Tensor Network Loop Cluster Expansions for Quantum Many-Body Problems

TL;DR

This work analyzes the tensor network loop cluster expansion as a systematic correction to belief propagation for quantum many-body problems, focusing on accurate ground-state observables in PEPS across 2D and 3D lattices with open or periodic boundaries and both spin and fermion degrees of freedom. It presents a practical algorithm that generates loop clusters up to size $C$, uses inclusion-exclusion to assign counting numbers, and computes observables via a controlled product (or sum) of cluster contributions, all anchored in the BP framework. The results show near-exponential convergence in $C$ and significant improvement over single-cluster contractions, with Wynn epsilon extrapolation enabling robust infinite-$C$ estimates across TFIM, Heisenberg, and Fermi-Hubbard models, though 3D fermionic cases can converge more slowly. The approach extends the reach of tensor-network contraction to challenging geometries (PBC, 3D) and large bond dimensions, and suggests future directions in environment approximation and generalized BP-inspired message passing.

Abstract

We analyze the tensor network loop cluster expansion, introduced in arXiv:2504.07344 as a systematic correction to belief propagation, in the context of general quantum many-body problems. We provide numerical examples of the accuracy and practical applicability of the approach for the computation of ground-state observables for high bond dimension tensor networks, in two- and three-dimensions, with open and periodic boundary conditions, and for spin and fermion problems.

Figures (6)

• Figure 1: Overview of loop cluster expansion calculation of an observable $\hat{O}$ acting on two neighboring sites. (a) Local patch of the tensor network $\langle \psi | \hat{O} | \psi \rangle$, with the operator acting on the central orange sites. (b) The loop cluster expansion combines results from all relevant clusters up to size $C=5$. (i--vi) are the largest clusters, each with counting number $c(r)=1$. (vii--xii) are generated by the intersections of the largest clusters, each has $c(r)=-1$. By the fixed-point condition, all of these are equivalent to the zeroth order term (xiii.), itself with $c(r)=1$. (c) Full tensor network example of cluster (iv.), with boundary messages $m$ and bond dimension $D$.
• Figure 2: (a) Simple update (SU) gauging scheme with the Vidal gauge. (b) SU local canonical condition. (c) BP fixed-point condition.
• Figure 3: Example convergence of the loop cluster expansion for two square lattice OBC PEPS SU states on two different models, compared against the 'single cluster' method and reference boundary contraction. The main loop cluster expansion data uses the product formula, whilst the thin darker line shows the (almost identical) sum formula for comparison.
• Figure 4: Relative energy contraction error of the loop cluster expansion, without extrapolation, as a function of cluster size $C$ and bond dimension $D$, for PEPS+SU ground-states of three different models defined on a $10{\times}10$ square OBC lattice. The top and bottom rows use Eq. \ref{['eq:product_cluster_expansion']} and Eq. \ref{['eq:sum_cluster_expansion']} respectively.
• Figure 5: Examples of Wynn extrapolation for the same two examples as Fig. \ref{['fig:energy-cluster-vs-expansion']}. $\epsilon_0$, $\epsilon_2$ and $\epsilon_4$ are the zeroth, 2nd order and 4th order sequences. We use the final value of $\epsilon_4$ as our extrapolated value, and the average final gradient (see main text) across $\epsilon_0$, $\epsilon_2$ and $\epsilon_4$ as an estimate of the error bar.