Implicitly Restarted Lanczos Enables Chemically-Accurate Shallow Neural Quantum States

Abstract

The variational optimization of high-dimensional neural network models, such as those used in neural quantum states (NQS), presents a significant challenge in machine intelligence. Conventional first-order stochastic methods (e.g., Adam) are plagued by slow convergence, sensitivity to hyperparameters, and numerical instability, preventing NQS from reaching the high accuracy required for fundamental science. We address this fundamental optimization bottleneck by introducing the implicitly restarted Lanczos (IRL) method as the core engine for NQS training. Our key innovation is an inherently stable second-order optimization framework that recasts the ill-conditioned parameter update problem into a small, well-posed Hermitian eigenvalue problem. By solving this problem efficiently and robustly with IRL, our approach automatically determines the optimal descent direction and step size, circumventing the need for demanding hyperparameter tuning and eliminating the numerical instabilities common in standard iterative solvers. We demonstrate that IRL enables shallow NQS architectures (with orders of magnitude fewer parameters) to consistently achieve extreme precision (1e-12 kcal/mol) in just 3 to 5 optimization steps. For the F2 molecule, this translates to an approximate 17,900-fold speed-up in total runtime compared to Adam. This work establishes IRL as a superior, robust, and efficient second-order optimization strategy for variational quantum models, paving the way for the practical, high-fidelity application of neural networks in quantum physics and chemistry.