Carving-width and contraction trees for tensor networks

TL;DR

This paper introduces contraction-trees for tensor networks, linking their structure to computational complexity, and demonstrates that carving-width heuristics effectively limit contraction time in physical simulations.

Contribution

It proposes contraction-trees as a new framework for tensor network contraction ordering and provides experimental validation of carving-width as a heuristic for complexity management.

Findings

Carving-width correlates with contraction time.

Implementation of Ratcatcher for planar networks.

Carving-width heuristic effectively limits contraction complexity.

Abstract

We study the problem of finding contraction orderings on tensor networks for physical simulations using a syncretic abstract data type, the $\textit{contraction-tree}$, and explain its connection to temporal and spatial measures of tensor contraction computational complexity (nodes express time; arcs express space). We have implemented the Ratcatcher of Seymour and Thomas for determining the carving-width of planar networks, in order to offer experimental evidence that this measure of spatial complexity makes a generally effective heuristic for limiting their total contraction time.