Implicitly Restarted Lanczos Enables Chemically Accurate Shallow Neural Quantum States

The variational optimization of neural quantum states (NQS) is hindered by the slow convergence and numerical instability of conventional optimizers. We address this bottleneck by introducing the implicitly restarted Lanczos (IRL) method as the core engine for NQS training. Our approach recasts the parameter update problem into a Hermitian eigenvalue problem that IRL solves efficiently, automatically determining the optimal descent direction and step size without hyperparameter tuning. We demonstrate that IRL enables shallow NQS architectures to achieve extreme precision in just 3-5 optimization steps, dramatically outperforming existing methods, including minimum-step stochastic reconfiguration (MinSR). For the N₂ dissociation curve, IRL successfully trains a shallow NQS to match full configuration interaction accuracy, validating its robust convergence in strongly correlated regimes. This work establishes IRL as an efficient and robust second-order optimization strategy for variational quantum models.