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arXiv:2604.18357

Momentum Stability and Adaptive Control in Stochastic Reconfiguration

Yuyang Wang, Xin Liu

math.OCmath.NAphysics.comp-phquant-phSubmitted Apr 20, 2026• Announced Apr 21, 2026• 0 citations
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Abstract

Variational Monte Carlo (VMC) combined with expressive neural network wavefunctions has become a powerful route to high-accuracy ground-state calculations, yet its practical success hinges on efficient and stable wavefunction optimization. While stochastic reconfiguration (SR) provides a geometry-aware preconditioner motivated by imaginary-time evolution, its Kaczmarz-inspired variant, subsampled projected-increment natural gradient descent (SPRING), achieves state-of-the-art empirical performance. However, the effectiveness of SPRING is highly sensitive to the choice of a momentum-like parameter $μ$. The original sensitivity of $μ$ and the instability observed at $μ=1$, have remained unclear. In this work, we clarify the distinct mechanisms governing the regimes $μ<1$ and $μ=1$. We establish convergence guarantees for $0\leμ<1$ under mild assumptions, and construct counterexamples showing that $μ=1$ can induce divergence via uncontrolled growth along kernel-related directions when the step-size is not summable. Motivated by these theoretical insights and numerical observations, we further propose \textit{Principal Range Informed MomEntum SR} (PRIME-SR), a tuning-free momentum-adaptive SR method based on effective spectral dimension and subspace overlap. PRIME-SR achieves performance comparable to optimally tuned SPRING while significantly improving robustness in VMC optimization.

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Neural network wavefunctions optimized using the variational Monte Carlo method have been shown to produce highly accurate results for the electronic structure of atoms and small molecules, but the high cost of optimizing such wavefunctions prevents their application to larger systems. We propose the Subsampled Projected-Increment Natural Gradient Descent (SPRING) optimizer to reduce this bottleneck. SPRING combines ideas from the recently introduced minimum-step stochastic reconfiguration optimizer (MinSR) and the classical randomized Kaczmarz method for solving linear least-squares problems. We demonstrate that SPRING outperforms both MinSR and the popular Kronecker-Factored Approximate Curvature method (KFAC) across a number of small atoms and molecules, given that the learning rates of all methods are optimally tuned. For example, on the oxygen atom, SPRING attains chemical accuracy after forty thousand training iterations, whereas both MinSR and KFAC fail to do so even after one hundred thousand iterations.

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The combination of the variational Monte Carlo (VMC) method with deep learning wave function architectures has led to several successes in ground-state calculations of quantum many-body systems in recent years. However, commonly used stochastic gradient-based methods often perform poorly on these parameter training problems and typically lack convergence guarantees. The stochastic reconfiguration (SR) method provides a robust preconditioner of the stochastic gradient, whose computational cost becomes prohibitive for large parameter spaces owing to the repeated inversion of large covariance matrices. To overcome this bottleneck, we propose a warm-started stochastic reconfiguration (WSSR) method, which integrates warm-start techniques from singular value decomposition (SVD) to refine low-rank approximations of the preconditioning matrix iteratively. Numerical experiments on typical atomic and molecular systems highlight the effectiveness of the WSSR method within VMC calculations.

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Rayleigh-Gauss-Newton optimization with enhanced sampling for variational Monte Carlo

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Variational Monte Carlo (VMC) is an approach for computing ground-state wave functions that has recently become more powerful due to the introduction of neural network-based wave-function parametrizations. However, efficiently training neural wave functions to converge to an energy minimum remains a difficult problem. In this work, we analyze optimization and sampling methods used in VMC and introduce alterations to improve their performance. First, based on theoretical convergence analysis in a noiseless setting, we motivate a new optimizer that we call the Rayleigh-Gauss-Newton method, which can improve upon gradient descent and natural gradient descent to achieve superlinear convergence at no more than twice the computational cost. Second, to realize this favorable comparison in the presence of stochastic noise, we analyze the effect of sampling error on VMC parameter updates and experimentally demonstrate that it can be reduced by the parallel tempering method. In particular, we demonstrate that RGN can be made robust to energy spikes that occur when the sampler moves between metastable regions of configuration space. Finally, putting theory into practice, we apply our enhanced optimization and sampling methods to the transverse-field Ising and $XXZ$ models on large lattices, yielding ground-state energy estimates with remarkably high accuracy after just 200 parameter updates.

