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arXiv:2601.04172

Stochastic Path Compression for Spectral Tensor Networks on Cyclic Graphs

Ryan T. Grimm, Joel David Eaves

Published Jan 7, 2026• 0 citations
arXivHTMLPDF

Abstract

We develop a new approach to compress cyclic tensor networks called stochastic path compression (SPC) that uses an iterative importance sampling procedure to target edges with large bond-dimensions. Closed random walks in SPC form compression pathways that spatially localize large bond-dimensions in the tensor network. Analogous to the phase separation of two immiscible liquids, SPC separates the graph of bond-dimensions into spatially distinct high and low density regions. When combined with our integral decimation algorithm, SPC facilitates the accurate compression of cyclic tensor networks with continuous degrees of freedom. To benchmark and illustrate the methods, we compute the absolute thermodynamics of $q$-state clock models on two-dimensional square lattices and an XY model on a Watts-Strogatz graph, which is a small-world network with random connectivity between spins.

DOI
10.48550/arxiv.2601.04172

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We develop a new approach to compress cyclic tensor networks called stochastic path compression (SPC) that uses an iterative importance sampling procedure to target edges with large bond-dimensions. Closed random walks in SPC form compression pathways that spatially localize large bond-dimensions in the tensor network. Analogous to the phase separation of two immiscible liquids, SPC separates the graph of bond-dimensions into spatially distinct high and low density regions. When combined with our integral decimation algorithm, SPC facilitates the accurate compression of cyclic tensor networks with continuous degrees of freedom. To benchmark and illustrate the methods, we compute the absolute thermodynamics of qqq-state clock models on two-dimensional square lattices and an XY model on a Watts-Strogatz graph, which is a small-world network with random connectivity between spins.
Motivated by statistical physics models connected to computation problems, we devise a tensor network technique that is suited to problems with or without translation invariance and with arbitrary boundary conditions. We introduce a compression-decimation algorithm as an efficient iterative scheme to optimize tensor networks that encode generalized vertex models on regular lattices. The algorithm first propagates local constraints to longer ranges via repeated contraction-decomposition sweeps over all lattice bonds, thus achieving compression on a given length scale. It then decimates the lattice via coarse-graining tensor contractions. Repeated iterations of these two steps allow us to gradually collapse the tensor network while keeping the tensor dimensions under control, such that ultimately the full tensor trace can be taken for relatively large systems. As a benchmark, we demonstrate the efficiency of the algorithm by computing the ground state entropy density of the planar ice model and the eight-vertex model. We then apply it to reversible classical computational problems based on a recently proposed vertex model representation of classical computations [Nat. Commun. 8, 15303 (2017)]. Our protocol allows us to obtain the exact number of solutions for computations where a naive enumeration would take astronomically long times, suggesting that the algorithm is a promising practical tool for the solution of a plethora of problems in physics and computer science.
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We develop a tensor network technique that can solve universal reversible classical computational problems, formulated as vertex models on a square lattice [Nat. Commun. 8, 15303 (2017)2041-172310.1038/ncomms15303]. By encoding the truth table of each vertex constraint in a tensor, the total number of solutions compatible with partial inputs and outputs at the boundary can be represented as the full contraction of a tensor network. We introduce an iterative compression-decimation (ICD) scheme that performs this contraction efficiently. The ICD algorithm first propagates local constraints to longer ranges via repeated contraction-decomposition sweeps over all lattice bonds, thus achieving compression on a given length scale. It then decimates the lattice via coarse-graining tensor contractions. Repeated iterations of these two steps gradually collapse the tensor network and ultimately yield the exact tensor trace for large systems, without the need for manual control of tensor dimensions. Our protocol allows us to obtain the exact number of solutions for computations where a naive enumeration would take astronomically long times.
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Tensor network contraction is central to problems ranging from many-body physics to computer science. We describe how to approximate tensor network contraction through bond compression on arbitrary graphs. In particular, we introduce a hyperoptimization over the compression and contraction strategy itself to minimize error and cost. We demonstrate that our protocol outperforms both handcrafted contraction strategies in the literature as well as recently proposed general contraction algorithms on a variety of synthetic and physical problems on regular lattices and random regular graphs. We further showcase the power of the approach by demonstrating approximate contraction of tensor networks for frustrated three-dimensional lattice partition functions, dimer counting on random regular graphs, and to access the hardness transition of random tensor network models, in graphs with many thousands of tensors. Published by the American Physical Society 2024
Disordered lattice spin systems are crucial in both theoretical and applied physics. However, understanding their properties poses significant challenges to Monte Carlo simulations. In this work, we investigate the two-dimensional random-bond Ising model using the recently proposed tensor network Monte Carlo (TNMC) method. This method generates biased samples from conditional probabilities computed via tensor network contractions and corrects the bias using the Metropolis scheme. Consequently, the proposals provided by tensor networks function as block updates for Monte Carlo simulations. Through extensive numerical experiments, we demonstrate that TNMC simulations can be performed on lattices as large as $1024\ifmmode\times\else\texttimes\fi{}1024$ spins with moderate computational resources, a substantial increase from the previous maximum size of $64\ifmmode\times\else\texttimes\fi{}64$ in Markov-chain Monte Carlo. Notably, we observe an almost complete absence of critical slowing down, enabling the efficient collection of unbiased samples and averaging over a large number of random realizations of bond disorders. We successfully pinpoint the multicritical point along the Nishimori line with significant precision and accurately determined the bulk and surface critical exponents. Our findings suggest that TNMC is a highly efficient algorithm for exploring disordered and frustrated systems in two dimensions.