Variational Monte Carlo for Ab Initio Electronic Structure and Quantum Chemistry with Neural Quantum States, Natural-Gradient Optimization, and Selective State-Space Models
TitleVariational Monte Carlo for Ab Initio Electronic Structure with Neural Quantum States and Selective State-Space Models Overview This repository provides a research implementation of Variational Monte Carlo (VMC) for ab initio electronic-structure calculations using Neural Quantum States (NQS). The work focuses on stable natural-gradient optimization, physically motivated wavefunction constraints, and the exploration of alternative architectural inductive biases for modeling electronic correlation. The project is released as an exploratory research artifact. All results, limitations, and open questions are stated explicitly. The work is motivated by a central question in modern neural quantum chemistry:can alternative sequence-based architectures reduce the quadratic scaling bottleneck of electron–electron interaction modeling while preserving physical fidelity? Scientific Scope The framework targets: Atomic systems (H through Ne) Small molecules (e.g., H₂, LiH, H₂O) Exploratory extensions, including: Periodic boundary conditions Excited-state considerations Berry-phase estimation Entanglement diagnostics Time-dependent VMC via McLachlan’s variational principle At present, numerical validation is limited to a single-electron benchmark (Hydrogen) and is explicitly presented as a sanity check of the physics engine and optimization loop, not as evidence of improved expressivity for correlated fermions. Core Methodology The implementation follows the standard Variational Monte Carlo formalism, in which a parameterized wavefunctionψ is optimized by minimizing the energy expectation value Optimization is performed using Stochastic Reconfiguration (SR), a natural-gradient method that accounts for the geometry of the variational manifold and significantly improves stability compared to first-order optimizers. Key numerical features include: Log-domain arithmetic for wavefunction evaluation Stable determinant computation using slogdet Hutchinson trace estimation for Laplacian evaluation Explicit enforcement of Kato cusp conditions via analytic Jastrow factors Multi-determinant antisymmetric ansatz to satisfy fermionic exchange symmetry SSM-Backflow Hypothesis The primary research contribution explored in this repository is the SSM-Backflow hypothesis. Most contemporary NQS models (e.g., FermiNet, PauliNet) model electron–electron interactions using explicit quadratic aggregation, leading to O(Ne2)O(N_e^2)O(Ne2) computational complexity. This work investigates an alternative inductive bias: For each electron, neighboring electrons are sorted by inter-electron distance The resulting sequence is processed using a selective state-space model (SSM) inspired by Mamba architectures Correlation information is accumulated sequentially through a learned recurrence Aggregated messages are used to parameterize backflow transformations or orbital functions This approach yields a theoretical complexity of O(NelogNe)O(N_e \log N_e)O(NelogNe) per layer (dominated by sorting) and is motivated by the physical observation that electronic correlation typically decays approximately exponentially with distance. Importantly, this is treated strictly as a hypothesis. The manuscript explicitly notes that: SSMs are not permutation-invariant by construction Sequential ordering may introduce inductive biases misaligned with fermionic symmetry Empirical validation on multi-electron systems is required before drawing conclusions Results and Current Validation Status At present, the repository reports results for atomic hydrogen (Ne = 1): Exact energy: −0.500000 Ha Achieved energy: −0.5001 Ha Error: 0.12 mHa (well below chemical accuracy) Local-energy variance: 0.0454 Optimizer: Stochastic Reconfiguration Convergence: ~100 optimization steps Because hydrogen has no electron–electron correlation, this result is interpreted solely as a sanity check of: Hamiltonian implementation Local-energy estimator Natural-gradient optimization loop Numerical stability mechanisms The manuscript clearly states that multi-electron benchmarks (e.g., He, Li, or H₂) are required to meaningfully evaluate the SSM-Backflow hypothesis and are planned for future versions. Stability and Diagnostics The implementation includes multiple practical safeguards designed to stabilize VMC optimization: Dynamic energy-based divergence detection Variance-based walker reset mechanisms Controlled respawning of walkers near nuclear centers In addition, an exploratory commutator-based diagnostic (informally termed the “Noether Inverse”) is included as a monitoring signal during training. This quantity is explicitly not claimed to represent a new conserved symmetry and is treated purely as an empirical observation requiring further theoretical and numerical investigation. Intended Use and Limitations This repository is released as: A research implementation A transparent record of an ongoing investigation A foundation for future validation and benchmarking It is not presented as a production-ready electronic-structure solver, nor as evidence of state-of-the-art performance. All limitations are explicitly documented in the manuscript. Licensing and Reuse The work is shared to encourage: Reproducibility Methodological discussion Extension and validation by the community If you use or build upon this work, please cite the Zenodo DOI associated with this release.
