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

TL;DR

This paper introduces a linear estimator for the Schrödinger equation's evolution operator that maintains unitarity, offers theoretical error bounds, and demonstrates superior accuracy over existing neural network surrogates.

Contribution

A novel linear estimator that preserves unitarity and provides theoretical guarantees for error bounds and time generalization in Schrödinger equation modeling.

Findings

Achieves up to 100x smaller relative errors than existing methods.

Provides uniform upper and lower bounds on prediction error.

Demonstrates effective time extrapolation across various quantum systems.

Abstract

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 bounds and lower bounds on the prediction error of the proposed estimator that hold uniformly over classes of 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 up to two orders of magnitude smaller than state-of-the-art methods such as the Fourier Neural Operator and DeepONet.