Efficient dynamics : reduced-order modeling of the time-dependent Schrödinger equation

Abstract

This work develops and rigorously analyzes reduced-order modeling (ROM) techniques for the time-dependent Schrödinger equation (TDSE), with the goal of efficiently capturing essential quantum dynamics at significantly reduced computational cost. Three major ROM frameworks–Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD), and Reduced Basis Methods (RBM) are explored and compared–in the context of quantum wavefunction evolution. Comprehensive mathematical formulations are presented, including projection-based Galerkin approximations, a priori and a posteriori error estimates, stability analyses, and convergence guarantees. Numerical experiments are conducted for canonical quantum systems such as the infinite square well, harmonic oscillator, and tunneling through potential barriers, as well as a time-dependent controlled two-level system. It is demonstrated that ROMs can achieve orders-of-magnitude dimensionality reduction while maintaining high fidelity with full-order model (FOM) solutions. Furthermore, the framework is extended to higher-dimensional problems, nonlinear potentials, and multi-particle systems, with applications in quantum control and entanglement dynamics. Visualization of ROM accuracy through mesh, surface, and isosurface plots, as well as convergence studies, confirms the robustness of the proposed methods. To support reproducibility and further research, all MATLAB code used to generate the numerical experiments is made publicly available via GitHub. These results establish ROM as a powerful tool for real-time simulation, control, and optimization in computational quantum mechanics.