Beyond Belief Propagation: Cluster-Corrected Tensor Network Contraction with Exponential Convergence

TL;DR

This work tackles the computational challenge of contracting tensor networks on general graphs by diagnosing the limitations of belief propagation (BP) and introducing a cluster expansion that converges for the logarithm of the partition function. By recasting the problem in the abstract polymer model and leveraging the Kotecký–Preiss criterion, the authors prove exponential convergence of the cluster expansion under exponential decay of loop contributions and provide a practical algorithm to compute corrections to BP to arbitrary order. Benchmarking on the 2D Ising model demonstrates that cluster-corrected BP yields significant improvements near criticality and remains robust to system size, outperforming conventional loop expansions which diverge thermodynamically. The framework enables systematic, polynomial-time improvements to BP for tensor networks with broad potential applications in classical/quantum error-correcting code decoding and quantum system simulations, offering a principled path beyond Belief Propagation.

Abstract

Tensor network contraction on arbitrary graphs is a fundamental computational challenge with applications ranging from quantum simulation to error correction. While belief propagation (BP) provides a powerful approximation algorithm for this task, its accuracy limitations are poorly understood and systematic improvements remain elusive. Here, we develop a rigorous theoretical framework for BP in tensor networks, leveraging insights from statistical mechanics to devise a \emph{cluster expansion} that systematically improves the BP approximation. We prove that the cluster expansion converges exponentially fast if an object called the \emph{loop contribution} decays sufficiently fast with the loop size, giving a rigorous error bound on BP. We also provide a simple and efficient algorithm to compute the cluster expansion to arbitrary order. We demonstrate the efficacy of our method on the two-dimensional Ising model, where we find that our method significantly improves upon BP and existing corrective algorithms such as loop series expansion. Our work opens the door to a systematic theory of BP for tensor networks and its applications in decoding classical and quantum error-correcting codes and simulating quantum systems.

Figures (13)

• Figure 1: Loop series expansion. The contraction of a five-vertex tensor network can be exactly represented as the sum of the BP vacuum and all the generalized loop excitations on the graph.
• Figure 1: Fixed points: (a) Fixed point 'energy' landscape for the message $\mu_\theta = (\cos\theta,\sin\theta)$ (b) Free energy density error $\delta f_w(\beta)$ for the fixed point from message passing dynamics (blue, solid) and the infinite-temperature fixed point ($\theta=0$, green, dashed) for $\beta \in \{0.3,0.4,0.5\}$ and system size $L=20$.
• Figure 2: (a) Disconnectedness and incompatibility. Example of a disconnected loop. (b) Example of two incompatible loops.
• Figure 2: Message correlation: Effect of a localized $z-$perturbation on the fixed point messages (plotted in log-scale) for the Ising model with system size $L=20$ across the phase transition, $\beta \in \{0.2,0.3,\beta_{\text{BP}}, 0.4,0.5\}$
• Figure 3: Pseudocode of Cluster Expansion. Computationally expensive steps are colored in red. When there are more than one tensor networks, we assume they are defined on the same graph so that they share the same clusters.

Theorems & Definitions (27)

• Definition 2.1: Generalized loops
• Lemma 2.1
• Definition 2.2: Loop correction
• Lemma 2.2: Loop series expansion
• Definition 3.1: Compatible loops
• Definition 3.2: Clusters
• Definition 3.3: Cluster correction
• Definition 3.4: Interaction graph
• Lemma 3.1: Connected clusters only
• Lemma 3.2