Transforming Generalized Ising Model into Boltzmann Machine

Abstract

We find an exact mapping from the generalized Ising models with many-spin interactions to equivalent Boltzmann machines, i.e., the models with only two-spin interactions between physical and auxiliary binary variables accompanied by local external fields. More precisely, the appropriate combination of the algebraic transformations, namely the star-triangle and decoration-iteration transformations, allows one to express the model in terms of fewer-spin interactions at the expense of the degrees of freedom. Furthermore, the benefit of the mapping in Monte Carlo simulations is discussed. In particular, we demonstrate that the application of the method in conjunction with the Swendsen-Wang algorithm drastically reduces the critical slowing down in a model with two- and three-spin interactions on the Kagomé lattice.

12 pages, 8 figures

References in corpus (40)

• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• Theoretical perspective on the glass transition and amorphous materials
• Machine learning phases of matter
• Learning phase transitions by confusion
• Discovering Phase Transitions with Unsupervised Learning
• Unsupervised learning of phase transitions: from principal component analysis to variational autoencoders
• Quantum Entanglement in Neural Network States
• Worm algorithms for classical statistical models
• Efficient Representation of Quantum Many-body States with Deep Neural Networks
• Restricted-Boltzmann-Machine Learning for Solving Strongly Correlated Quantum Systems
• Machine Learning Topological States
• The Loop Algorithm
• Learning Thermodynamics with Boltzmann Machines
• Machine Learning Phases of Strongly Correlated Fermions
• Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design
• Accelerate Monte Carlo Simulations with Restricted Boltzmann Machines
• Equivalence of restricted Boltzmann machines and tensor network states
• Neural-Network Quantum States, String-Bond States, and Chiral Topological States
• Lattice Glass Models
• Constructing exact representations of quantum many-body systems with deep neural networks
• Non-equilibrium relaxation and interface energy of the Ising model
• Deep Learning the Quantum Phase Transitions in Random Two-Dimensional Electron Systems
• Identifying Quantum Phase Transitions with Adversarial Neural Networks
• Non-perturbative k-body to two-body commuting conversion Hamiltonians and embedding problem instances into Ising spins
• Generalized Transformation for Decorated Spin Models
• Generalized algebraic transformations and exactly solvable classical-quantum models
• Phase diagram of disordered higher-order topological insulator: A machine learning study
• Learning Disordered Topological Phases by Statistical Recovery of Symmetry
• Deep Learning the Quantum Phase Transitions in Random Electron Systems: Applications to Three Dimensions
• Exactly solvable Ising--Heisenberg chain with triangular XXZ-Heisenberg plaquettes
• Exact solution of the geometrically frustrated spin-1/2 Ising-Heisenberg model on the triangulated Kagome (triangles-in-triangles) lattice
• Cluster representations and the Wolff algorithm in arbitrary external fields
• Can Boltzmann Machines Discover Cluster Updates ?
• Efficient Machine Learning Representations of Surface Code with Boundaries, Defects, Domain Walls and Twists
• Exact results for a generalized spin-1/2 Ising-Heisenberg diamond chain with the second-neighbor interaction between nodal spins
• Exact evidence for the spontaneous antiferromagnetic long-range order in the two-dimensional hybrid model of localized Ising spins and itinerant electrons
• Neural Network Operations and Susuki-Trotter evolution of Neural Network States
• Soluble kagome Ising model in a magnetic field
• Exactly Solvable Lattice Models with Crossing Symmetry
• Strong- and weak-universal critical behaviour of a mixed-spin Ising model with triplet interactions on the Union Jack (centered square) lattice

Cited by in corpus (11)

• Constructing neural stationary states for open quantum many-body systems
• Large-scale Ising Emulation with Four-Body Interaction and All-to-All Connection
• Drawing Phase Diagrams of Random Quantum Systems by Deep Learning the Wave Functions
• Purifying Deep Boltzmann Machines for Thermal Quantum States
• Exact representations of many body interactions with RBM neural networks
• Barriers and Dynamical Paths in Alternating Gibbs Sampling of Restricted Boltzmann Machines
• Disentangling representations in Restricted Boltzmann Machines without adversaries
• Convolutional restricted Boltzmann machine aided Monte Carlo: An application to Ising and Kitaev models
• Deep reinforcement learning for preparation of thermal and prethermal quantum states
• Variational Autoencoder Analysis of Ising Model Statistical Distributions and Phase Transitions
• Conservation Laws and Spin System Modeling through Principal Component Analysis