination of unitaries. In this work, we apply a recently proposed decomposition-free method from linear systems to eigenvalue problems in structural engineering. Specifically, we develop two algorithms based on this method, the decomposition-free variational quantum eigensolver (DF-VQE) and generalized eigensolver (DF-VQGE) for the standard eigenvalue problems and the generalized eigenvalue problems, respectively. In these algorithms, the Rayleigh Quotient and generalized Rayleigh Quotient cost evaluations are reformulated so that the need for matrix decomposition is bypassed. The central technical step in this application is to use vectorization to encode the system matrices directly into quantum state vectors, allowing their properties to be computed via a standard swap test quantum circuit. The proposed methods are validated on a quantum simulator with three representative structural engineering problems, including a 1D bar, a 2D beam, and a 3D truss bridge, obtaining results that align well with reference solutions from classical finite element analysis. We also discuss the quantum state preparation challenge that must be addressed before this formulation can be connected to meaningful quantum advantage.