We propose a Restricted Boltzmann Machine (RBM) neural network using a quantum thermodynamics formalism and the maximization of entropy as the cost function for the optimization problem. We verify the possibility of using an entropy maximization approach for modeling physical systems by applying it to the Inverse Ising Problem, which describes a system of interacting spins. The temperature-dependent behaviors of physical quantities, such as magnetization and Gibbs energy, were calculated from the Ising Hamiltonian whose coefficients were obtained through entropy maximization and were found to be in good agreement to literature results. These results suggest that RBM neural networks using the principle of maximum entropy can be applied to modeling physical systems that can be described by discrete states, including fundamental quantum physics such as topological systems, and biological systems such as the correlated spiking of neurons.
We propose a Restricted Boltzmann Machine (RBM) neural network using a quantum thermodynamics formalism and the maximization of entropy as the cost function for the optimization problem. We verify the possibility of using an entropy maximization approach for modeling physical systems by applying it to the Inverse Ising Problem, which describes a system of interacting spins. The temperature-dependent behaviors of physical quantities, such as magnetization and Gibbs energy, were calculated from the Ising Hamiltonian whose coefficients were obtained through entropy maximization and were found to be in good agreement to literature results. These results suggest that RBM neural networks using the principle of maximum entropy can be applied to modeling physical systems that can be described by discrete states, including fundamental quantum physics such as topological systems, and biological systems such as the correlated spiking of neurons.
Recently, quantum-state representation using artificial neural networks has started to be recognized as a powerful tool. However, due to the black-box nature of machine learning, it is difficult to analyze what machine learns or why it is powerful. Here, by applying one of the simplest neural networks, the restricted Boltzmann machine (RBM), to the ground-state representation of the one-dimensional (1D) transverse-field Ising (TFI) model, we make an attempt to directly analyze the optimized network parameters. In the RBM optimization, a zero-temperature quantum state is mapped onto a finite-temperature classical state of the extended Ising spins that constitute the RBM. We find that the quantum phase transition from the ordered phase to the disordered phase in the 1D TFI model by increasing the transverse field is clearly reflected in the behaviors of the optimized RBM parameters and hence in the finite-temperature phase diagram of the classical RBM Ising system. The present finding of a correspondence between the neural-network parameters and quantum phases suggests that a careful investigation of the neural-network parameters may provide a new route to extracting nontrivial physical insights from the neural-network wave functions.
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.
The restricted Boltzmann machine (RBM) is a two-layer energy-based model that uses its hidden-visible connections to learn the underlying distribution of visible units, whose interactions are often complicated by high-order correlations. Previous studies on the Ising model of small system sizes have shown that RBMs are able to accurately learn the Boltzmann distribution and reconstruct thermal quantities at temperatures away from the critical point Tc. How the RBM encodes the Boltzmann distribution and captures the phase transition are, however, not well explained. In this work, we perform RBM learning of the 2d and 3d Ising model and carefully examine how the RBM extracts useful probabilistic and physical information from Ising configurations. We find several indicators derived from the weight matrix that could characterize the Ising phase transition. We verify that the hidden encoding of a visible state tends to have an equal number of positive and negative units, whose sequence is randomly assigned during training and can be inferred by analyzing the weight matrix. We also explore the physical meaning of the visible energy and loss function (pseudo-likelihood) of the RBM and show that they could be harnessed to predict the critical point or estimate physical quantities such as entropy.
The connection between the maximum entropy (MaxEnt) formalism and restricted Boltzmann machines (RBMs) is natural as both give rise to a Boltzmann-like distribution with constraints enforced by Lagrange multipliers, which correspond to RBM parameters. We integrate RBMs into quantum state tomography by using them as probabilistic models to approximate quantum states while satisfying MaxEnt constraints. Additionally, we employ polynomially efficient quantum sampling techniques to enhance RBM training, enabling scalable and high-fidelity quantum state reconstruction. This approach provides a computationally efficient framework for applying RBMs to MaxEnt-based quantum tomography. Furthermore, our method applies to the general and previously unaddressed case of reconstructing arbitrary mixed quantum states from incomplete and potentially noncommuting sets of expectations of observables while still ensuring maximal entropy.
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
A Boltzmann machine is a stochastic neural network that has been extensively used in the layers of deep architectures for modern machine learning applications. In this paper, we develop a Boltzmann machine that is capable of modeling thermodynamic observables for physical systems in thermal equilibrium. Through unsupervised learning, we train the Boltzmann machine on data sets constructed with spin configurations importance sampled from the partition function of an Ising Hamiltonian at different temperatures using Monte Carlo (MC) methods. The trained Boltzmann machine is then used to generate spin states, for which we compare thermodynamic observables to those computed by direct MC sampling. We demonstrate that the Boltzmann machine can faithfully reproduce the observables of the physical system. Further, we observe that the number of neurons required to obtain accurate results increases as the system is brought close to criticality.