Stochastic Path Compression for Spectral Tensor Networks on Cyclic Graphs

TL;DR

A new approach to compress cyclic tensor networks called stochastic path compression (SPC) is developed that uses an iterative importance sampling procedure to target edges with large bond-dimensions to facilitate the accurate compression of cyclic tensor networks with continuous degrees of freedom.

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.

Figures (3)

• Figure 1: Building a spectral tensor network using integral decimation and compressing it using stochastic path compression. (a) A tensor network on a square lattice and its representation in a compressed form where vertical lines represent the physical bonds that are not contracted over. The Boltzmann weight of a many-body problem on the lattice decomposes into sets of body-ordered gates, and an iterative procedure, starting from a simple product state, forms a computationally tractable partition function. The two-body gate (green rectangle) acts on the network, enlarging the bond-dimension between the two sites. (b) To compress the resulting network, implement a compression cycle. First, draw a random subset of the elementary cycles on the graph to maximize coverage of edges with large bond-dimensions. Generate the compression cycle from symmetric difference (XOR) of the elementary cycles (yellow squares with a filled/unfilled dot denoting the presence of the elementary cycle). (c) In a tensor network with cycles, each edge may need to be compressed more than once. To facilitate this, convert the compression cycle into a directed cycle, where the edge is traversable in both directions. (d) After many iterations, SPC generates an emergent information topology where the network separates into regions of high and low bond-dimensions -- similar to phase separation in two immiscible liquids. The dark line between the red and blue regions represents the interface.
• Figure 2: Computing the absolute thermodynamics of the $q$-state clock model on a square lattice using SPC. Energy is in units of $J$, and temperature is in units of $J / k_B.$ We report free energy per site. (a) The storage requirements of a spectral tensor network representation of the Boltzmann distribution for chain and grid representation graphs. We compress the chain with the standard algorithm and the grid with simple update (SU), compression around four-point cycles (Loop), and stochastic path compression (SPC). (b) The final bond-dimension distribution of an $8 \times 8$ lattice. (c) The free energy of a $16 \times 16$ clock model at $q \in \{4, 8, 16, 32\}$ computed with ID and SPC. (d) Relative error in the free energy of an $11 \times 11$ lattice at $q = 4$ compared to the exact transfer matrix solution levin2007-bs. The green line shows SPC, and the black line SU. (e) The specific heat of the clock model for $q \in {2, \dots, 11}$ values computed via differentiation of the spectral tensor network.
• Figure 3: Computing the thermodynamics of the XY model on a Watts-Strogatz graph with $16$, (d), and $32$, (a-c), nodes using ID. Energy is in units of $J$, and temperature is in units of $J / k_B.$ We report free energy per site. (a) The free energy computed with three compression modes, where we run SPC after contracting each edge. The modes are exact, no compression applied, simple, SU, and compressed, SPC. (b) The connectivity of the random graph. (c) Storage requirements for each intermediate network during the contraction for $\beta = 1$: without compression, by simple truncation of each edge, and SPC. (d) Relative error compared to a quasi Monte Carlo integration of the partition function. The red dashed line denotes the relative SVD cutoff $\epsilon_{\mathrm{SVD}}$.