Machine learning is becoming widely used in analyzing the thermodynamics of many-body condensed matter systems. Restricted Boltzmann Machine (RBM) aided Monte Carlo simulations have sparked interest recently, as they manage to speed up classical Monte Carlo simulations. Here we employ the Convolutional Restricted Boltzmann Machine (CRBM) method and show that its use helps to reduce the number of parameters to be learned drastically by taking advantage of translation invariance. Furthermore, we show that it is possible to train the CRBM at smaller lattice sizes, and apply it to larger lattice sizes. To demonstrate the efficiency of CRBM we apply it to the paradigmatic Ising and Kitaev models in two-dimensions.
Machine learning is becoming widely used in analyzing the thermodynamics of many-body condensed matter systems. Restricted Boltzmann Machine (RBM) aided Monte Carlo simulations have sparked interest recently, as they manage to speed up classical Monte Carlo simulations. Here we employ the Convolutional Restricted Boltzmann Machine (CRBM) method and show that its use helps to reduce the number of parameters to be learned drastically by taking advantage of translation invariance. Furthermore, we show that it is possible to train the CRBM at smaller lattice sizes, and apply it to larger lattice sizes. To demonstrate the efficiency of CRBM we apply it to the paradigmatic Ising and Kitaev models in two-dimensions.
Machine learning is becoming widely used in analyzing the thermodynamics of many-body condensed matter systems. Restricted Boltzmann machine (RBM) aided Monte Carlo simulations have sparked interest recently, as they manage to speed up classical Monte Carlo simulations. Here we employ the convolutional restricted Boltzmann machine (CRBM) method and show that its use helps to reduce the number of parameters to be learned drastically by taking advantage of translation invariance. Furthermore, we show that it is possible to train the CRBM at smaller lattice sizes, and apply it to larger lattice sizes. To demonstrate the efficiency of CRBM we apply it to the paradigmatic Ising and Kitaev models in two dimensions.
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
Conventionally, the study of phases in statistical mechan- ics is performed with the help of random sampling tools. Among the most powerful are Monte Carlo simulations consisting of a stochastic importance sampling over state space and evaluation of estimators for physical quantities. The ability of modern machine learning techniques to classify, identify, or in- terpret massive data sets provides a complementary paradigm to the above approach to analyze the exponentially large number of states in statistical physics. In this report, it is demonstrated by application on Ising-type models that deep learning has potential wide applications in solving many-body statis- tical physics problems. In application of supervised learning, we showed that the feed-forward neural network can identify phases and phase transitions in the ferromagnetic Ising model and the convolutional neural network (CNN) is extremely powerful in classifying T = 0 and T = ∞ phases in the Ising gauge model; In application of unsupervised learning, we illustrated that a deep auto-encoder constructed by stacked restricted Boltzmann machines (RBM)\nis closely related to the renormalization group (RG) method well understood in modern physics and our reconstruction of Ising spin configurations in the ferromagnetic Ising model is similar to the hand-written digits reconstruction.
Geometric frustration gives rise to emergent quantum phenomena and exotic phases of matter. While Monte Carlo methods are traditionally used to simulate such systems, their sampling efficiency is limited by the complexity of interactions and ground-state properties. Restricted Boltzmann Machines (RBMs), a class of probabilistic neural networks, offer improved sampling by incorporating machine learning techniques. However, fully-connected bipartite RBMs are inefficient for representing physical lattices with sparse interactions. To address this, we implement Convolutional Restricted Boltzmann Machines (CRBMs) that leverage translational symmetry inherent to lattices. Using the classical Shastry-Sutherland (SS) Ising lattice, we demonstrate (i) CRBM formulation that captures SS interactions, and (ii) digital hardware accelerator to enhance sampling performance. We simulate lattices with up to 324 spins, recovering all known phases of the SS Ising model, including the long range ordered fractional plateau. Our hardware characterizes spin behavior at critical points and within spin liquid phases. This implementation achieves a speedup of 3 to 5 orders of magnitude (33 ns to 120 ms) over GPU-based implementations. Moreover, the time-to-solution is within two orders of magnitude of quantum annealers, while offering superior scalability, room-temperature operation and reprogrammability. This work paves a pathway for scalable digital hardware that embeds physical symmetries to enable large scale simulations of material systems.
Restricted Boltzmann machine (RBM) has been proposed as a powerful variational ansatz to represent the ground state of a given quantum many-body system. On the other hand, as a shallow neural network, it is found that the RBM is still hardly able to capture the characteristics of systems with large sizes or complicated interactions. In order to find a way out of the dilemma, here, we propose to adopt the Green’s function Monte Carlo (GFMC) method for which the RBM is used as a guiding wave function. To demonstrate the implementation and effectiveness of the proposal, we have applied the proposal to study the frustrated J 1–J 2 Heisenberg model on a square lattice, which is considered as a typical model with sign problem for quantum Monte Carlo simulations. The calculation results demonstrate that the GFMC method can significantly further reduce the relative error of the ground-state energy on the basis of the RBM variational results. This encourages to combine the GFMC method with other neural networks like convolutional neural networks for dealing with more models with sign problem in the future.
We train a set of restricted Boltzmann machines (RBMs) on one- and two-dimensional Ising spin configurations at various values of temperature, generated using Monte Carlo simulations. We validate the training procedure by monitoring several estimators, including measurements of the log likelihood, with the corresponding partition functions estimated using annealed importance sampling. The effects of various choices of hyperparameters on training the RBM are discussed in detail, with a generic prescription provided. Finally, we present a closed-form expression for extracting the values of couplings, for every n-point interaction between the visible nodes of an RBM, in a binary system such as the Ising model. We aim at using this study as the foundation for further investigations of less well-known systems.