Quantum-enhanced spectral analysis for materials and networked systems

Eigenvalue computation underpins applications from structural mechanics to network science, yet classical methods struggle with dense, ill-conditioned, or highly constrained matrices. We develop a quantized Rayleigh-quotient framework that reformulates eigenvalue problems as binary optimization, enabling quantum annealers to outperform classical mixed-integer programming. Eigenvector components are discretized via binary encoding, transforming the Rayleigh quotient into a problem solvable by D-Wave quantum processors. Comparative benchmarks on 50-node dense stiffness matrices and graph Laplacians show quantum annealing achieves 0.10% relative error for minimum eigenvalues (vs 31.78% for classical), 2.51% for Fiedler values (vs 21.36% for classical), and 0.0074% for maximum eigenvalues (vs 7.19% for classical)—spanning five orders of magnitude. Application to a 25 × 25 flexible lattice yields eigenvalues closely matching classical results (λminclassical=1.18, λminquantum=1.24) while capturing spatially extended mode structures. This formulation generalizes to symmetric positive semi-definite matrices across physics, engineering, and data science, establishing practical quantum advantage in spectral computation and expanding the applicability of quantum optimization to fundamental problems in numerical linear algebra.