Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2509.12365 · Physical Review Research

Entanglement and optimization within autoregressive neural quantum states

Andrew Jreissaty, Hang Zhang, Jairo C. Quijano, Juan Carrasquilla, Roeland Wiersema

Published Jan 12, 2026• 0 citations
arXivHTMLPDF

Abstract

Neural quantum states (NQSs) are powerful variational ansätze capable of representing highly entangled quantum many-body wavefunctions. While the average entanglement properties of ensembles of restricted Boltzmann machines are well understood, the entanglement structure of autoregressive NQSs such as recurrent neural networks and transformers remains largely unexplored. We perform large-scale simulations of ensembles of random autoregressive wavefunctions for chains of up to 256 spins and uncover signatures of transitions in their average entanglement scaling, entanglement spectra, and correlation functions. We show that the standard softmax normalization of the wavefunction suppresses entanglement and fluctuations, and introduce a square modulus normalization function that restores them. Finally, we connect the insights gained from our entanglement and activation function analysis to initialization strategies for finding the ground states of strongly correlated Hamiltonians via variational Monte Carlo.

DOI
10.1103/t2cg-kr7y

Related papers

Autoregressive neural quantum states of Fermi Hubbard models

Eduardo Ibarra-García-Padilla, Hannah Lange, Roger G. Melko +4 more

Neural quantum states (NQSs) have emerged as a powerful ansatz for variational quantum Monte Carlo studies of strongly correlated systems. Here, we apply recurrent neural networks (RNNs) and autoregressive transformer neural networks to the Fermi-Hubbard and the (non-Hermitian) Hatano-Nelson-Hubbard models in one and two dimensions. In both cases, we observe that the convergence of the RNN ansatz is challenged when increasing the interaction strength. We present a physically motivated and easy-to-implement strategy for improving the optimization, namely, by ramping of the model parameters. Furthermore, we investigate the advantages and disadvantages of the autoregressive sampling property of both network architectures. For the Hatano-Nelson-Hubbard model, we identify convergence issues that stem from the autoregressive sampling scheme in combination with the non-Hermitian nature of the model. Our findings provide insights into the challenges of the NQS approach and make the first step towards exploring strongly correlated electrons using this ansatz.

arXiv:2411.07144•Physical Review Research•Feb 3, 2025•9 citations

Quantum-Enhanced Neural Networks for Quantum Many-Body Simulations

Zongkang Zhang, Ying Li, Xiaosi Xu

Neural quantum states (NQS) have gained prominence in variational quantum Monte Carlo methods in approximating ground-state wavefunctions. Despite their success, they face limitations in optimization, scalability, and expressivity in addressing certain problems. In this work, we propose a quantum-neural hybrid framework that combines parametrized quantum circuits with neural networks to model quantum many-body wavefunctions. This approach combines the efficient sampling and optimization capabilities of autoregressive neural networks with the enhanced expressivity provided by quantum circuits. Numerical simulations demonstrate the scalability and accuracy of the hybrid ansatz in spin systems and quantum chemistry problems. Our results reveal that the hybrid method achieves notably lower relative energy compared to stand-alone NQS. These findings underscore the potential of quantum-neural hybrid methods for tackling challenging problems in quantum many-body simulations.

arXiv:2501.12130•PRX Intelligence•Jul 28, 2026•1 citation

Parallel Scan Recurrent Neural Quantum States for Scalable Variational Monte Carlo

Ejaaz Merali, Mohamed Hibat-Allah, Mohammad Kohandel +2 more

Neural-network quantum states have emerged as a powerful variational framework for quantum many-body systems, with recent progress often driven by massively parallel architectures such as transformers. Recurrent neural network quantum states, however, are frequently regarded as intrinsically sequential and therefore less scalable. Here we revisit this view by showing that modern recurrent architectures can support fast, accurate, and computationally accessible neural quantum state simulations. Using autoregressive recurrent wave functions together with recent advances in parallelizable recurrence, we develop variational ansätze, called parallel scan recurrent neural quantum states (PSR-NQS), which can be trained efficiently within variational Monte Carlo in one and two spatial dimensions. We demonstrate accurate benchmark results and show that, with iterative retraining, our approach reaches two-dimensional spin lattices as large as $52\times52$ while remaining in agreement with available quantum Monte Carlo data. Our results establish recurrent architectures as a practical and promising route toward scalable neural quantum state simulations with modest computational resources.

