Recent research has demonstrated the usefulness of neural networks as variational ansatz functions for quantum many-body states. However, high-dimensional sampling spaces and transient autocorrelations confront these approaches with a challenging computational bottleneck. Compared to conventional neural networks, physical-model devices offer a fast, efficient and inherently parallel substrate capable of related forms of Markov chain Monte Carlo sampling. Here, we demonstrate the ability of a neuromorphic chip to represent the ground states of quantum spin models by variational energy minimization. We develop a training algorithm and apply it to the transverse field Ising model, showing good performance at moderate system sizes ( $N\leq 10$ ). A systematic hyperparameter study shows that scalability to larger system sizes mainly depends on sample quality, which is limited by temporal parameter variations on the analog neuromorphic chip. Our work thus provides an important step towards harnessing the capabilities of neuromorphic hardware for tackling the curse of dimensionality in quantum many-body problems.Recent research has demonstrated the usefulness of neural networks as variational ansatz functions for quantum many-body states. However, high-dimensional sampling spaces and transient autocorrelations confront these approaches with a challenging computational bottleneck. Compared to conventional neural networks, physical-model devices offer a fast, efficient and inherently parallel substrate capable of related forms of Markov chain Monte Carlo sampling. Here, we demonstrate the ability of a neuromorphic chip to represent the ground states of quantum spin models by variational energy minimization. We develop a training algorithm and apply it to the transverse field Ising model, showing good performance at moderate system sizes ($N\leq 10$). A systematic hyperparameter study shows that scalability to larger system sizes mainly depends on sample quality, which is limited by temporal parameter variations on the analog neuromorphic chip. Our work thus provides an important step towards harnessing the capabilities of neuromorphic hardware for tackling the curse of dimensionality in quantum many-body problems.

arXiv:2109.15169Sep 30, 20210 citations

Probabilistic Computers for Neural Quantum States

Shuvro Chowdhury, Jasper Pieterse, Navid Anjum Aadit +3 more

Neural quantum states efficiently represent many-body wavefunctions with neural networks, but the cost of Monte Carlo sampling limits their scaling to large system sizes. Here we address this challenge by combining sparse Boltzmann machine architectures with probabilistic computing hardware. We implement a probabilistic computer on field-programmable gate arrays (FPGAs) and use it as a fast sampler for energy-based neural quantum states. For the two-dimensional transverse-field Ising model at criticality, we obtain accurate ground-state energies for lattices up to 80$\times$80 (6400 spins) using a custom multi-FPGA cluster. Furthermore, we introduce a dual-sampling algorithm to train deep Boltzmann machines, replacing intractable marginalization with conditional sampling over auxiliary layers. This enables the training of sparse deep models and improves parameter efficiency relative to shallow networks. We further implement this algorithm on a single FPGA, demonstrating the training of deep Boltzmann machines for systems as large as $30 \times 30$ (900 spins). Together, these results demonstrate that probabilistic hardware can overcome the sampling bottleneck in variational simulation of quantum many-body systems, opening a path to larger system sizes and deeper variational architectures.Neural quantum states efficiently represent many-body wavefunctions with neural networks, but the cost of Monte Carlo sampling limits their scaling to large system sizes. Here we address this challenge by combining sparse Boltzmann machine architectures with probabilistic computing hardware. We implement a probabilistic computer on field-programmable gate arrays (FPGAs) and use it as a fast sampler for energy-based neural quantum states. For the two-dimensional transverse-field Ising model at criticality, we obtain accurate ground-state energies for lattices up to 80$\times$80 (6400 spins) using a custom multi-FPGA cluster. Furthermore, we introduce a dual-sampling algorithm to train deep Boltzmann machines, replacing intractable marginalization with conditional sampling over auxiliary layers. This enables the training of sparse deep models and improves parameter efficiency relative to shallow networks. We further implement this algorithm on a single FPGA, demonstrating the training of deep Boltzmann machines for systems as large as $30 \times 30$ (900 spins). Together, these results demonstrate that probabilistic hardware can overcome the sampling bottleneck in variational simulation of quantum many-body systems, opening a path to larger system sizes and deeper variational architectures.

quant-pharXiv:2512.24558Jan 1, 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 neural network quantum state

Xin Pang, Yanan Li, Zhimin Wang

Quantum many-body computation is a central challenge across multiple domains of scientific research, and Monte Carlo method has emerged as one of the most effective numerical techniques for investigating large-scale many-body systems. In recent years, the integration of Monte Carlo methods with machine learning has led to the development of neural quantum state approach. Quantum neural networks (QNNs) offer superior expressiveness compared to classical neural networks. In this paper, we propose a novel Quantum Neural Network Quantum State (QNNQS) method, based on the variational Monte Carlo framework. This approach leverages a quantum neural network to represent the many-body quantum state, and optimizes it iteratively through gradient descent to determine the ground-state energy. To assess the performance of this method, we conduct simulations of the one-dimensional transverse-field Ising model, investigating the influence of quantum circuit architecture on circuit expressibility, entangling capability, and computational accuracy. The results demonstrate that the QNNQS method achieves comparable simulation accuracy while utilizing significantly fewer parameters and requiring less training resources than classical network quantum state methods.Quantum many-body computation is a central challenge across multiple domains of scientific research, and Monte Carlo method has emerged as one of the most effective numerical techniques for investigating large-scale many-body systems. In recent years, the integration of Monte Carlo methods with machine learning has led to the development of neural quantum state approach. Quantum neural networks (QNNs) offer superior expressiveness compared to classical neural networks. In this paper, we propose a novel Quantum Neural Network Quantum State (QNNQS) method, based on the variational Monte Carlo framework. This approach leverages a quantum neural network to represent the many-body quantum state, and optimizes it iteratively through gradient descent to determine the ground-state energy. To assess the performance of this method, we conduct simulations of the one-dimensional transverse-field Ising model, investigating the influence of quantum circuit architecture on circuit expressibility, entangling capability, and computational accuracy. The results demonstrate that the QNNQS method achieves comparable simulation accuracy while utilizing significantly fewer parameters and requiring less training resources than classical network quantum state methods.

