A Variance-Based Convergence Criterion in Neural Variational Monte Carlo for Quantum Systems

The optimization of neural wave functions in variational Monte Carlo crucially relies on a robust convergence criterion. While the energy variance is theoretically a definitive measure, its practical application as a primary convergence criterion has been underexplored. In this work, we develop a lightweight, general-purpose solver that utilizes the energy variance as a convergence criterion. We apply it to several systems-including the harmonic oscillator, hydrogen atom, and charmonium hadron-for validating the variance as a reliable diagnostic, and using a empirical threshold $10^{-3}$ as the energy variance convergence values for performing rapid parameter scans to enable preliminary physical verification. To clarify the scope of our approach, we derive an inequality that delineates the limitations of variance-based optimization in nodal systems. Despite these limitations, the energy variance proves to be a highly valuable tool, guiding our solver to efficient and reliable results across a range of quantum problems.