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 (2D) tensor networks and approximating the ground state of 2D systems. The advantages of our method are threefold: (1) the same algorithm works for both finite and infinite systems; (2) it allows natural and efficient parallelization; and (3) given its flexibility, it would allow us 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.
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 (2D) tensor networks and approximating the ground state of 2D systems. The advantages of our method are threefold: (1) the same algorithm works for both finite and infinite systems; (2) it allows natural and efficient parallelization; and (3) given its flexibility, it would allow us 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.
Simulating many-body quantum systems on a classical computer is difficult due to the large number of degrees of freedom, causing the computational complexity to grow exponentially with system size. Tensor Networks (TN) is a framework that breaks down large tensors into a network of smaller tensors, enabling efficient simulation of certain many-body quantum systems. To calculate expectation values of local observables or simulate nearest-neighbor interactions, a contraction of the entire network is needed. This is a known hard problem, which cannot be done exactly for systems with spatial dimension D>1 and is the major bottleneck in all tensor-network based algorithms. Various approximate-contraction algorithms have been suggested, all with their strengths and weaknesses. Nevertheless, contracting a 2D TN remains a major numerical challenge, limiting the use of TN techniques for many interesting systems.
Recently, a close connection between TN and Probabilistic Graphical Models (PGM) has been shown. In the PGM framework, marginals of complicated probability distributions can be approximated using iterative message passing algorithms such as Belief Propagation (BP). The BP algorithm can be adapted to the TN framework as an efficient contraction algorithm. While BP is extremely efficient and easy to parallelize, it often yields inaccurate results for highly correlated quantum states or frustrated systems. To overcome this, we suggest the BlockBP algorithm, which coarse-grains the system into blocks and performs BP between them.
This thesis focuses on: (i) development and implementation of the BlockBP algorithm for infinite lattices; (ii) using this algorithm to study the anti-ferromagnetic Heisenberg model on the Kagome lattice in the thermodynamic limit - a frustrated 2D model that is difficult to simulate using existing numerical methods.
Tensor networks have attracted much attention as a powerful tool for modeling quantum many-body systems. Their contraction is a significant challenge, however, especially in highly connected networks, as memory requirements become prohibitive and the optimal contraction order is increasingly hard to find. The belief propagation (BP) algorithm has emerged as an alternative to exact contraction. Being formulated in a graph-agnostic way, it offers great flexibility, but its accuracy suffers in the presence of loops. In this work, we combine BP with the DMRG algorithm to solve ground-state problems, thereby extending DMRG to higher dimensions and arbitrary lattices. We demonstrate the viability of BP-DMRG on the transverse-field Ising model on a 2×2 hexagonal lattice, finding that it produces states with a fidelity between 0.9 and 0.99 to the true ground state, and energy estimates with a relative error between 10−2 and 10−3. Additionally, BP-DMRG can find ground states on randomly generated lattices, with fidelity improving as the transverse field increases. We conclude with a discussion of the limitations we encounter when using belief propagation, highlighting that the TFI tensor network operators lead to larger errors during BP iterations in BP-DMRG.
Recent years have seen a growing interest in the use of belief propagation - an algorithm originally introduced for performing statistical inference on graphical models - for approximate, but highly efficient, tensor network contraction. Here, we detail how to apply generalized belief propagation (GBP) - where messages are passed within a hierarchy of overlapping regions of the tensor network - to approximately contract tensor networks and obtain accurate results. The original belief propagation algorithm is a corner case of this approach, corresponding to a particularly simple choice of regions of the tensor network. We implement the GBP algorithm for a number of different region choices on a range of two- and three-dimensional, infinite and finite tensor networks, solving the corresponding fixed point equations both numerically and, in certain tractable cases, analytically. Our examples include calculating the partition function of the fully frustrated Ising model, computing the ground state degeneracy of three-dimensional ice models, measuring observables on the deformed AKLT quantum state and evaluating the norm of randomly generated tensor network states.
Belief propagation (BP) provides a scalable heuristic for contracting tensor networks on loopy graphs, but its success in quantum many-body settings has largely rested on empirical evidence. Developing upon a recently introduced cluster-expansion framework for tensor networks, we rigorously study the applicability of BP to many-body quantum systems. For a state represented as a PEPS satisfying a ``loop-decay" condition, we prove that BP supplemented by cluster corrections approximates local observables with exponentially small relative error, and we give explicit formulas expressing local expectation values as BP predictions dressed by connected clusters intersecting the observable region. This representation establishes a direct link between cluster corrections and physical correlation functions. As a result, we show that ``loop-decay" \emph{necessarily implies} exponential decay of connected correlations, yielding sharp, rigorous criteria for when BP can and cannot succeed, and ruling out its validity at critical points. Numerical simulations of the two- and three-dimensional transverse field Ising model at zero and finite temperature confirm our analytical predictions, demonstrating quantitative accuracy deep in gapped phases and systematic failure near criticality.
We introduce "matrix product belief propagation" (MP-BP) as a controlled method for the approximate contraction of two-dimensional tensor networks and graphical models, which generically leads to a quadratic reduction of errors relative to existing methods: at similar computational effort, the number of accurate digits is asymptotically doubled. For infinite systems, MP-BP can be understood as a practical implementation of Baxter's corner-transfer-matrix method; for reflection-symmetric networks, it further coincides with the boundary matrix-product-state and corner-transfer-matrix renormalization group algorithms as conventionally implemented. In the limit of unit matrix-product rank, MP-BP is equivalent to belief propagation, and more generally, can be understood as a generalization of BP in which the "messages" live on surfaces and are assumed to have a matrix-product factorization. We further extend the quadratic improvement to the computation of expectation values, and numerically demonstrate the qualitatively improved convergence for a variety of classical and quantum 2D tensor network problems.
Tensor Networks have recently received much attention in the physics community, thanks to their capabilities in modeling quantum-mechanical operations.Their contraction often poses a significant challenge due to di!culties in contraction order search and memory constraints.Belief Propagation o"ers a way around these problems: it represents a memory e!cient contraction scheme, that is completely independent of the topology of the Tensor Network.This thesis aims to investigate the applicability of Belief Propagation to one of Computational Physics' most prominent problems: the ground state problem.Given a Hamiltonian in an arbitrary (possibly high-dimensional) lattice, it is not at all obvious how one can e!ciently find its ground state.Since the Belief Propagation algorithm is geometry-independent, I combine it with the Density Matrix Renormalization Group algorithm, thereby introducing BP-DMRG.I show that BP-DMRG is able to recover ground states of the transverse-field Ising model and the Heisenberg model on random lattices.Moreover, BP-DMRG o"ers insight into the e!cacy of Belief Propagation on physical states; I show that ground states of the TFI are not amenable to accurate contraction by Belief Propagation.