Physics-informed transformers for electronic quantum states

Neural-network-based variational quantum states, particularly autoregressive models, are powerful tools for describing complex many-body wave functions. However, their performance depends on the computational basis chosen and they often lack physical interpretability. We propose a modified variational Monte-Carlo framework which leverages prior physical information to construct a complete computational many-body basis containing a reference state that serves as a rough approximation to the true ground state. A Transformer is used to parametrize and autoregressively sample corrections to this reference state, giving rise to a more interpretable and computationally efficient representation of the ground state. We demonstrate this approach in a fermionic model featuring a metal-insulator transition by employing Hartree-Fock and a strong-coupling limit to define physics-informed bases. We also show that the Transformer's hidden representation captures the natural energetic order of the different basis states. This work paves the way for more efficient and interpretable neural quantum-state representations.Neural-network-based variational quantum states, particularly autoregressive models, are powerful tools for describing complex many-body wave functions. However, their performance depends on the computational basis chosen and they often lack physical interpretability. We propose a modified variational Monte-Carlo framework which leverages prior physical information to construct a complete computational many-body basis containing a reference state that serves as a rough approximation to the true ground state. A Transformer is used to parametrize and autoregressively sample corrections to this reference state, giving rise to a more interpretable and computationally efficient representation of the ground state. We demonstrate this approach in a fermionic model featuring a metal-insulator transition by employing Hartree-Fock and a strong-coupling limit to define physics-informed bases. We also show that the Transformer's hidden representation captures the natural energetic order of the different basis states. This work paves the way for more efficient and interpretable neural quantum-state representations.

Nature CommunicationsNov 28, 20253 citations

Physics-informed Transformers for Electronic Quantum States

João Augusto Sobral, Michael Perle, Mathias S. Scheurer

Neural-network-based variational quantum states in general, and more recently autoregressive models in particular, have proven to be powerful tools to describe complex many-body wave functions. However, their performance crucially depends on the computational basis chosen and they often lack physical interpretability. To mitigate these issues, we here propose a modified variational Monte-Carlo framework which leverages prior physical information to construct a computational second-quantized basis containing a reference state that serves as a rough approximation to the true ground state. In this basis, a Transformer is used to parametrize and autoregressively sample the corrections to the reference state, giving rise to a more interpretable and computationally efficient representation of the ground state. We demonstrate this approach using a non-sparse fermionic model featuring a metal-insulator transition and employing Hartree-Fock and a strong-coupling limit to define physics-informed bases. We also show that the Transformer's hidden representation captures the natural energetic order of the different basis states. This work paves the way for more efficient and interpretable neural quantum-state representations.Neural-network-based variational quantum states in general, and more recently autoregressive models in particular, have proven to be powerful tools to describe complex many-body wave functions. However, their performance crucially depends on the computational basis chosen and they often lack physical interpretability. To mitigate these issues, we here propose a modified variational Monte-Carlo framework which leverages prior physical information to construct a computational second-quantized basis containing a reference state that serves as a rough approximation to the true ground state. In this basis, a Transformer is used to parametrize and autoregressively sample the corrections to the reference state, giving rise to a more interpretable and computationally efficient representation of the ground state. We demonstrate this approach using a non-sparse fermionic model featuring a metal-insulator transition and employing Hartree-Fock and a strong-coupling limit to define physics-informed bases. We also show that the Transformer's hidden representation captures the natural energetic order of the different basis states. This work paves the way for more efficient and interpretable neural quantum-state representations.