TitleVariational Monte Carlo for Ab Initio Electronic Structure with Neural Quantum States and Selective State-Space Models Overview This repository provides a research implementation of Variational Monte Carlo (VMC) for ab initio electronic-structure calculations using Neural Quantum States (NQS). The work focuses on stable natural-gradient optimization, physically motivated wavefunction constraints, and the exploration of alternative architectural inductive biases for modeling electronic correlation. The project is released as an exploratory research artifact. All results, limitations, and open questions are stated explicitly. The work is motivated by a central question in modern neural quantum chemistry:can alternative sequence-based architectures reduce the quadratic scaling bottleneck of electron–electron interaction modeling while preserving physical fidelity? Scientific Scope The framework targets: Atomic systems (H through Ne) Small molecules (e.g., H₂, LiH, H₂O) Exploratory extensions, including: Periodic boundary conditions Excited-state considerations Berry-phase estimation Entanglement diagnostics Time-dependent VMC via McLachlan’s variational principle At present, numerical validation is limited to a single-electron benchmark (Hydrogen) and is explicitly presented as a sanity check of the physics engine and optimization loop, not as evidence of improved expressivity for correlated fermions. Core Methodology The implementation follows the standard Variational Monte Carlo formalism, in which a parameterized wavefunctionψ is optimized by minimizing the energy expectation value Optimization is performed using Stochastic Reconfiguration (SR), a natural-gradient method that accounts for the geometry of the variational manifold and significantly improves stability compared to first-order optimizers. Key numerical features include: Log-domain arithmetic for wavefunction evaluation Stable determinant computation using slogdet Hutchinson trace estimation for Laplacian evaluation Explicit enforcement of Kato cusp conditions via analytic Jastrow factors Multi-determinant antisymmetric ansatz to satisfy fermionic exchange symmetry SSM-Backflow Hypothesis The primary research contribution explored in this repository is the SSM-Backflow hypothesis. Most contemporary NQS models (e.g., FermiNet, PauliNet) model electron–electron interactions using explicit quadratic aggregation, leading to O(Ne2)O(N_e^2)O(Ne2) computational complexity. This work investigates an alternative inductive bias: For each electron, neighboring electrons are sorted by inter-electron distance The resulting sequence is processed using a selective state-space model (SSM) inspired by Mamba architectures Correlation information is accumulated sequentially through a learned recurrence Aggregated messages are used to parameterize backflow transformations or orbital functions This approach yields a theoretical complexity of O(NelogNe)O(N_e \log N_e)O(NelogNe) per layer (dominated by sorting) and is motivated by the physical observation that electronic correlation typically decays approximately exponentially with distance. Importantly, this is treated strictly as a hypothesis. The manuscript explicitly notes that: SSMs are not permutation-invariant by construction Sequential ordering may introduce inductive biases misaligned with fermionic symmetry Empirical validation on multi-electron systems is required before drawing conclusions Results and Current Validation Status At present, the repository reports results for atomic hydrogen (Ne = 1): Exact energy: −0.500000 Ha Achieved energy: −0.5001 Ha Error: 0.12 mHa (well below chemical accuracy) Local-energy variance: 0.0454 Optimizer: Stochastic Reconfiguration Convergence: ~100 optimization steps Because hydrogen has no electron–electron correlation, this result is interpreted solely as a sanity check of: Hamiltonian implementation Local-energy estimator Natural-gradient optimization loop Numerical stability mechanisms The manuscript clearly states that multi-electron benchmarks (e.g., He, Li, or H₂) are required to meaningfully evaluate the SSM-Backflow hypothesis and are planned for future versions. Stability and Diagnostics The implementation includes multiple practical safeguards designed to stabilize VMC optimization: Dynamic energy-based divergence detection Variance-based walker reset mechanisms Controlled respawning of walkers near nuclear centers In addition, an exploratory commutator-based diagnostic (informally termed the “Noether Inverse”) is included as a monitoring signal during training. This quantity is explicitly not claimed to represent a new conserved symmetry and is treated purely as an empirical observation requiring further theoretical and numerical investigation. Intended Use and Limitations This repository is released as: A research implementation A transparent record of an ongoing investigation A foundation for future validation and benchmarking It is not presented as a production-ready electronic-structure solver, nor as evidence of state-of-the-art performance. All limitations are explicitly documented in the manuscript. Licensing and Reuse The work is shared to encourage: Reproducibility Methodological discussion Extension and validation by the community If you use or build upon this work, please cite the Zenodo DOI associated with this release.
