Structure-Preserving Reduced-Order Modeling of the Time-Dependent Schrödinger Equation

This paper develops and rigorously analyzes structure-preserving reduced-order modeling (ROM) strategies for the time-dependent Schrödinger equation (TDSE), aiming to efficiently approximate quantum dynamics while maintaining the fundamental geometric and physical properties of the system. Within a unified framework, we formulate and compare three ROM approaches—Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD), and Reduced Basis Methods (RBM)—with emphasis on their ability to preserve Hamiltonian structure, unitary evolution, and norm conservation. Projection-based Galerkin reductions are systematically derived, and their stability and structure-preserving properties are validated via a priori and a posteriori error estimates. Benchmark numerical experiments—including the infinite square well, harmonic oscillator, potential barrier scattering, and a time-dependent controlled two-level system—demonstrate that the ROMs achieve substantial dimensional reduction while accurately reproducing full-order model (FOM) dynamics and essential physical features. The framework is further extended to higher-dimensional systems, nonlinear potentials, and multi-particle scenarios, with potential applications in quantum control and entanglement dynamics. Convergence studies and visual diagnostics confirm the robustness and reliability of the reduced models. To support reproducibility, all MATLAB codes used for the simulations are publicly available. Overall, this work establishes structure-preserving ROM as a practical and efficient tool for real-time quantum simulation, control, and optimization.