Neural Networks Boost Accuracy of Quantum Simulations for Complex Materials

Researchers are developing neural quantum states (NQS) as a promising method for modelling complex quantum systems, but current evaluations of electronic energies largely rely on variational Monte Carlo (VMC), which can struggle with accuracy and efficiency. Marco Julian Solanki, Lexin Ding, and Markus Reiher, all from the Department of Chemistry and Applied Biosciences at ETH Z urich, have now assessed the potential of NQS-VMC and compared it to a novel NQS-based selected configuration (NQS-SC) approach. Their systematic comparison of molecular systems exhibiting both static and dynamical correlation reveals a significant advantage for NQS-SC in terms of energy accuracy and wave-function coefficients, especially for molecules with strong static correlation, and demonstrates its potential for systematic improvement. These findings establish NQS-SC as a superior method to NQS-VMC for electronic structure calculations, while also identifying limitations in capturing dynamical correlation and suggesting avenues for future hybrid methodologies.

This work centres on neural quantum states (NQS), a method that uses artificial neural networks to represent the wave functions describing the behaviour of electrons within a molecule.

While NQS holds promise for tackling strongly correlated problems, practical implementation previously relied heavily on variational Monte Carlo (VMC) sampling, a technique hampered by inefficiencies when dealing with electronic Hamiltonians. The team introduces a refined NQS-based selected configuration (NQS-SC) method and systematically compares its performance against conventional NQS-VMC.

NQS-SC leverages the predictive power of the neural network to intelligently select relevant configurations, leading to a more stable and accurate optimisation process for determining ground-state energies. The comparison reveals a significant advantage for NQS-SC, particularly in accurately representing molecules where electrons are strongly static, meaning their interactions dictate a fixed arrangement.

NQS-SC not only delivers more accurate energies and wave-function coefficients but also demonstrates a capacity for systematic improvement, a feature lacking in NQS-VMC. Optimizations using NQS-SC consistently yielded lower energies than those obtained with NQS-VMC. Specifically, for molecular systems exhibiting strong static correlation, NQS-SC demonstrated an average energy reduction of 0.027 Hartree compared to NQS-VMC.

Wave-function coefficients generated by NQS-SC exhibited significantly reduced variance compared to those from NQS-VMC, with analysis revealing a decrease in the magnitude of higher-order terms, indicating a more compact and physically meaningful representation of the electronic structure. This is particularly notable in statically correlated molecules, where NQS-SC consistently produced coefficients with a smaller spread, suggesting a more efficient capture of the essential correlation effects.

A central challenge in applying neural quantum states to molecular systems lies in efficiently evaluating energies, and this work addresses this through a detailed comparison of NQS-VMC and NQS-SC. Initial NQS implementations relied on Metropolis-Hastings sampling for energy evaluation, but this proved inefficient due to low acceptance rates, as low as 0.1% for water in a 6-31G basis set, and the need for extensive pruning and burn-in periods to ensure accurate sampling.

Autoregressive sampling was then adopted to circumvent these limitations, generating uncorrelated samples by sequentially predicting conditional probabilities using a neural network, but this required specific network architectures and struggled to enforce fundamental symmetries without ad hoc masking. To overcome these issues, the research team implemented NQS-SC, drawing inspiration from selected configuration interaction (SCI) methods traditionally used in quantum chemistry.

NQS-SC selects configurations not from a pre-defined variational space, but dynamically from outside this space based on probability amplitudes predicted by the NQS ansatz itself. This selection process utilizes a ‘local energy’ metric, enabling a non-stochastic evaluation of the electronic ground-state energy and potentially leading to more stable optimizations and improved accuracy compared to NQS-VMC.

The methodology deliberately focuses on capturing static correlation by selecting a limited, but highly relevant, subset of configurations from the vast Hilbert space. These findings establish NQS-SC as a preferred method for electronic structure calculations, offering a more robust and reliable pathway to understanding molecular behaviour. The research highlights a critical limitation of both NQS-SC and NQS-VMC in capturing dynamical correlation, the rapid fluctuations in electron interactions due to their movement.

This suggests the need for future hybrid methods that combine the strengths of NQS with techniques like multiconfigurational perturbation theory, potentially unlocking even more accurate and efficient simulations of complex chemical systems. The persistent challenge of accurately modelling complex molecular systems has long vexed computational chemists, with traditional methods struggling with static correlation and demanding computational resources that scale unfavourably with system size.

This work signals a potential shift, demonstrating a clear advantage for NQS-SC over NQS-VMC in capturing these difficult correlations. The improvement suggests a pathway towards systematically improving accuracy, something that has proven elusive with many existing techniques. However, the study also reveals that both NQS-SC and NQS-VMC falter when faced with dynamical correlation.

This highlights the need for hybrid methods, combining the strengths of neural networks with established techniques like multiconfigurational perturbation theory. The next phase will likely involve integrating these approaches, creating a tiered system where neural networks handle static correlation and perturbation theory refines the description of dynamical effects.