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This research presents a novel numerical framework for solving time-dependent Schrödinger equations using residual minimization within Gaussian approximation manifolds. By achieving second-order convergence without spatial discretization, the method offers significant efficiency gains for simulating quantum dynamics through explicit closed-form expressions. The work provides critical theoretical bounds that separate time discretization effects from residual accuracy, advancing the computational toolkit for quantum simulation.
Residual Minimization Achieves Second-Order Convergence for Schrödinger Equations
This work analyzes time discretization via residual minimization for evolution equations, including the Schrödinger equation, achieving second-order convergence.
Eddy de León and Caroline Lasser from Technical University of Munich have analyzed a technique to solve time-dependent Schrödinger equations that achieves first- and second-order convergence rates under certain assumptions, offering potential improvements over existing methods. Their work centers on residual minimization, a technique for approximating solutions within low-dimensional spaces, and analyzes two distinct formulations for time-stepping: one that discretizes then parametrizes the equation, and another that discretizes the Dirac, Frenkel variational principle. The researchers demonstrate a unified error analysis for both approaches, yielding bounds that separate the effects of time discretization and residual minimization. Importantly, when applied to Gaussian…
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