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arXiv:2601.01437

Implicitly Restarted Lanczos Enables Chemically-Accurate Shallow Neural Quantum States

Wei Liu, Wenjie Dou

quant-phphysics.chem-phSubmitted Jan 4, 2026• Announced Jan 6, 2026• 0 citations
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Abstract

The variational optimization of high-dimensional neural network models, such as those used in neural quantum states (NQS), presents a significant challenge in machine intelligence. Conventional first-order stochastic methods (e.g., Adam) are plagued by slow convergence, sensitivity to hyperparameters, and numerical instability, preventing NQS from reaching the high accuracy required for fundamental science. We address this fundamental optimization bottleneck by introducing the implicitly restarted Lanczos (IRL) method as the core engine for NQS training. Our key innovation is an inherently stable second-order optimization framework that recasts the ill-conditioned parameter update problem into a small, well-posed Hermitian eigenvalue problem. By solving this problem efficiently and robustly with IRL, our approach automatically determines the optimal descent direction and step size, circumventing the need for demanding hyperparameter tuning and eliminating the numerical instabilities common in standard iterative solvers. We demonstrate that IRL enables shallow NQS architectures (with orders of magnitude fewer parameters) to consistently achieve extreme precision (1e-12 kcal/mol) in just 3 to 5 optimization steps. For the F2 molecule, this translates to an approximate 17,900-fold speed-up in total runtime compared to Adam. This work establishes IRL as a superior, robust, and efficient second-order optimization strategy for variational quantum models, paving the way for the practical, high-fidelity application of neural networks in quantum physics and chemistry.

Related papers

Implicitly Restarted Lanczos Enables Chemically Accurate Shallow Neural Quantum States

Wei Liu, Wenjie Dou

Abstract The variational optimization of neural quantum states (NQS) is hindered by the slow convergence and numerical instability of conventional optimizers. We address this bottleneck by introducing the implicitly restarted Lanczos (IRL) method as the core engine for NQS training. Our approach recasts the parameter update problem into a Hermitian eigenvalue problem that IRL solves efficiently, automatically determining the optimal descent direction and step size without hyperparameter tuning. We demonstrate that IRL enables shallow NQS architectures to achieve extreme precision in just 3–5 optimization steps, dramatically outperforming existing methods, including minimum-step stochastic reconfiguration (MinSR). For the N2 dissociation curve, IRL successfully trains a shallow NQS to match full configuration interaction accuracy, validating its robust convergence in strongly correlated regimes. This work establishes IRL as an efficient and robust second-order optimization strategy for variational quantum models.

Precision Chemistry•Mar 30, 2026•1 citation

Generalized Lanczos method for systematic optimization of neural-network quantum states

Jia-Qi Wang, Rong-Qiang He, Zhong-Yi Lu

Recently, artificial intelligence for science has made significant inroads into various fields of natural science research. In the field of quantum many-body computation, researchers have developed numerous ground state solvers based on neural-network quantum states (NQSs), achieving ground state energies with accuracy comparable to or surpassing traditional methods such as variational Monte Carlo methods, density matrix renormalization group, and quantum Monte Carlo methods. Here, we combine supervised learning, variational Monte Carlo (VMC), and the Lanczos method to develop a systematic approach to improving the NQSs of many-body systems, which we refer to as the NQS Lanczos method. The algorithm mainly consists of two parts: the supervised learning part and the VMC optimization part. Through supervised learning, the Lanczos states are represented by the NQSs. Through VMC, the NQSs are further optimized. We analyze the reasons for the underfitting problem and demonstrate how the NQS Lanczos method systematically improves the energy in the highly frustrated regime of the two-dimensional Heisenberg $J_1$-$J_2$ model. Compared to the existing method that combines the Lanczos method with the restricted Boltzmann machine, the primary advantage of the NQS Lanczos method is its linearly increasing computational cost.

arXiv:2502.01264•Physical review. B./Physical review. B•Jan 13, 2026•1 citation

A Nonstochastic Optimization Algorithm for Neural-Network Quantum States

Xiang Li, Jia-Cheng Huang, Guang-Ze Zhang +4 more

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.

