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arXiv:2603.13304

A Block Belief-Propagation Algorithm for the Contraction of Tensor-Networks

Nir Gutman

quant-phSubmitted Mar 2, 2026• Announced Mar 4, 2026• 0 citations
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Abstract

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.

Comments
Master's thesis submitted to the Technion -- Israel Institute of Technology, supervised by Anna Keselman and Itai Arad. 56 pages, 44 figures

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[eng] Quantum many-body dynamics is a vast field of physics at the intersection of statistical mechanics, condensed matter, quantum information, and computational physics. Understanding the time evolution of observable correlations—such as those involved in material responses—is both a milestone for technological advancement and a corner-stone for generating fundamental knowledge. At the intersection of emergent collective behavior in many-body systems, quantum physics, and information theory lies the classical simulation of out-of-equilibrium phenomena. Without numerical tools, progress in disciplines that require heavy computations with high-dimensional vectors and matrices would be limited to purely analytical predictions and constrained by the available expertise and resources needed to build benchmarking experiments. The goal of this thesis is to advance the classical simulation of quantum many-body systems using Tensor Networks. This technique has proven fruitful for predicting equilibrium ground states and thermal properties of one-dimensional systems, yet it still faces challenges when applied to out-of-equilibrium scenarios. In brief, the dynamics of experimentally accessible expectation values can be represented as a 2D Tensor Network, which must be contracted using iterative methods such as Time-Evolving Block Decimation. However, the memory and computational time required for these calculations—which reflect the methods’ complexity—are known to grow exponentially with the simulated physical time in generic systems. This limitation is often referred to in the literature as the entanglement barrier. This manuscript tackles the problem through the three inequivalent ways in which the 2D Tensor Network can be contracted. In Chapter 3, we study how Schrödinger-picture contraction from the state edge of the network can be modified to improve the convergence of iterative contraction and reduce truncation errors in small systems. To this end, we explore physical decoherence in optimized bases and artificial generalized decoherence, finding that both alternatives show promise for new protocols that overcome the entanglement barrier. In Chapter 4, we explore Heisenberg-picture contraction from the observable edge, combining the concept of generalized decoherence with Pauli Weight truncation. By thoroughly characterizing the Pauli weight requirements to accurately describe the time evolution of local expectation values from initial product states, we conclude that truncating off-diagonal Pauli Weights is an effective scheme for simulating long-time dynamics—and we successfully implement it. We conclude with a spatial characterization of Pauli Weight spreading, bridging Tensor Network simulations and the field of operator hydrodynamics. Finally, in Chapter 5, we focus on the transverse contraction of the 2D Tensor Net-work using the light-cone algorithm, analyzing its variants in terms of truncation strategies and convergence across different models. This leads to a systematic characterization of the method based on the integrability of the underlying Hamiltonians and single-spin noise models.

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•Jan 1, 2026•0 citations
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.
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 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.
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 2×22\times 22×2 hexagonal lattice, finding that it produces states with a fidelity between 0.90.90.9 and 0.990.990.99 to the true ground state, and energy estimates with a relative error between 10−210^{-2}10−2 and 10−310^{-3}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.
[eng] Quantum many-body dynamics is a vast field of physics at the intersection of statistical mechanics, condensed matter, quantum information, and computational physics. Understanding the time evolution of observable correlations—such as those involved in material responses—is both a milestone for technological advancement and a corner-stone for generating fundamental knowledge. At the intersection of emergent collective behavior in many-body systems, quantum physics, and information theory lies the classical simulation of out-of-equilibrium phenomena. Without numerical tools, progress in disciplines that require heavy computations with high-dimensional vectors and matrices would be limited to purely analytical predictions and constrained by the available expertise and resources needed to build benchmarking experiments. The goal of this thesis is to advance the classical simulation of quantum many-body systems using Tensor Networks. This technique has proven fruitful for predicting equilibrium ground states and thermal properties of one-dimensional systems, yet it still faces challenges when applied to out-of-equilibrium scenarios. In brief, the dynamics of experimentally accessible expectation values can be represented as a 2D Tensor Network, which must be contracted using iterative methods such as Time-Evolving Block Decimation. However, the memory and computational time required for these calculations—which reflect the methods’ complexity—are known to grow exponentially with the simulated physical time in generic systems. This limitation is often referred to in the literature as the entanglement barrier. This manuscript tackles the problem through the three inequivalent ways in which the 2D Tensor Network can be contracted. In Chapter 3, we study how Schrödinger-picture contraction from the state edge of the network can be modified to improve the convergence of iterative contraction and reduce truncation errors in small systems. To this end, we explore physical decoherence in optimized bases and artificial generalized decoherence, finding that both alternatives show promise for new protocols that overcome the entanglement barrier. In Chapter 4, we explore Heisenberg-picture contraction from the observable edge, combining the concept of generalized decoherence with Pauli Weight truncation. By thoroughly characterizing the Pauli weight requirements to accurately describe the time evolution of local expectation values from initial product states, we conclude that truncating off-diagonal Pauli Weights is an effective scheme for simulating long-time dynamics—and we successfully implement it. We conclude with a spatial characterization of Pauli Weight spreading, bridging Tensor Network simulations and the field of operator hydrodynamics. Finally, in Chapter 5, we focus on the transverse contraction of the 2D Tensor Net-work using the light-cone algorithm, analyzing its variants in terms of truncation strategies and convergence across different models. This leads to a systematic characterization of the method based on the integrability of the underlying Hamiltonians and single-spin noise models.
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.
Strongly correlated quantum systems are one of the most interesting and challenging phenomena in modern condensed matter physics. Unfortunately, the theoretical access even to their low energy properties is limited by the exponential growth of the Hilbert space in a many-body setting. To mitigate this computational barrier, tensor network methods are one of the most competitive approaches to classically simulate complex quantum systems. In this thesis we will discuss recent developments in the field of tensor networks to efficiently and accurately simulate two-dimensional quantum many-body states in the thermodynamic limit. Hereby, we introduce the variational update globally optimizing the wave-function, encoded as tensor network in the form of an infinite projected entangled pair state (iPEPS), in order to obtain ground states of many-body Hamiltonian models. We focus in the discussion on the well-established method to approximately contract this two-dimensional structure, the corner transfer matrix renormalization group (CTMRG), and discuss its native application on square lattice and triangular lattice based models. Furthermore, we report the development of a refined numerical algorithm, the split-CTMRG, to significantly reduce its computational complexity, pushing the limits of feasible simulations within modern hardware capabilities. Finally, the method advancements are applied to a currently widely studied model of frustrated magnetism, the Heisenberg maple-leaf antiferromagnet. We are able to find a large phase of vanishing magnetization in contradiction to previous works and, to the best of our knowledge, the lowest reported variational ground state energy at the isotropic point of the model. As an outlook, we describe a potential adaption of the previously developed methods to variationally simulate two-dimensional many-body thermal states.