Convergence to the fixed-node limit in deep variational Monte Carlo

TL;DR

This paper investigates how deep neural network-based variational quantum Monte Carlo methods approach the fixed-node limit, demonstrating their ability to overcome basis set limitations and improve correlation energy recovery.

Contribution

It provides a detailed analysis of the convergence behavior of deep variational QMC ansatzes, showing their potential to reach the fixed-node limit and outperform traditional wavefunctions.

Findings

Deep neural networks can reach the mean-field complete-basis-set limit.

Large networks can recover fixed-node correlation energies.

Single-determinant Slater--Jastrow--backflow ansatz overcomes fixed-node limitations.

Abstract

Variational quantum Monte Carlo (QMC) is an ab-initio method for solving the electronic Schr\"odinger equation that is exact in principle, but limited by the flexibility of the available ansatzes in practice. The recently introduced deep QMC approach, specifically two deep-neural-network ansatzes PauliNet and FermiNet, allows variational QMC to reach the accuracy of diffusion QMC, but little is understood about the convergence behavior of such ansatzes. Here, we analyze how deep variational QMC approaches the fixed-node limit with increasing network size. First, we demonstrate that a deep neural network can overcome the limitations of a small basis set and reach the mean-field complete-basis-set limit. Moving to electron correlation, we then perform an extensive hyperparameter scan of a deep Jastrow factor for LiH and H$_4$ and find that variational energies at the fixed-node limit can be obtained with a sufficiently large network. Finally, we benchmark mean-field and many-body ansatzes on H$_2$O, increasing the fraction of recovered fixed-node correlation energy of single-determinant Slater--Jastrow-type ansatzes by half an order of magnitude compared to previous variational QMC results and demonstrate that a single-determinant Slater--Jastrow--backflow version of the ansatz overcomes the fixed-node limitations. This analysis helps understanding the superb accuracy of deep variational ansatzes in comparison to the traditional trial wavefunctions at the respective level of theory, and will guide future improvements of the neural network architectures in deep QMC.