Stochastic optimization of Neural Network Quantum States (NQS) in discrete Fock spaces is limited by sampling variance and slow mixing. We present a deterministic framework that optimizes a neural backflow ansatz within dynamically adaptive configuration subspaces, corrected by second-order perturbation theory. This approach eliminates Monte Carlo noise and, through a hybrid CPU-GPU implementation, exhibits sub-linear scaling with respect to subspace size. Benchmarks on bond dissociation in H2O and N2, and the strongly correlated chromium dimer Cr2, validate the method's accuracy and stability in large Hilbert spaces.
Stochastic optimization of Neural Network Quantum States (NQS) in discrete Fock spaces is limited by sampling variance and slow mixing. We present a deterministic framework that optimizes a neural backflow ansatz within dynamically adaptive configuration subspaces, corrected by second-order perturbation theory. This approach eliminates Monte Carlo noise and, through a hybrid CPU-GPU implementation, exhibits sub-linear scaling with respect to subspace size. Benchmarks on bond dissociation in H2O and N2, and the strongly correlated chromium dimer Cr2, validate the method's accuracy and stability in large Hilbert spaces.
Neural-network quantum states (NQS) employ artificial neural networks to encode many-body wave functions in a second quantization through variational Monte Carlo (VMC). They have recently been applied to accurately describe electronic wave functions of molecules and have shown the challenges in efficiency compared with traditional quantum chemistry methods. Here, we introduce a general nonstochastic optimization algorithm for NQS in chemical systems, which deterministically generates a selected set of important configurations simultaneously with energy evaluation of NQS. This method bypasses the need for Markov-chain Monte Carlo within the VMC framework, thereby accelerating the entire optimization process. Furthermore, this newly developed nonstochastic optimization algorithm for NQS offers comparable or superior accuracy compared to its stochastic counterpart and ensures more stable convergence. The application of this model to test molecules exhibiting strong electron correlations provides further insight into the performance of NQS in chemical systems and opens avenues for future enhancements.
The ground state of second-quantized quantum chemistry Hamiltonians is key to determining molecular properties. Neural quantum states (NQS) offer flexible and expressive wavefunction ansatze for this task but face two main challenges: highly peaked ground-state wavefunctions hinder efficient sampling, and local energy evaluations scale quartically with system size, incurring significant computational costs. In this work, we overcome these challenges by introducing a suite of algorithmic enhancements, which includes efficient periodic compact subspace construction, truncated local energy evaluations, improved stochastic sampling, and physics-informed modifications. Applying these techniques to the neural network backflow (NNBF) ansatz, we demonstrate significant gains in both accuracy and scalability. Our enhanced method surpasses traditional quantum chemistry methods like CCSD and CCSD(T), outperforms other NQS approaches, and achieves competitive energies with state-of-the-art ab initio techniques such as HCI, ASCI, FCIQMC, and DMRG. A series of ablation and comparative studies quantifies the contribution of each enhancement to the observed improvements in accuracy and efficiency. Furthermore, we investigate the representational capacity of the ansatz, finding that its performance correlates with the inverse participation ratio (IPR), with more delocalized states being more challenging to approximate.
The ground state of second-quantized quantum chemistry Hamiltonians is key to determining molecular properties. Neural quantum states (NQSs) offer flexible and expressive wave-function Ans\"atze for this task but face two main challenges: highly peaked ground-state wave functions hinder efficient sampling, and local energy evaluations scale quartically with system size, incurring significant computational costs. In this work, we overcome these challenges by introducing a suite of algorithmic enhancements, which includes efficient periodic compact subspace construction, truncated local energy evaluations, improved stochastic sampling, and physics-informed modifications. Applying these techniques to the neural network backflow Ansatz, we demonstrate significant gains in both accuracy and scalability. Our enhanced method surpasses traditional quantum chemistry methods like coupled cluster singles and doubles (CCSD) and CCSD and perturbative triples [CCSD(T)], outperforms other NQS approaches, and achieves competitive energies with state-of-the-art ab initio techniques such as heat-bath configuration interaction, adaptive sampling configuration interaction, full configuration interaction quantum Monte Carlo, and density matrix renormalization group. A series of ablation and comparative studies quantifies the contribution of each enhancement to the observed improvements in accuracy and efficiency. Furthermore, we investigate the representational capacity of the Ansatz, finding that its performance correlates with the inverse participation ratio, with more delocalized states being more challenging to approximate.
Accurately simulating extended periodic systems is a central challenge in condensed matter physics. Neural quantum states (NQS) offer expressive wavefunctions for this task but face issues with scalability. In this work, we successfully extend the neural network backflow (NNBF) approach to ab-initio solid-state materials. Building on our scalable optimization framework for molecules [Liu et al., PRB 112, 155162 (2025)], we introduce a two-stage pruning strategy to manage the massive configuration space expansions: by utilizing a computationally cheap, physics-informed importance proxy, we devote exact NNBF amplitude evaluations solely to the most relevant determinants, significantly improving optimization efficiency, energy estimation, and convergence. Our framework achieves state-of-the-art accuracy across diverse solid-state benchmarks. For 1D hydrogen chains, NNBF matches or surpasses DMRG and AFQMC, remains robust in strongly correlated bond-breaking regimes where coupled-cluster methods fail, and smoothly extrapolates to the thermodynamic limit. We further demonstrate its scalability by computing ground-state potential energy curves for 2D graphene and 3D silicon. Finally, ablation studies validate the computational savings of our pruning strategy and highlight the dependence of the NNBF energies on basis sets.
The ground state of second-quantized quantum chemistry Hamiltonians provides access to an important set of chemical properties. Wave functions based on machine-learning architectures have shown promise in approximating these ground states in a variety of physical systems. In this paper, we show how to achieve state-of-the-art energies for molecular Hamiltonians using the the neural network backflow (NNBF) wave function. To accomplish this, we optimize this ansatz with a variant of the deterministic optimization scheme based on selected configuration interaction introduced by Li et al., [J. Chem. Theory Comput. 19, 8156 (2023)], which we find works better than standard Markov chain Monte Carlo sampling. For the molecules we studied, NNBF gives lower energy states than both Coupled Cluster with Single and Double excitations and other neural network quantum states. We systematically explore the role of network size as well as optimization parameters in improving the energy. We find that, while the number of hidden layers and determinants play a minor role in improving the energy, there are significant improvements in the energy from increasing the number of hidden units as well as the batch size used in optimization, with the batch size playing a more important role.
Neural quantum states (NQS) provide a flexible and highly expressive parametrization 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.