cond-mat.str-elCross-listedarXiv:2605.13807•May 14, 2026•0 citations

Adaptive neural quantum states: a recurrent neural network perspective

Jake McNaughton, Mohamed Hibat-Allah

Abstract Neural-network quantum states (NQS) are powerful neural-network ansätzes that have emerged as promising tools for studying quantum many-body physics through the lens of the variational principle. These architectures are known to be systematically improvable by increasing the number of parameters. Here we demonstrate an Adaptive scheme to optimize NQSs, through the example of recurrent neural networks (RNNs), using a fraction of the computation cost while reducing training fluctuations and improving the quality of variational calculations targeting ground states of prototypical models in one- and two-spatial dimensions. This Adaptive framework reduces the computational cost through training small RNNs and reusing them to initialize larger RNNs. This work opens up the possibility for optimizing graphical processing unit resources deployed in large-scale NQS simulations.

arXiv:2507.18700•Machine Learning Science and Technology•Sep 15, 2026•0 citations

Autoregressive neural-network wavefunctions for ab initio quantum chemistry

Thomas D. Barrett, Aleksei Malyshev, Alexander I. Lvovsky

In recent years, neural network quantum states (NNQS) have emerged as powerful tools for the study of quantum many-body systems. Electronic structure calculations are one such canonical many-body problem that have attracted significant research efforts spanning multiple decades, whilst only recently being attempted with NNQS. However, the complex non-local interactions and high sample complexity are significant challenges that call for bespoke solutions. Here, we parameterise the electronic wavefunction with a novel autoregressive neural network (ARN) that permits highly efficient and scalable sampling, whilst also embedding physical priors reflecting the structure of molecular systems without sacrificing expressibility. This allows us to perform electronic structure calculations on molecules with up to 30 spin-orbitals -- at least an order of magnitude more Slater determinants than previous applications of conventional NNQS -- and we find that our ansatz can outperform the de-facto gold-standard coupled cluster methods even in the presence of strong quantum correlations. With a highly expressive neural network for which sampling is no longer a computational bottleneck, we conclude that the barriers to further scaling are not associated with the wavefunction ansatz itself, but rather are inherent to any variational Monte Carlo approach.