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Expectation–Maximization-Based Optimization of Neural Quantum States for Ab Initio Quantum Chemistry

Shen Fang, Hongkun Dou, Zeyu Li +3 more

Accurate solutions for ab initio quantum chemistry are critical for understanding chemistry on the atomic scale. Recently, optimization methods for neural quantum states (NQSs) based on deep neural networks have shown great potential for improving computational accuracy over traditional methods, but they inevitably introduce the challenge of substantial computational costs. To address this challenge, this study proposed expectation–maximization-based optimization of NQS (EMO-NQS), a novel method developed to accelerate optimizations through cost amortization. Specifically, the EMO-NQS method reformulates time-intensive operations as the E-step, amortizing their costs through replays of efficient network-updating operations in the M-step to reduce the overall computational load. Furthermore, an active learning (AL)-based sampling method was introduced to enhance the robustness of EMO-NQS. The performance of EMO-NQS was validated by optimizing the NQS of the equilibrium structures for various molecules. The results demonstrated that EMO-NQS achieved optimization speeds at least twice as fast as those of the vanilla NQS method without compromising accuracy. Furthermore, the consistent acceleration of optimization and the robustness of accuracy facilitated by the AL-based EMO-NQS method were demonstrated by an additional optimization of the construction of the potential energy surface of the N 2 molecule.

Journal of Chemical Theory and Computation•May 20, 2025•1 citation

One More Time: Revisiting Neural Quantum States from a Reinforcement Learning Perspective

Juan Agustín Duque, Sergio García Heredia, Vinicius Hernandes +4 more

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.

cs.LGCross-listedarXiv:2607.02292•Jul 3, 2026•0 citations

Exploration of Quantum-Inspired Deep Learning Architectures for Fundamental Applications in Scientific Computing

Oliver Knitter

Neural-network quantum states (NQS) is an exciting area of quantum-inspired machine learning, wherein a fully classical neural network is used to efficiently learn the optimal state vector for a quantum unsupervised learning problem, bypassing the need to model the full Hilbert space, which is exponentially large. In theory, NQS architectures are broadly applicable, and have been demonstrated successfully on several fundamental applications. We seek to further explore the practical utility of NQS as a black box solver for various applications in the scientific computing domain. We first present NQS as an explicit avenue for de-quantizing variational quantum eigensolvers, demonstrating with the variational quantum linear solver (VQLS). We then explore the utility of VQLS and its de-quantization, VNLS, as black box matrix inversion tools for a Newton-based linear complementarity solver, used to model the time evolution of a rudimentary granular medium. Turning our attention toward applications of NQS in quantum chemistry, we introduce a new retentive network (RetNet) autoregressive NQS ansatz to solve electronic ground state problems, and explore the incorporation of neural annealing techniques to these problems as a means of improving accuracy and ease of training. We finish with a rudimentary scaling law analysis of the RetNet architecture and its immediate autoregressive predecessors as applied to electronic structure calculations.

