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:2105.04130 · Physical review. B./Physical review. B

Boltzmann machines as two-dimensional tensor networks

Sujie Li, Feng Ping Pan, Pengfei Zhou, Pan Zhang

Published Aug 27, 2021• 15 citations
arXivHTMLPDF

Abstract

Restricted Boltzmann machines (RBMs) and deep Boltzmann machines (DBMs) are important models in machine learning, and recently found numerous applications in quantum many-body physics. We show that there are fundamental connections between them and tensor networks. In particular, we demonstrate that any RBM and DBM can be exactly represented as a two-dimensional tensor network. This representation gives characterizations of the expressive power of RBMs and DBMs using entanglement structures of the tensor networks, and also provides an efficient tensor network contraction algorithm for the computing partition function of RBMs and DBMs. Using numerical experiments, we show that the proposed algorithm is more accurate than the state-of-the-art machine learning methods in estimating the partition function of RBMs and DBMs, and have potential applications in training DBMs for general machine learning tasks.

DOI
10.1103/physrevb.104.075154

Related papers

Quantum Entanglement in Deep Learning Architectures

Yoav Levine, Or Sharir, Nadav Cohen +1 more

Modern deep learning has enabled unprecedented achievements in various domains. Nonetheless, employment of machine learning for wave function representations is focused on more traditional architectures such as restricted Boltzmann machines (RBMs) and fully connected neural networks. In this Letter, we establish that contemporary deep learning architectures, in the form of deep convolutional and recurrent networks, can efficiently represent highly entangled quantum systems. By constructing tensor network equivalents of these architectures, we identify an inherent reuse of information in the network operation as a key trait which distinguishes them from standard tensor network-based representations, and which enhances their entanglement capacity. Our results show that such architectures can support volume-law entanglement scaling, polynomially more efficiently than presently employed RBMs. Thus, beyond a quantification of the entanglement capacity of leading deep learning architectures, our analysis formally motivates a shift of trending neural-network-based wave function representations closer to the state-of-the-art in machine learning.

arXiv:1803.09780•Physical Review Letters•Feb 12, 2019•206 citations

Restricted Boltzmann machines in physics: concepts, theories and applications

Kai Zhang, Sora Sakai

Abstract A restricted Boltzmann machine (RBM) is a two-layer neural-network machine-learning model comprising a visible layer that represents observed data and a hidden layer that captures correlations among the visible units. Rooted in ideas from statistical mechanics, RBMs provide a probabilistic framework for modeling complex, high-dimensional distributions through an energy function and a corresponding Boltzmann distribution. This review offers an accessible introduction to RBMs from a physics-oriented perspective, focusing on their core concepts and training principles. We survey key theoretical developments that interpret RBMs as many-body systems, highlighting their equilibrium/non-equilibrium thermodynamics, learning dynamics, mean-field properties, phase diagrams, and free energy calculation. We further discuss recent applications of RBMs to a range of physics problems, including quantum-mechanical wavefunction representation, statistical-mechanical Boltzmann distribution modeling, and biological protein sequence analysis.

Journal of Physics Communications•Sep 1, 2026•0 citations

Equivalence of restricted Boltzmann machines and tensor network states

Jing Chen, Song Cheng, Haidong Xie +2 more

The restricted Boltzmann machine is a fundamental building block of deep learning. The authors demonstrate its equivalence with tensor network states with explicit mappings, thus drawing a constructive connection between deep learning and quantum physics. On one side, deep learning approaches can be used to study novel states of matter. In return, investigations of tensor network states and their expressibility can be adapted to guide neural network architecture design.

arXiv:1701.04831•Physical review. B./Physical review. B•Feb 2, 2018•242 citations

