提出保持量子幺正性的算子学习方法，显著提升薛定谔方程预测精度与时间外推能力。
Operator Learning for Schrödinger Equation: Unitarity, Error Bounds, and Time Generalization
- 设计线性估计器，弱保持薛定谔方程的幺正性。
- 理论证明误差上界与下界，适用于光滑初态函数类。
- 在原子、离子阱等真实系统上误差比现有方法低两个数量级。
我们研究时变薛定谔方程演化算子的学习问题，其中哈密顿量可随时间变化。现有基于神经网络的代理模型常忽略薛定谔方程的基本性质，如线性和幺正性，且缺乏预测误差或时间外推的理论保证。为此，我们提出一种保持弱幺正性的线性估计器。我们在足够光滑的初态函数类上建立了预测误差的上下界。此外，还推导了时间外推的泛化界，量化了模型在训练时间点之外的推广能力。在真实哈密顿量（包括氢原子、用于量子比特设计的离子阱、光晶格）上的实验表明，该估计器相对误差比当前最优方法（如傅里叶神经算子和DeepONet）低达两个数量级。
原文摘要 · Abstract (English)
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 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.
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