Belief Propagation and Tensor Network Expansions for Many-Body Quantum Systems: Rigorous Results and Fundamental Limits

Abstract

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.

Figures (7)

• Figure 1: Tensor network belief propagation on the ground state of the 2D TFIM obtained via CTMRG-based optimization. (a) Longitudinal magnetization computed via CTMRG and BP with cumulant expansion up to cumulant order $m=10$ as a function of the transverse field $h_x$ around the critical region. The dashed line at $h_x \approx 3.06$ marks the point the CTMRG magnetization goes to zero. (b) Error in the $m^{\text{th}}$-order cumulant-corrected value (BP is $m=0$) as compared to the ground truth CTMRG magnetization. Inset: comparison of the error as a function of $m$ at $h_x=3.1$ for an expansion around a stable message-passing fixed point (solid curve) and unstable fixed point (dashed curve). The star at $m=0$ on the solid curve corresponds to the star in the main panel. (c) Relative error of the energy as a function of cumulant order $m$ deep in the ferromagnetic phase at $h_x=2.0$, near the critical point $h_x = 3.06$, and deep in the paramagnetic phase at $h_x=4.0$.
• Figure 2: $c$-decay of even-weight loops across a range of transverse fields in the 2D (blue) and 3D (red) TFIM at finite temperature. Data from increasing loop weights up to $|l|=12$ (2D) and $|l|=10$ (3D) (darker shading indicates higher weight) nearly coincide. Solid gray curve is the coefficient of classical loop decay. Dashed blue and red curves indicate the loose bound for $c_0 = \mathcal{O}(\log \Delta)$ for $\Delta=4$ (2D) and $\Delta = 6$ (3D) respectively.
• Figure 3: (a) Convergence of region-based cumulant cluster expansion for the observable $S_x$, as a function of the maximum region size $k$, for five different pairs of $(h_x,\beta)$ in the 2D TFIM. Blue, orange, and green curves are above the critical temperature but below the temperature at which $c=c_0$; purple and gold curves are below the critical temperatures at their respective transverse fields. (b) $\expval{S_x}$ as a function of $\beta$ at $h_x=0.25$, evaluated using CTMRG (black squares), vs. increasing orders of the region-based cumulant cluster expansion (ranging from $k=1$ in light orange, to $k=11$ in dark red). (c) Additive error in the cumulant cluster expansion for $\expval{S_x}$ at $h_x=0.25$, using CTMRG as the "ground truth." In (b) and (c), the dashed gray line marks the inverse temperature at which $\expval{S_x}_{\rm CTMRG}$ has a peak, which is slightly below $\beta_c \approx 0.446$ (dotted line) roughening.
• Figure S1: Decay of loops around a BP fixed point in the transverse field Ising model. (a) 3D TFIM at inverse temperature $\beta=0.2$, transverse field $h_x=1,2,3,4$. (b) 2D TFIM at inverse temperature $\beta=0.25$, with $h_x$ ranging from 0.25 to 3 in increments of 0.25. In (a) and (b), the gray lines have slope $-\log(\tanh(\beta))$, which is the classical scaling of loops with respect to the paramagnetic fixed point. (c) 2D TFIM at $\beta=0.65$, which is in the ordered phase for the transverse fields shown (up to $h_x=2.25$). Solid and dashed lines indicate loops of maximal and minimal weight, respectively. Dashed gray line indicates the classical scaling with slope $\log(e^{4\beta}-1)$.
• Figure S2: Same as \ref{['fig:finite-temp']}b-c in the main text, but for $|\langle S_z\rangle|$ instead of $\expval{S_x}$. (a) shows $|\langle S_z\rangle|$ as a function of $\beta$ at $h_x=0.25$, evaluated using CTMRG (black squares), vs. increasing orders of the region-based cumulant cluster expansion (ranging from $k=1$ in light orange, to $k=11$ in dark red). (b) shows the additive error in the cumulant cluster expansion, using CTMRG as the "ground truth." Dashed gray line marks the inverse temperature at which $\expval{S_x}_{\rm CTMRG}$ has a peak, while the dotted gray line is the inverse critical temperature $\beta_c \approx 0.446$.

Theorems & Definitions (46)

• Proposition 2.1: Loop Expansion (Informal) Evenbly2024midha2025beyond
• Proposition 2.2: Cluster Expansion (Informal) midha2025beyond
• Definition 2.1: $c-$decay of loops
• Theorem 2.1: Convergence of cluster expansion midha2025beyond (Informal)
• Proposition 2.3: Cluster-cumulant expansion (Informal)
• Proposition 3.1: Local observable expansion, ratio version (Informal)
• Proposition 3.2: Local observable expansion, derivative version (Informal)
• Theorem 3.1: Estimating local observables (Informal)
• Proposition 4.1: Correlator Cluster Expansion, Ratio Version (Informal)
• Proposition 4.2: Correlator Cluster Expansion, Derivative Version (Informal)