International Conference on Algorithms, Network, and Communication Technology (ICANCT)Mar 3, 20252 citations

Improved Ground State Estimation in Quantum Field Theories via Normalising Flow-Assisted Neural Quantum States

Vishal S. Ngairangbam, Michael Spannowsky, Timur Sypchenko

We propose a hybrid variational framework that enhances Neural Quantum States (NQS) with a Normalising Flow-based sampler to improve the expressivity and trainability of quantum many-body wavefunctions. Our approach decouples the sampling task from the variational ansatz by learning a continuous flow model that targets a discretised, amplitude-supported subspace of the Hilbert space. This overcomes limitations of Markov Chain Monte Carlo (MCMC) and autoregressive methods, especially in regimes with long-range correlations and volume-law entanglement. Applied to the transverse-field Ising model with both short- and long-range interactions, our method achieves comparable ground state energy errors with state-of-the-art matrix product states and lower energies than autoregressive NQS. For systems up to 50 spins, we demonstrate high accuracy and robust convergence across a wide range of coupling strengths, including regimes where competing methods fail. Our results showcase the utility of flow-assisted sampling as a scalable tool for quantum simulation and offer a new approach toward learning expressive quantum states in high-dimensional Hilbert spaces.We propose a hybrid variational framework that enhances Neural Quantum States (NQS) with a Normalising Flow-based sampler to improve the expressivity and trainability of quantum many-body wavefunctions. Our approach decouples the sampling task from the variational ansatz by learning a continuous flow model that targets a discretised, amplitude-supported subspace of the Hilbert space. This overcomes limitations of Markov Chain Monte Carlo (MCMC) and autoregressive methods, especially in regimes with long-range correlations and volume-law entanglement. Applied to the transverse-field Ising model with both short- and long-range interactions, our method achieves comparable ground state energy errors with state-of-the-art matrix product states and lower energies than autoregressive NQS. For systems up to 50 spins, we demonstrate high accuracy and robust convergence across a wide range of coupling strengths, including regimes where competing methods fail. Our results showcase the utility of flow-assisted sampling as a scalable tool for quantum simulation and offer a new approach toward learning expressive quantum states in high-dimensional Hilbert spaces.

arXiv:2506.12128Jun 13, 20250 citations

Quantum many-body states: A novel neuromorphic application

Andreas Baumbach, Robert Klassert, Stefanie Czischek +2 more

Emergent phenomena in condensed matter physics, such as superconductivity, are rooted in the interaction of many quantum particles. These phenomena remain poorly understood in part due to the computational demands of their simulation. In recent years variational representations based on artificial neural networks, so called neural quantum states (NQS), have been shown to be efficient, ie. sub-exponentially scaling, representations. However, the computational complexity of such representations scales not only with the size of the physical system, but also with the size of the neural network. In this work, we use the analog neuromorphic BrainScaleS-2 platform to implement probabilistic representations of two particular types of quantum states. The physical nature of the neuromorphic system enforces an inherent parallelism of the compuation, rendering the emulation time independent of the used network size. We show the effectiveness of our scheme in two settings: First, we consider a hallmark test for ”quantumness” by representing a quantum state that violates the classical bounds of the Bell inequality. Second, we show that we can represent the large class of stoquastic quantum states with fidelities above 98% for moderate system sizes. This offers a novel application for spike-based neuromorphic hardware which departs from the more traditional neuroscience-inspired use cases.Emergent phenomena in condensed matter physics, such as superconductivity, are rooted in the interaction of many quantum particles. These phenomena remain poorly understood in part due to the computational demands of their simulation. In recent years variational representations based on artificial neural networks, so called neural quantum states (NQS), have been shown to be efficient, ie. sub-exponentially scaling, representations. However, the computational complexity of such representations scales not only with the size of the physical system, but also with the size of the neural network. In this work, we use the analog neuromorphic BrainScaleS-2 platform to implement probabilistic representations of two particular types of quantum states. The physical nature of the neuromorphic system enforces an inherent parallelism of the compuation, rendering the emulation time independent of the used network size. We show the effectiveness of our scheme in two settings: First, we consider a hallmark test for ”quantumness” by representing a quantum state that violates the classical bounds of the Bell inequality. Second, we show that we can represent the large class of stoquastic quantum states with fidelities above 98% for moderate system sizes. This offers a novel application for spike-based neuromorphic hardware which departs from the more traditional neuroscience-inspired use cases.

Neuro Inspired Computational Elements Conference (NICE)Mar 28, 20220 citations

N\leq 10

). A systematic hyperparameter study shows that scalability to larger system sizes mainly depends on sample quality, which is limited by temporal parameter variations on the analog neuromorphic chip. Our work thus provides an important step towards harnessing the capabilities of neuromorphic hardware for tackling the curse of dimensionality in quantum many-body problems.

30 \times 30

(900 spins). Together, these results demonstrate that probabilistic hardware can overcome the sampling bottleneck in variational simulation of quantum many-body systems, opening a path to larger system sizes and deeper variational architectures.