Stochastic Loop Corrections to Belief Propagation for Tensor Network Contraction

TL;DR

A hybrid method is introduced that achieves exact results by stochastically sampling loop corrections to BP and is showcased by applying it to the two-dimensional ferromagnetic Ising model.

Abstract

Tensor network contraction is a fundamental computational challenge underlying quantum many-body physics, statistical mechanics, and machine learning. Belief propagation (BP) provides an efficient approximate solution, but introduces systematic errors on graphs with loops. Here, we introduce a hybrid method that achieves exact results by stochastically sampling loop corrections to BP and showcase our method by applying it to the two-dimensional ferromagnetic Ising model. For any pairwise Markov random field with symmetric edge potentials, our approach exploits an exact factorization of the partition function into the BP contribution and a loop correction factor summing over all valid loop configurations, weighted by edge weights derived directly from the potentials. We sample this sum using Markov chain Monte Carlo with moves that preserve the loop constraint, combined with umbrella sampling to ensure efficient exploration across all correlation strengths. Our stochastic approach provides unbiased estimates with controllable statistical error in any parameter regime.

Figures (5)

• Figure 1: MCMC moves via symmetric difference ($\oplus$) on loop configurations. Blue solid lines show the current configuration $G$ and orange/green dashed lines show the cycle to flip. Shaded regions highlight the active loops. (a) Merging loops. Flipping an adjacent plaquette merges it with the existing loop by removing the shared edge. (b) Splitting a loop. Flipping a plaquette inside a larger loop splits off that portion. (c) Winding cycle move on a torus with periodic boundary conditions. XOR with the horizontal winding cycle $W_h$ creates a winding loop.
• Figure 2: Comparison of exact enumeration (black solid), BP (blue dashed), and BPLMC (red line) for the ($3 \times 3$) ferromagnetic Ising model with periodic boundary conditions. (a) Free energy per site $F/N$. (b) Energy per site $E/N$. (c) Specific heat per site $C/N$. All quantities are plotted versus inverse temperature $\beta$. BPLMC matches the exact solution across all temperatures, while BP shows systematic errors that grow at low temperature (high $\beta$).
• Figure 3: Free energy per site for the ($10 \times 10$) ferromagnetic Ising model with periodic boundary conditions. (a) Full inverse temperature range showing BP (blue dashed), BPLMC (red), and Onsager exact solution (black solid). (b) Magnified view around the critical temperature $\beta_c \approx 0.44$ (orange dotted line). BPLMC accurately tracks the exact solution across all temperatures, while BP shows systematic deviations that persist even at low temperature (high $\beta$). The continued deviation of BP at low temperature arises because we initialize messages uniformly, which corresponds to the high-temperature fixed point. For $\beta > \beta_{\mathrm{BP}} = \ln(2)/2 \approx 0.347$, BP admits multiple fixed points midha2025beyond, and the correct low-temperature fixed point requires symmetry-broken initialization to capture the ferromagnetic ordering.
• Figure 4: Temperature dependence of loop configuration statistics for the $10 \times 10$ ferromagnetic Ising model. (a) Mean edge count $\langle |G| \rangle$ increases with inverse temperature $\beta$, reflecting the growing statistical weight of configurations with more edges as temperature decreases. (b) Winding fraction, the percentage of loop configurations containing topologically non-trivial cycles that wrap around the torus. The winding fraction increases sharply near the critical temperature $\beta_c \approx 0.44$ (vertical dotted line), indicating a transition from locally fluctuating plaquettes to system-spanning correlations. (c) Logarithm of the loop partition function $Z_{\mathrm{loop}} = Z/Z_{\mathrm{BP}}$, which quantifies the correction factor from BP to the exact partition function. Near $\beta_c$, $\log_{10} Z_{\mathrm{loop}}$ grows rapidly, indicating that loop corrections become increasingly important as the system approaches criticality.
• Figure 5: Distribution of loop configurations for the ($10 \times 10$) ferromagnetic Ising model at three representative temperatures. Distribution of total edge count $|G|$ per configuration at (a) high temperature $\beta = 0.3125$, (b) near critical temperature $\beta = 0.4545 \approx \beta_c$, and (c) low temperature $\beta = 0.7143$. The grey dashed line indicates the reweighted mean $\langle |G| \rangle$. Distribution of individual loop sizes $\ell$ (number of edges per connected component) at the same temperatures. At high temperature (d), small contractible loops of size 4 (plaquettes) dominate, with a secondary peak at larger sizes corresponding to system-spanning winding loops. As temperature decreases (e, f), the winding loop peak shifts to larger sizes and becomes more prominent, reflecting the increasing importance of winding configurations in the ordered phase.