arXiv:2412.12248Dec 16, 20241 citation

Enhancing Neural-Network Variational Monte Carlo through Basis Transformation

Zhixuan Liu, Dongheng Qian, Jing Wang

Neural-network variational Monte Carlo (NNVMC) has emerged as a powerful tool for solving quantum many-body problems, yet systematic pathways for improving its accuracy remain largely heuristic. Here, we introduce a physically motivated basis transformation for NNVMC that enhances variational expressivity without increasing the complexity of the neural-network ansatz itself. By formulating the many-body wave function in a Gaussian basis, we introduce a single learnable locality parameter, $α$ , that reshapes the target ground state into a more learnable representation. This approach introduces minimal computational overhead and can be readily combined with existing neural-network architectures. Using the three-dimensional homogeneous electron gas as a benchmark, we show that the optimized basis transformation consistently lowers the variational energy for both FermiNet and message-passing neural-network architectures. Notably, for the latter, it enables a more precise determination of the Fermi liquid to Wigner crystal phase transition. More broadly, our results highlight basis transformation as a new route to improving NNVMC in continuous space, showing that accuracy can be enhanced not only by refining the ansatz but also by making the target ground state easier to represent.Neural-network variational Monte Carlo (NNVMC) has emerged as a powerful tool for solving quantum many-body problems, yet systematic pathways for improving its accuracy remain largely heuristic. Here, we introduce a physically motivated basis transformation for NNVMC that enhances variational expressivity without increasing the complexity of the neural-network ansatz itself. By formulating the many-body wave function in a Gaussian basis, we introduce a single learnable locality parameter, $α$, that reshapes the target ground state into a more learnable representation. This approach introduces minimal computational overhead and can be readily combined with existing neural-network architectures. Using the three-dimensional homogeneous electron gas as a benchmark, we show that the optimized basis transformation consistently lowers the variational energy for both FermiNet and message-passing neural-network architectures. Notably, for the latter, it enables a more precise determination of the Fermi liquid to Wigner crystal phase transition. More broadly, our results highlight basis transformation as a new route to improving NNVMC in continuous space, showing that accuracy can be enhanced not only by refining the ansatz but also by making the target ground state easier to represent.

cond-mat.str-elCross-listedarXiv:2604.15888Apr 20, 20260 citations

Deep Autoregressive Models for the Efficient Variational Simulation of Many-Body Quantum Systems

Or Sharir, Yoav Levine, Noam Wies +2 more

Artificial neural networks were recently shown to be an efficient representation of highly entangled many-body quantum states. In practical applications, neural-network states inherit numerical schemes used in variational Monte Carlo method, most notably the use of Markov-chain Monte Carlo (MCMC) sampling to estimate quantum expectations. The local stochastic sampling in MCMC caps the potential advantages of neural networks in two ways: (i) Its intrinsic computational cost sets stringent practical limits on the width and depth of the networks, and therefore limits their expressive capacity; (ii) its difficulty in generating precise and uncorrelated samples can result in estimations of observables that are very far from their true value. Inspired by the state-of-the-art generative models used in machine learning, we propose a specialized neural-network architecture that supports efficient and exact sampling, completely circumventing the need for Markov-chain sampling. We demonstrate our approach for two-dimensional interacting spin models, showcasing the ability to obtain accurate results on larger system sizes than those currently accessible to neural-network quantum states.Artificial neural networks were recently shown to be an efficient representation of highly entangled many-body quantum states. In practical applications, neural-network states inherit numerical schemes used in variational Monte Carlo method, most notably the use of Markov-chain Monte Carlo (MCMC) sampling to estimate quantum expectations. The local stochastic sampling in MCMC caps the potential advantages of neural networks in two ways: (i) Its intrinsic computational cost sets stringent practical limits on the width and depth of the networks, and therefore limits their expressive capacity; (ii) its difficulty in generating precise and uncorrelated samples can result in estimations of observables that are very far from their true value. Inspired by the state-of-the-art generative models used in machine learning, we propose a specialized neural-network architecture that supports efficient and exact sampling, completely circumventing the need for Markov-chain sampling. We demonstrate our approach for two-dimensional interacting spin models, showcasing the ability to obtain accurate results on larger system sizes than those currently accessible to neural-network quantum states.

arXiv:1902.04057Physical Review LettersJan 16, 2020257 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.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:2501.12130PRX IntelligenceJul 28, 20261 citation

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.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:2109.12606Sep 26, 20211 citation

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, that reshapes the target ground state into a more learnable representation. This approach introduces minimal computational overhead and can be readily combined with existing neural-network architectures. Using the three-dimensional homogeneous electron gas as a benchmark, we show that the optimized basis transformation consistently lowers the variational energy for both FermiNet and message-passing neural-network architectures. Notably, for the latter, it enables a more precise determination of the Fermi liquid to Wigner crystal phase transition. More broadly, our results highlight basis transformation as a new route to improving NNVMC in continuous space, showing that accuracy can be enhanced not only by refining the ansatz but also by making the target ground state easier to represent.