Operator Learning for Schrödinger Equation: Unitarity, Error Bounds, and Time Generalization
This addresses the problem of accurate and theoretically grounded operator learning for quantum systems, with incremental improvements in preserving physical properties and error guarantees.
The paper tackles learning the evolution operator for the time-dependent Schrödinger equation by introducing a linear estimator that preserves unitarity and provides theoretical error bounds, achieving relative errors 10^-2 to 10^-3 times smaller than state-of-the-art methods in experiments on real-world Hamiltonians.
We consider the problem of learning the evolution operator for the time-dependent Schrödinger equation, where the Hamiltonian may vary with time. Existing neural network-based surrogates often ignore fundamental properties of the Schrödinger 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.