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Variational Monte Carlo for Ab Initio Electronic Structure and Quantum Chemistry with Neural Quantum States, Natural-Gradient Optimization, and Selective State-Space Models

Devanik Debnath

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(Nelog⁡Ne)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.

•Feb 17, 2026•0 citations

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Neural quantum states (NQS) provide a flexible and scalable framework for approximating quantum many-body wavefunctions. Among NQS parameterizations, autoregressive models are especially attractive because they enable exact, independent sampling from the Born distribution, avoiding the autocorrelation and mixing issues of Markov chain methods. Yet their optimization remains comparatively underexplored: Adam is a scalable method but ignores function space geometry, while stochastic reconfiguration is principled but costly and numerically fragile in large models. To address this gap, we show that variational energy minimization can be viewed as an advantage policy-gradient problem over the Born distribution, motivating trust-region optimization for NQS training. We introduce Proximal Wavefunction Optimization (PWO), a principled trust-region algorithm that clips probability-ratio changes in the amplitude channel and phase increments in the phase channel. PWO avoids explicit matrix inversion, reuses samples across multiple updates, and combines the scalability of first-order optimization with theoretical guarantees. Across Ising and frustrated $J_1$-$J_2$ one- and two-dimensional spin systems, PWO improves stability and wall-clock convergence over Adam, minSR, and SPRING. Finally, we fine-tune a $1.5$B-parameter RWKV-7 model, demonstrating NQS optimization at a scale over three orders of magnitude beyond prior work.

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Abstract Neural-network architectures have been increasingly used to represent quantum many-body wave functions. These networks require a large number of variational parameters and are challenging to optimize using traditional methods, as gradient descent. Stochastic reconfiguration (SR) has been effective with a limited number of parameters, but becomes impractical beyond a few thousand parameters. Here, we leverage a simple linear algebra identity to show that SR can be employed even in the deep learning scenario. We demonstrate the effectiveness of our method by optimizing a Deep Transformer architecture with 3 × 105parameters, achieving state-of-the-art ground-state energy in theJ1–J2Heisenberg model atJ2/J1 = 0.5 on the 10 × 10 square lattice, a challenging benchmark in highly-frustrated magnetism. This work marks a significant step forward in the scalability and efficiency of SR for neural-network quantum states, making them a promising method to investigate unknown quantum phases of matter, where other methods struggle.