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.
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.
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.
Describing the electronic structure of molecules using computational methods is of central importance in many areas of chemical research and industry. To this day, the variational quantum Monte Carlo (VMC) algorithm, which utilizes the variational principle of quantum mechanics to optimize an ansatz towards the true electronic wave function, is one of the lesser employed of such computational methods. While it is generally applicable, and provides direct access to the wave function, its accuracy has been limited by insufficiently expressive ansatzes. This barrier was recently lifted with the introduction of deep-learning-based ansatzes, providing unprecedented flexibility and corresponding accuracy, at the cost of increased computational requirements. In this thesis, our implementation of the VMC algorithm with deep-learning ansatzes is described, along with a number of developments facilitating the computation of electronic excited states, extraction of various observables, and characterization of potential energy surfaces, at a reduced computational cost. The method is implemented as open-source software, employing a common framework to encompass most ansatzes present in the literature. Utilizing the direct wave function access granted by VMC, an approach is developed to extract one-electron reduced densities, that are free from Gaussian basis set effects. Significant advances in the methodology of computing electronic excited states via the ladder-of-states approach are made, including an improved form of the overlap penalty term with an automatically determined prefactor, and a new loss term with which the total spin of the solution can be selected. The efficiency of computing excited potential energy surfaces are enhanced by optimizing a single ansatz for a range of molecular configurations. To enable such transferable training in the presence of state crossings, a dynamic, geometry-wise ordering in the ladder of states is introduced. Geometry-optimization-based tasks are identified as natural fits for such transferable ansatzes, and a transferability-native approach for extracting the necessary information from VMC data is devised. Finally, transferable training is employed to simultaneously model different chemical species, resulting in a foundation model predicting bond breaking at a cost-accuracy ratio competitive with mature multi-reference methods. The work presented here contributes significantly to the development of a practically useful deep-learning-based VMC method. The described approaches have the potential to become tools for a wide community of computational chemists, as a source of accurate data, describing strong correlation in both ground and excited states.
Abstract Neural-network quantum states (NQSs) offer a powerful and expressive ansatz for representing quantum many-body wave functions. However, their training via Variational Monte Carlo (VMC) methods remains challenging. It is well known that some scenarios—such as sharply peaked wave functions emerging in quantum chemistry—lead to high-variance gradient estimators hindering the effectiveness of variational optimizations. In this work we investigate a systematic strategy to tackle those sampling issues by means of adaptively tuned importance sampling. Our approach is explicitly designed to target the gradient estimator instead of the loss function, and be computationally inexpensive. We benchmarked our approach across the ground-state search of a wide variety of hamiltonians, including frustrated spin systems and ab-initio quantum chemistry. Overall, our approach can reduce the computational cost of vanilla VMC considerably, up to a factor of 100× when targeting highly peaked quantum chemistry wavefunctions.
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.