Block belief propagation algorithm for two-dimensional tensor networks

Abstract

Belief propagation is a well-studied algorithm for approximating local marginals of multivariate probability distribution over complex networks, while tensor network states are powerful tools for quantum and classical many-body problems. Building on a recent connection between the belief propagation algorithm and the problem of tensor network contraction, we propose a block belief propagation algorithm for contracting two-dimensional tensor networks and approximating the ground state of $2D$ systems. The advantages of our method are three-fold: 1) the same algorithm works for both finite and infinite systems; 2) it allows natural and efficient parallelization; 3) given its flexibility it would allow to deal with different unit cells. As applications, we use our algorithm to study the $2D$ Heisenberg and transverse Ising models, and show that the accuracy of the method is on par with state-of-the-art results.

arXiv comment12 pages, 13 figures

References in corpus (44)

• The density-matrix renormalization group in the age of matrix product states
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Tensor renormalization group approach to 2D classical lattice models
• Stripe order in the underdoped region of the two-dimensional Hubbard model
• Tensor networks for complex quantum systems
• Finite-Size Scaling of the Ground State Parameters of the Two-Dimensional Heisenberg Model
• Accurate determination of tensor network state of quantum lattice models in two dimensions
• Competing states in the t-J model: uniform d-wave state versus stripe state
• Corner Transfer Matrix Renormalization Group Method
• Coarse-graining renormalization by higher-order singular value decomposition
• Simulation of two dimensional quantum systems on an infinite lattice revisited: corner transfer matrix for tensor contraction
• Gapless spin-liquid ground state in the $S=1/2$ kagome antiferromagnet
• A message passing approach for general epidemic models
• Tensor Network Renormalization
• The computational complexity of PEPS
• Percolation on sparse networks
• Second Renormalization of Tensor-Network States
• Differentiable Programming Tensor Networks
• Variational optimization with infinite projected entangled-pair states
• The iPEPS algorithm, improved: fast full update and gauge fixing
• Algorithms for finite Projected Entangled Pair States
• Tensor network study of the Shastry-Sutherland model in zero magnetic field
• Renormalization of tensor-network states
• Variational quantum Monte Carlo simulations with tensor-network states
• Loop series for discrete statistical models on graphs
• Crystals of bound states in the magnetization plateaus of the Shastry-Sutherland model
• Cluster Variation Method in Statistical Physics and Probabilistic Graphical Models
• Unifying Projected Entangled Pair States contractions
• Message passing on networks with loops
• Monte Carlo simulation with Tensor Network States
• How to Compute Loop Corrections to Bethe Approximation
• Belief-propagation algorithm and the Ising model on networks with arbitrary distributions of motifs
• Belief propagation for networks with loops
• Region graph partition function expansion and approximate free energy landscapes: Theory and some numerical results
• Tensor Networks contraction and the Belief Propagation algorithm
• Magnetization of the spin-1/2 Heisenberg antiferromagnet on the triangular lattice
• Replica Cluster Variational Method: the Replica Symmetric solution for the 2D random bond Ising model
• Finite Projected Entangled Pair States for the Hubbard model
• Simplifying Generalized Belief Propagation on Redundant Region Graphs
• Loop-corrected belief propagation for lattice spin models

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• Dynamics of disordered quantum systems with two- and three-dimensional tensor networks
• Loop Series Expansions for Tensor Networks
• Nonperturbative decay of bipartite discrete time crystals
• Simulating dynamics of the two-dimensional transverse-field Ising model: a comparative study of large-scale classical numerics
• Nonequilibrium steady-state dynamics of Markov processes on graphs
• Exploring the performance of superposition of product states: from 1D to 3D quantum spin systems