Abstract
Variational quantum Monte Carlo (QMC) is an ab initio method for solving the electronic Schrödinger equation that is exact in principle, but limited by the flexibility of the available Ansätze in practice. The recently introduced deep QMC approach, specifically two deep-neural-network Ansätze PauliNet and FermiNet, allows variational QMC to reach the accuracy of diffusion QMC, but little is understood about the convergence behavior of such Ansätze. 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 (MF) complete-basis-set limit. Moving to electron correlation, we then perform an extensive hyperparameter scan of a deep Jastrow factor for LiH and H4 and find that variational energies at the fixed-node limit can be obtained with a sufficiently large network. Finally, we benchmark MF and many-body Ansätze on H2O, increasing the fraction of recovered fixed-node correlation energy of single-determinant Slater-Jastrow-type Ansätze 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 understand the superb accuracy of deep variational Ansätze 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.
Highlights
The fundamental problem in quantum chemistry is to solve the electronic Schrödinger equation as accurately as possible at a manageable cost
Variational quantum Monte Carlo (QMC) is an ab-initio method for solving the electronic Schrödinger equation that is exact in principle, but limited by the flexibility of the available ansatzes in practice
The recently introduced deep QMC approach, 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
Summary
The fundamental problem in quantum chemistry is to solve the electronic Schrödinger equation as accurately as possible at a manageable cost. We identify a hierarchy of model ansatzes based on the traditional VMC methodology (Fig. 1) that enables us to distinguish the effects of improving single-particle orbitals and adding correlation in the symmetric part of the wavefunction ansatz This is of particular interest in the context of discriminating these improvements from reducing the energy by solving the intricate problem of missing many-body effects in the nodal surface. The nodal surface of the trial wavefunctions can be improved by increasing the number of determinants or by applying the backflow technique, transforming single-particle orbitals to many-body orbitals under consideration of the symmetry constraints These are key concepts to efficiently reach very high accuracy with VMC and integral features of deep QMC.
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