Partitioned Expansions for Approximate Tensor Network Contractions

TL;DR

Partitioned Network Expansions (PNE) provide a general, flexible framework to approximate tensor-network contractions by decomposing the target into a sum of cheaper subnet contractions using projector-based partitions. Unlike BP-loop-corrections, PNE does not require a known BP fixed point and can utilize higher-rank projections to handle degenerate fixed points, enabling accurate results across 2D, 3D, and infinite networks. Across Ising, AKLT, and random tensors, PNE consistently improves over BP and often outperforms SVD truncations, while offering a scalable path via recursive partitioning and higher-rank projections. The approach is straightforward to implement and readily parallelizable, presenting a practical, complementary tool for contracting challenging tensor networks in quantum many-body simulations and related domains.

Abstract

We propose a method for approximating the contraction of a tensor network by partitioning the network into a sum of computationally cheaper networks. This method, which we call a partitioned network expansion (PNE), builds upon recent work that systematically improves belief propagation (BP) approximations using loop corrections. However, in contrast to previous approaches, our expansion does not require a known BP fixed point to be implemented and can still yield accurate results even in cases where BP fails entirely. The flexibility of our approach is demonstrated through applications to a variety of example networks, including finite 2D and 3D networks, infinite networks, networks with open indices, and networks with degenerate BP fixed points. Benchmark numerical results for networks composed of Ising, AKLT, and random tensors typically show an improvement in accuracy over BP by several orders of magnitude (when BP solutions are obtainable) and also demonstrate improved performance over traditional network approximations based on singular value decomposition (SVD) for certain tasks.

Figures (20)

• Figure 1: (a) The identity $I$ is decomposed as the sum of a projector $P$ and its compliment $Q$. (b) The the linear form of the partitioned network expansion for a network of three tensors, see also Eq. \ref{['eq:A3']}. (c) The combinatorial form of the expansion, see also Eq. \ref{['eq:A5']}
• Figure 2: (a) A network of six tensors, where indices are assumed to be $\chi$-dimensional, has an exact contraction cost of $O(\chi^4)$. (b) The BP approximation, whose contraction cost scales as $O(\chi^3)$, is obtained by inserting fixed-point message pairs $\hbox{$| \mu_i \rangle$}$ on all indices $i$. (c) A network with contraction cost $O(\chi^3)$ can also be obtained by inserting the BP messages on a single index. (d) Rank $r=1$ projectors $P$ can be formed from the outer product of the fixed-point BP messages. (e) Complementary projectors $Q$ can be formed by subtracting $P$ from the identity $I$. (f) An $O(\chi^3)$ approximation is obtained via the linear form of a PNE based on partitioning the three vertical indices.
• Figure 3: Relative errors $\varepsilon$, as defined in Eq. \ref{['eq:B1']}, for the contraction of the network shown in Fig. \ref{['fig:2']}(a). Results are compared across the BP approximation from Fig. \ref{['fig:2']}(b), the single-index cut from Fig. \ref{['fig:2']}(c), and the PNE approximation from Fig. \ref{['fig:2']}(f). (left) Results obtained using tensors derived from either the $2D$ classical Ising model (at criticality) or the $2D$ square-lattice AKLT model, both blocked into networks of bond dimension $\chi=64$. (right) Aggregate results over 100 independent instances of random tensors with bond dimension $\chi=64$.
• Figure 4: (a) Expansion of the network into the BP fixed point plus loop-correction terms, where thick red lines denote indices projected into the subspace orthogonal to the fixed-point messages. (b) Depiction of the residue arising from the approximation in Fig. \ref{['fig:2']}(c), which includes contributions of degrees $\{4,6,7\}$. (c) The residue from the PNE in Fig. \ref{['fig:2']}(f) contains only a single degree-7 term.
• Figure 5: (a) The $3\times 3$ network, which has an exact contraction cost of $O(\chi^6)$, is approximated as a sum of cost $O(\chi^5)$ networks via a PNE using four single-index partitions in total. (b) The network is approximated as a sum of cost $O(\chi^4)$ networks via the combinatorial form of a PNE using six multi-index partitions. Sub-leading terms arising from combinations of these partitions are not shown.