arXiv:2109.12606•Sep 26, 2021•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
Neural quantum states (NQSs) are powerful variational ansätze capable of representing highly entangled quantum many-body wavefunctions. While the average entanglement properties of ensembles of restricted Boltzmann machines are well understood, the entanglement structure of autoregressive NQSs such as recurrent neural networks and transformers remains largely unexplored. We perform large-scale simulations of ensembles of random autoregressive wavefunctions for chains of up to 256 spins and uncover signatures of transitions in their average entanglement scaling, entanglement spectra, and correlation functions. We show that the standard softmax normalization of the wavefunction suppresses entanglement and fluctuations, and introduce a square modulus normalization function that restores them. Finally, we connect the insights gained from our entanglement and activation function analysis to initialization strategies for finding the ground states of strongly correlated Hamiltonians via variational Monte Carlo.
Neural quantum states (NQSs) have emerged as a powerful ansatz for variational quantum Monte Carlo studies of strongly correlated systems. Here, we apply recurrent neural networks (RNNs) and autoregressive transformer neural networks to the Fermi-Hubbard and the (non-Hermitian) Hatano-Nelson-Hubbard models in one and two dimensions. In both cases, we observe that the convergence of the RNN ansatz is challenged when increasing the interaction strength. We present a physically motivated and easy-to-implement strategy for improving the optimization, namely, by ramping of the model parameters. Furthermore, we investigate the advantages and disadvantages of the autoregressive sampling property of both network architectures. For the Hatano-Nelson-Hubbard model, we identify convergence issues that stem from the autoregressive sampling scheme in combination with the non-Hermitian nature of the model. Our findings provide insights into the challenges of the NQS approach and make the first step towards exploring strongly correlated electrons using this ansatz.
Neural quantum states (NQS) have gained prominence in variational quantum Monte Carlo methods in approximating ground-state wavefunctions. Despite their success, they face limitations in optimization, scalability, and expressivity in addressing certain problems. In this work, we propose a quantum-neural hybrid framework that combines parametrized quantum circuits with neural networks to model quantum many-body wavefunctions. This approach combines the efficient sampling and optimization capabilities of autoregressive neural networks with the enhanced expressivity provided by quantum circuits. Numerical simulations demonstrate the scalability and accuracy of the hybrid ansatz in spin systems and quantum chemistry problems. Our results reveal that the hybrid method achieves notably lower relative energy compared to stand-alone NQS. These findings underscore the potential of quantum-neural hybrid methods for tackling challenging problems in quantum many-body simulations.
Neural-network quantum states have emerged as a powerful variational framework for quantum many-body systems, with recent progress often driven by massively parallel architectures such as transformers. Recurrent neural network quantum states, however, are frequently regarded as intrinsically sequential and therefore less scalable. Here we revisit this view by showing that modern recurrent architectures can support fast, accurate, and computationally accessible neural quantum state simulations. Using autoregressive recurrent wave functions together with recent advances in parallelizable recurrence, we develop variational ansätze, called parallel scan recurrent neural quantum states (PSR-NQS), which can be trained efficiently within variational Monte Carlo in one and two spatial dimensions. We demonstrate accurate benchmark results and show that, with iterative retraining, our approach reaches two-dimensional spin lattices as large as 52×5252\times5252×52 while remaining in agreement with available quantum Monte Carlo data. Our results establish recurrent architectures as a practical and promising route toward scalable neural quantum state simulations with modest computational resources.
Abstract Neural-network quantum states (NQS) are powerful neural-network ansätzes that have emerged as promising tools for studying quantum many-body physics through the lens of the variational principle. These architectures are known to be systematically improvable by increasing the number of parameters. Here we demonstrate an Adaptive scheme to optimize NQSs, through the example of recurrent neural networks (RNNs), using a fraction of the computation cost while reducing training fluctuations and improving the quality of variational calculations targeting ground states of prototypical models in one- and two-spatial dimensions. This Adaptive framework reduces the computational cost through training small RNNs and reusing them to initialize larger RNNs. This work opens up the possibility for optimizing graphical processing unit resources deployed in large-scale NQS simulations.
In recent years, neural network quantum states (NNQS) have emerged as powerful tools for the study of quantum many-body systems. Electronic structure calculations are one such canonical many-body problem that have attracted significant research efforts spanning multiple decades, whilst only recently being attempted with NNQS. However, the complex non-local interactions and high sample complexity are significant challenges that call for bespoke solutions. Here, we parameterise the electronic wavefunction with a novel autoregressive neural network (ARN) that permits highly efficient and scalable sampling, whilst also embedding physical priors reflecting the structure of molecular systems without sacrificing expressibility. This allows us to perform electronic structure calculations on molecules with up to 30 spin-orbitals -- at least an order of magnitude more Slater determinants than previous applications of conventional NNQS -- and we find that our ansatz can outperform the de-facto gold-standard coupled cluster methods even in the presence of strong quantum correlations. With a highly expressive neural network for which sampling is no longer a computational bottleneck, we conclude that the barriers to further scaling are not associated with the wavefunction ansatz itself, but rather are inherent to any variational Monte Carlo approach.
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.