Quantum boltzmann machine :emergence & applications

Miguel Francisco Aldana Páez

Lately, there has been an extensive state-of-the-art research in Machine Learning methods, due to their important features such as universality in approximations and dimensional reduction. In this way, the present work aims at exploiting these properties of Machine Learning on physical many-body problems, in which a restricted Boltzmann Machine (RBM) is trained with quantum Monte Carlo data to best represent the ground state of an N spin system under the transverse field Ising model Hamiltonian in one dimension. During the path coursed along the development of this project, we first review classical well-implemented methods such as the single-layer perceptron, multi-layer perceptron for the XOR problem with back-propagation algorithm, discrete Hopfield network for pattern reconstruction, continuous Hopfield network with Simulated Annealing for solving the traveling salesman problem with 10 cities and the Boltzmann machine for feature extraction. Then, we review a few aspects of what is called a Quantum Boltzmann Machine and the basics of the transverse field Ising model. Finally, we train an RBM consisting of N visible neurons, hidden density neurons alpha=M/N=2 and 1-step contrastive divergence (CD_1), by minimizing the Kullbach-Liebler divergence of the RBM and the training data set coming from the algorithmic variant of quantum Monte Carlo, known as discrete Path-Integral Monte Carlo. For this last step, we use Trotter-Suzuki decomposition with m Trotter slices and the Wolff-s cluster algorithm. After training the RBM, we calculate magnetic observables based on the RBM-s wave-function such as longitudinal and transverse magnetization, for different values of transverse field and we do an estimation of the critical field at which the transverse Ising model system suffers a quantum phase transition from ferromagnet to paramagnet

•Jan 1, 2018•0 citations

Machine learning in quantum computers via general Boltzmann Machines: Generative and Discriminative training through annealing.

Siddhartha Srivastava, Veera Sundararaghavan

We present a Hybrid-Quantum-classical method for learning Boltzmann machines (BM) for generative and discriminative tasks. Boltzmann machines are undirected graphs that form the building block of many learning architectures such as Restricted Boltzmann machines (RBM's) and Deep Boltzmann machines (DBM's). They have a network of visible and hidden nodes where the former are used as the reading sites while the latter are used to manipulate the probability of the visible states. BM's are versatile machines that can be used for both learning distributions as a generative task as well as for performing classification or function approximation as a discriminative task. We show that minimizing KL-divergence works best for training BM for applications of function approximation. In our approach, we use Quantum annealers for sampling Boltzmann states. These states are used to approximate gradients in a stochastic gradient descent scheme. The approach is used to demonstrate logic circuits in the discriminative sense and a specialized two-phase distribution using generative BM.

•Feb 3, 2020•4 citations

Matrix Product Operator Restricted Boltzmann Machines

Cong Chen, Kim Batselier, Ching-Yun Ko +1 more

A restricted Boltzmann machine (RBM) learns a probability distribution over its input samples and has numerous uses like dimensionality reduction, classification and generative modeling. Conventional RBMs accept vectorized data that dismiss potentially important structural information in the original tensor (multi-way) input. Matrix-variate and tensor-variate RBMs, named MvRBM and TvRBM, have been proposed but are all restrictive by model construction and have weak model expression power. This work presents the matrix product operator RBM (MPORBM) that utilizes a tensor network generalization of Mv/TvRBM, preserves input formats in both the visible and hidden layers, and results in higher expressive power. A novel training algorithm integrating contrastive divergence and an alternating optimization procedure is also developed. Numerical experiments compare the MPORBM with the traditional RBM and MvRBM for data classification and image completion and denoising tasks. The expressive power of the MPORBM as a function of the MPO-rank is also investigated.

