Operator Learning for Schrödinger Equation: Unitarity, Error Bounds, and Time Generalization

TL;DR

This work introduces a linear, spectral-domain surrogate for learning the Schrödinger evolution operator $F$, designed to preserve a weak unitarity property and to yield uniform error guarantees over Sobolev-smooth initial states. By actively querying a PDE solver on a spectral basis up to bandwidth $K_n$ and constructing $\widehat{F}_n = \sum_{|k|_{\infty}\le K_n} w_k \otimes \varphi_k$, the authors obtain provable upper and lower bounds on the prediction error that depend on the solver accuracy $\varepsilon$, the smoothness $s$ of the initial wave, and the dimension $d$, with favorable rates $n^{-s/d}$ when $2s> d$. Time-generalization for time-independent Hamiltonians is established, showing controlled error growth over multiple steps under Sobolev-bounded norms, and the framework yields significant empirical advantages (up to $10^2-10^3$ in relative accuracy) over Fourier Neural Operator and DeepONet baselines on hydrogen, ion-trap, and optical-lattice systems. The results offer a principled, physics-informed approach to operator learning in quantum dynamics, with practical implications for rapid design and simulation in quantum devices and material sciences.

Abstract

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.

Figures (1)

• Figure 1: Squared amplitude $|\psi(x)|^2$ of the initial wave, the true wave at $T=0.1$, and the estimator's prediction for the barrier potential with double slits on $[0,2\pi)^2$.

Theorems & Definitions (14)

• Proposition 3.1
• Theorem 4.2: Upper Bound
• Theorem 4.3: Lower Bound
• Theorem 5.1
• Corollary 5.2
• proof
• proof
• proof
• Theorem F.2: Improved Upper Bound
• proof : Proof of Theorem \ref{['thm:improved']}