•Jan 1, 2024•0 citations
The variational optimization of high-dimensional neural network models, such as those used in neural quantum states (NQS), presents a significant challenge in machine intelligence. Conventional first-order stochastic methods (e.g., Adam) are plagued by slow convergence, sensitivity to hyperparameters, and numerical instability, preventing NQS from reaching the high accuracy required for fundamental science. We address this fundamental optimization bottleneck by introducing the implicitly restarted Lanczos (IRL) method as the core engine for NQS training. Our key innovation is an inherently stable second-order optimization framework that recasts the ill-conditioned parameter update problem into a small, well-posed Hermitian eigenvalue problem. By solving this problem efficiently and robustly with IRL, our approach automatically determines the optimal descent direction and step size, circumventing the need for demanding hyperparameter tuning and eliminating the numerical instabilities common in standard iterative solvers. We demonstrate that IRL enables shallow NQS architectures (with orders of magnitude fewer parameters) to consistently achieve extreme precision (1e-12 kcal/mol) in just 3 to 5 optimization steps. For the F2 molecule, this translates to an approximate 17,900-fold speed-up in total runtime compared to Adam. This work establishes IRL as a superior, robust, and efficient second-order optimization strategy for variational quantum models, paving the way for the practical, high-fidelity application of neural networks in quantum physics and chemistry.
Abstract The variational optimization of neural quantum states (NQS) is hindered by the slow convergence and numerical instability of conventional optimizers. We address this bottleneck by introducing the implicitly restarted Lanczos (IRL) method as the core engine for NQS training. Our approach recasts the parameter update problem into a Hermitian eigenvalue problem that IRL solves efficiently, automatically determining the optimal descent direction and step size without hyperparameter tuning. We demonstrate that IRL enables shallow NQS architectures to achieve extreme precision in just 3–5 optimization steps, dramatically outperforming existing methods, including minimum-step stochastic reconfiguration (MinSR). For the N2 dissociation curve, IRL successfully trains a shallow NQS to match full configuration interaction accuracy, validating its robust convergence in strongly correlated regimes. This work establishes IRL as an efficient and robust second-order optimization strategy for variational quantum models.
Recently, artificial intelligence for science has made significant inroads into various fields of natural science research. In the field of quantum many-body computation, researchers have developed numerous ground state solvers based on neural-network quantum states (NQSs), achieving ground state energies with accuracy comparable to or surpassing traditional methods such as variational Monte Carlo methods, density matrix renormalization group, and quantum Monte Carlo methods. Here, we combine supervised learning, variational Monte Carlo (VMC), and the Lanczos method to develop a systematic approach to improving the NQSs of many-body systems, which we refer to as the NQS Lanczos method. The algorithm mainly consists of two parts: the supervised learning part and the VMC optimization part. Through supervised learning, the Lanczos states are represented by the NQSs. Through VMC, the NQSs are further optimized. We analyze the reasons for the underfitting problem and demonstrate how the NQS Lanczos method systematically improves the energy in the highly frustrated regime of the two-dimensional Heisenberg J1J_1J1​-J2J_2J2​ model. Compared to the existing method that combines the Lanczos method with the restricted Boltzmann machine, the primary advantage of the NQS Lanczos method is its linearly increasing computational cost.
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.
Accurate solutions for ab initio quantum chemistry are critical for understanding chemistry on the atomic scale. Recently, optimization methods for neural quantum states (NQSs) based on deep neural networks have shown great potential for improving computational accuracy over traditional methods, but they inevitably introduce the challenge of substantial computational costs. To address this challenge, this study proposed expectation–maximization-based optimization of NQS (EMO-NQS), a novel method developed to accelerate optimizations through cost amortization. Specifically, the EMO-NQS method reformulates time-intensive operations as the E-step, amortizing their costs through replays of efficient network-updating operations in the M-step to reduce the overall computational load. Furthermore, an active learning (AL)-based sampling method was introduced to enhance the robustness of EMO-NQS. The performance of EMO-NQS was validated by optimizing the NQS of the equilibrium structures for various molecules. The results demonstrated that EMO-NQS achieved optimization speeds at least twice as fast as those of the vanilla NQS method without compromising accuracy. Furthermore, the consistent acceleration of optimization and the robustness of accuracy facilitated by the AL-based EMO-NQS method were demonstrated by an additional optimization of the construction of the potential energy surface of the N 2 molecule.
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.
Neural-network quantum states (NQS) is an exciting area of quantum-inspired machine learning, wherein a fully classical neural network is used to efficiently learn the optimal state vector for a quantum unsupervised learning problem, bypassing the need to model the full Hilbert space, which is exponentially large. In theory, NQS architectures are broadly applicable, and have been demonstrated successfully on several fundamental applications. We seek to further explore the practical utility of NQS as a black box solver for various applications in the scientific computing domain. We first present NQS as an explicit avenue for de-quantizing variational quantum eigensolvers, demonstrating with the variational quantum linear solver (VQLS). We then explore the utility of VQLS and its de-quantization, VNLS, as black box matrix inversion tools for a Newton-based linear complementarity solver, used to model the time evolution of a rudimentary granular medium. Turning our attention toward applications of NQS in quantum chemistry, we introduce a new retentive network (RetNet) autoregressive NQS ansatz to solve electronic ground state problems, and explore the incorporation of neural annealing techniques to these problems as a means of improving accuracy and ease of training. We finish with a rudimentary scaling law analysis of the RetNet architecture and its immediate autoregressive predecessors as applied to electronic structure calculations.