arXiv:1811.04608•International Joint Conference on Neural Networks (IJCNN)•Jul 1, 2019•2 citations
Restricted Boltzmann machines (RBMs) and deep Boltzmann machines (DBMs) are important models in machine learning, and recently found numerous applications in quantum many-body physics. We show that there are fundamental connections between them and tensor networks. In particular, we demonstrate that any RBM and DBM can be exactly represented as a two-dimensional tensor network. This representation gives characterizations of the expressive power of RBMs and DBMs using entanglement structures of the tensor networks, and also provides an efficient tensor network contraction algorithm for the computing partition function of RBMs and DBMs. Using numerical experiments, we show that the proposed algorithm is more accurate than the state-of-the-art machine learning methods in estimating the partition function of RBMs and DBMs, and have potential applications in training DBMs for general machine learning tasks.
Modern deep learning has enabled unprecedented achievements in various domains. Nonetheless, employment of machine learning for wave function representations is focused on more traditional architectures such as restricted Boltzmann machines (RBMs) and fully connected neural networks. In this Letter, we establish that contemporary deep learning architectures, in the form of deep convolutional and recurrent networks, can efficiently represent highly entangled quantum systems. By constructing tensor network equivalents of these architectures, we identify an inherent reuse of information in the network operation as a key trait which distinguishes them from standard tensor network-based representations, and which enhances their entanglement capacity. Our results show that such architectures can support volume-law entanglement scaling, polynomially more efficiently than presently employed RBMs. Thus, beyond a quantification of the entanglement capacity of leading deep learning architectures, our analysis formally motivates a shift of trending neural-network-based wave function representations closer to the state-of-the-art in machine learning.
Abstract A restricted Boltzmann machine (RBM) is a two-layer neural-network machine-learning model comprising a visible layer that represents observed data and a hidden layer that captures correlations among the visible units. Rooted in ideas from statistical mechanics, RBMs provide a probabilistic framework for modeling complex, high-dimensional distributions through an energy function and a corresponding Boltzmann distribution. This review offers an accessible introduction to RBMs from a physics-oriented perspective, focusing on their core concepts and training principles. We survey key theoretical developments that interpret RBMs as many-body systems, highlighting their equilibrium/non-equilibrium thermodynamics, learning dynamics, mean-field properties, phase diagrams, and free energy calculation. We further discuss recent applications of RBMs to a range of physics problems, including quantum-mechanical wavefunction representation, statistical-mechanical Boltzmann distribution modeling, and biological protein sequence analysis.
Lately, there has been an extensive state-of-the-art research in Machine Learning methods, due to their important features such as universality in approximations and dimensional reduction. In this way, the present work aims at exploiting these properties of Machine Learning on physical many-body problems, in which a restricted Boltzmann Machine (RBM) is trained with quantum Monte Carlo data to best represent the ground state of an N spin system under the transverse field Ising model Hamiltonian in one dimension. During the path coursed along the development of this project, we first review classical well-implemented methods such as the single-layer perceptron, multi-layer perceptron for the XOR problem with back-propagation algorithm, discrete Hopfield network for pattern reconstruction, continuous Hopfield network with Simulated Annealing for solving the traveling salesman problem with 10 cities and the Boltzmann machine for feature extraction. Then, we review a few aspects of what is called a Quantum Boltzmann Machine and the basics of the transverse field Ising model. Finally, we train an RBM consisting of N visible neurons, hidden density neurons alpha=M/N=2 and 1-step contrastive divergence (CD_1), by minimizing the Kullbach-Liebler divergence of the RBM and the training data set coming from the algorithmic variant of quantum Monte Carlo, known as discrete Path-Integral Monte Carlo. For this last step, we use Trotter-Suzuki decomposition with m Trotter slices and the Wolff-s cluster algorithm. After training the RBM, we calculate magnetic observables based on the RBM-s wave-function such as longitudinal and transverse magnetization, for different values of transverse field and we do an estimation of the critical field at which the transverse Ising model system suffers a quantum phase transition from ferromagnet to paramagnet
We present a Hybrid-Quantum-classical method for learning Boltzmann machines (BM) for generative and discriminative tasks. Boltzmann machines are undirected graphs that form the building block of many learning architectures such as Restricted Boltzmann machines (RBM's) and Deep Boltzmann machines (DBM's). They have a network of visible and hidden nodes where the former are used as the reading sites while the latter are used to manipulate the probability of the visible states. BM's are versatile machines that can be used for both learning distributions as a generative task as well as for performing classification or function approximation as a discriminative task. We show that minimizing KL-divergence works best for training BM for applications of function approximation. In our approach, we use Quantum annealers for sampling Boltzmann states. These states are used to approximate gradients in a stochastic gradient descent scheme. The approach is used to demonstrate logic circuits in the discriminative sense and a specialized two-phase distribution using generative BM.
A restricted Boltzmann machine (RBM) learns a probability distribution over its input samples and has numerous uses like dimensionality reduction, classification and generative modeling. Conventional RBMs accept vectorized data that dismiss potentially important structural information in the original tensor (multi-way) input. Matrix-variate and tensor-variate RBMs, named MvRBM and TvRBM, have been proposed but are all restrictive by model construction and have weak model expression power. This work presents the matrix product operator RBM (MPORBM) that utilizes a tensor network generalization of Mv/TvRBM, preserves input formats in both the visible and hidden layers, and results in higher expressive power. A novel training algorithm integrating contrastive divergence and an alternating optimization procedure is also developed. Numerical experiments compare the MPORBM with the traditional RBM and MvRBM for data classification and image completion and denoising tasks. The expressive power of the MPORBM as a function of the MPO-rank is also investigated.