Neural Quantum States Based on Selected Configurations

TL;DR

A systematic comparison of the ground-state optimizations obtained with NQS-VMC and NQS-SC for molecular systems dominated by either static or dynamical correlation demonstrates a clear advantage of NQS-SC over NQS-VMC in both energy accuracy and wave-function coefficients, particularly for statically correlated molecules.

Abstract

Neural quantum states (NQS) provide a flexible and highly expressive parameterization of wave functions for strongly correlated problems in quantum chemistry. Despite rapid advances in network architectures, the evaluation of electronic energies remains almost exclusively based on variational Monte Carlo (VMC). While VMC is effective for structured systems such as spin chains, its accuracy and efficiency for electronic Hamiltonians are hindered by sharply peaked distributions, stochastic gradient noise, and slow convergence with sample size. In this letter, we assess the capability of NQS-VMC to efficiently capture correlation in electronic ground states by comparing it to a recently developed NQS-based selected configuration (NQS-SC) approach. We set up a systematic comparison of the ground-state optimizations obtained with NQS-VMC and NQS-SC for molecular systems dominated by either static or dynamical correlation. The comparison demonstrates a clear advantage of NQS-SC over NQS-VMC in both energy accuracy and wave-function coefficients, particularly for statically correlated molecules. Moreover, NQS-SC exhibits robust systematic improvability, whereas NQS-VMC does not. These findings position NQS-SC as the new default approach over NQS-VMC for electronic structure calculations. We further observe that neither NQS-SC nor NQS-VMC can efficiently capture dynamical correlation, highlighting the need for future hybrid methods, such as multiconfigurational perturbation theories built on top of NQS solutions.

Figures (4)

• Figure 1: Sorted probability amplitudes from the NBF model trained with varying $n_\mathrm{select}$ and $n_\mathrm{sample}$ using selected configurations and exact Monte Carlo sampling compared to the FCI solution for stretched $\text{N}_\text{2}$ using an STO-3G basis set and canonical HF orbitals (14,400 total configurations).
• Figure 2: Energy errors for the NBF solutions plotted in Figure \ref{['fig:C-N2s-STO-3G-HF']}. For $n_\mathrm{select} = 2^{11}$, $\vert E^\mathrm{SCI}(n_\mathrm{select}) - E^\mathrm{FCI}\vert$ drops down to approx. $5\cdot 10^{-11}$ Ha where it remains for subsequent $n_\mathrm{select}$.
• Figure 3: Sorted probability amplitudes from the NBF model trained with varying $n_\mathrm{select}$ and $n_\mathrm{sample}$ using selected configurations and exact Monte Carlo sampling compared to the FCI solution for $\text{H}_\text{2}\text{O}$ using a 6-31G basis set and MP2 natural orbitals (1,656,369 total configurations).
• Figure 4: Energy errors for the NBF solutions plotted in Figure \ref{['fig:C-H2O-6-31G-NO']}. $\vert E_{\bm{\theta}}^\mathrm{MC}(n_\mathrm{sample}) - E^\mathrm{FCI}\vert$ for $n_\mathrm{sample} = 2^{17}$ is approx. $837$ mHa. The blue dashed trendline was calculated by linear least-squares in log-log space, ignoring the outlier value of $\vert E_{\bm{\theta}}^\mathrm{MC}(n_\mathrm{sample}) - E^\mathrm{FCI}\vert$ for $n_\mathrm{sample} = 2^{17}$.