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Variational Monte Carlo (VMC) combined with expressive neural network wavefunctions has become a powerful route to high-accuracy ground-state calculations, yet its practical success hinges on efficient and stable wavefunction optimization. While stochastic reconfiguration (SR) provides a geometry-aware preconditioner motivated by imaginary-time evolution, its Kaczmarz-inspired variant, subsampled projected-increment natural gradient descent (SPRING), achieves state-of-the-art empirical performance. However, the effectiveness of SPRING is highly sensitive to the choice of a momentum-like parameter μμμ. The original sensitivity of μμμ and the instability observed at μ=1μ=1μ=1, have remained unclear. In this work, we clarify the distinct mechanisms governing the regimes μ<1μ<1μ<1 and μ=1μ=1μ=1. We establish convergence guarantees for 0≤μ<10\leμ<10≤μ<1 under mild assumptions, and construct counterexamples showing that μ=1μ=1μ=1 can induce divergence via uncontrolled growth along kernel-related directions when the step-size is not summable. Motivated by these theoretical insights and numerical observations, we further propose \textit{Principal Range Informed MomEntum SR} (PRIME-SR), a tuning-free momentum-adaptive SR method based on effective spectral dimension and subspace overlap. PRIME-SR achieves performance comparable to optimally tuned SPRING while significantly improving robustness in VMC optimization.
Neural network wavefunctions optimized using the variational Monte Carlo method have been shown to produce highly accurate results for the electronic structure of atoms and small molecules, but the high cost of optimizing such wavefunctions prevents their application to larger systems. We propose the Subsampled Projected-Increment Natural Gradient Descent (SPRING) optimizer to reduce this bottleneck. SPRING combines ideas from the recently introduced minimum-step stochastic reconfiguration optimizer (MinSR) and the classical randomized Kaczmarz method for solving linear least-squares problems. We demonstrate that SPRING outperforms both MinSR and the popular Kronecker-Factored Approximate Curvature method (KFAC) across a number of small atoms and molecules, given that the learning rates of all methods are optimally tuned. For example, on the oxygen atom, SPRING attains chemical accuracy after forty thousand training iterations, whereas both MinSR and KFAC fail to do so even after one hundred thousand iterations.
The combination of the variational Monte Carlo (VMC) method with deep learning wave function architectures has led to several successes in ground-state calculations of quantum many-body systems in recent years. However, commonly used stochastic gradient-based methods often perform poorly on these parameter training problems and typically lack convergence guarantees. The stochastic reconfiguration (SR) method provides a robust preconditioner of the stochastic gradient, whose computational cost becomes prohibitive for large parameter spaces owing to the repeated inversion of large covariance matrices. To overcome this bottleneck, we propose a warm-started stochastic reconfiguration (WSSR) method, which integrates warm-start techniques from singular value decomposition (SVD) to refine low-rank approximations of the preconditioning matrix iteratively. Numerical experiments on typical atomic and molecular systems highlight the effectiveness of the WSSR method within VMC calculations.
Variational Monte Carlo (VMC) is an approach for computing ground-state wave functions that has recently become more powerful due to the introduction of neural network-based wave-function parametrizations. However, efficiently training neural wave functions to converge to an energy minimum remains a difficult problem. In this work, we analyze optimization and sampling methods used in VMC and introduce alterations to improve their performance. First, based on theoretical convergence analysis in a noiseless setting, we motivate a new optimizer that we call the Rayleigh-Gauss-Newton method, which can improve upon gradient descent and natural gradient descent to achieve superlinear convergence at no more than twice the computational cost. Second, to realize this favorable comparison in the presence of stochastic noise, we analyze the effect of sampling error on VMC parameter updates and experimentally demonstrate that it can be reduced by the parallel tempering method. In particular, we demonstrate that RGN can be made robust to energy spikes that occur when the sampler moves between metastable regions of configuration space. Finally, putting theory into practice, we apply our enhanced optimization and sampling methods to the transverse-field Ising and XXZXXZXXZ models on large lattices, yielding ground-state energy estimates with remarkably high accuracy after just 200 parameter updates.
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(Nelog⁡Ne)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 scalable framework for approximating quantum many-body wavefunctions. Among NQS parameterizations, autoregressive models are especially attractive because they enable exact, independent sampling from the Born distribution, avoiding the autocorrelation and mixing issues of Markov chain methods. Yet their optimization remains comparatively underexplored: Adam is a scalable method but ignores function space geometry, while stochastic reconfiguration is principled but costly and numerically fragile in large models. To address this gap, we show that variational energy minimization can be viewed as an advantage policy-gradient problem over the Born distribution, motivating trust-region optimization for NQS training. We introduce Proximal Wavefunction Optimization (PWO), a principled trust-region algorithm that clips probability-ratio changes in the amplitude channel and phase increments in the phase channel. PWO avoids explicit matrix inversion, reuses samples across multiple updates, and combines the scalability of first-order optimization with theoretical guarantees. Across Ising and frustrated J1J_1J1​-J2J_2J2​ one- and two-dimensional spin systems, PWO improves stability and wall-clock convergence over Adam, minSR, and SPRING. Finally, we fine-tune a 1.51.51.5B-parameter RWKV-7 model, demonstrating NQS optimization at a scale over three orders of magnitude beyond prior work.
Abstract Neural-network architectures have been increasingly used to represent quantum many-body wave functions. These networks require a large number of variational parameters and are challenging to optimize using traditional methods, as gradient descent. Stochastic reconfiguration (SR) has been effective with a limited number of parameters, but becomes impractical beyond a few thousand parameters. Here, we leverage a simple linear algebra identity to show that SR can be employed even in the deep learning scenario. We demonstrate the effectiveness of our method by optimizing a Deep Transformer architecture with 3 × 105parameters, achieving state-of-the-art ground-state energy in theJ1–J2Heisenberg model atJ2/J1 = 0.5 on the 10 × 10 square lattice, a challenging benchmark in highly-frustrated magnetism. This work marks a significant step forward in the scalability and efficiency of SR for neural-network quantum states, making them a promising method to investigate unknown quantum phases of matter, where other methods struggle.