Operator Learning for Schr\"{o}dinger Equation: Unitarity, Error Bounds, and Time Generalization

We consider the problem of learning the evolution operator for the time-dependent Schr\"{o}dinger equation, where the Hamiltonian may vary with time. Existing neural network-based surrogates often ignore fundamental properties of the Schr\"{o}dinger equation, such as linearity and unitarity, and lack theoretical guarantees on prediction error or time generalization. To address this, we introduce a linear estimator for the evolution operator that preserves a weak form of unitarity. We establish both upper and lower bounds on the prediction error that hold uniformly over all sufficiently smooth initial wave functions. Additionally, we derive time generalization bounds that quantify how the estimator extrapolates beyond the time points seen during training. Experiments across real-world Hamiltonians -- including hydrogen atoms, ion traps for qubit design, and optical lattices -- show that our estimator achieves relative errors 10⁻² to 10⁻³ times smaller than state-of-the-art methods such as the Fourier Neural Operator and DeepONet.