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Stochastic Path Compression for Spectral Tensor Networks on Cyclic Graphs

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.