# Parameterization of tensor network contraction

## TL;DR

This paper introduces a comprehensive framework for parameterizing tensor network contraction costs using contraction trees, enabling analysis of contraction complexity and parallelization strategies across arbitrary tensor networks.

## Contribution

It presents a general, graph-theoretic framework for tensor network contraction cost analysis, connecting contraction trees with existing graph algorithms and properties.

## Key findings

- Contraction tree properties directly relate to contraction time and space costs.
- The framework applies to tensor networks with arbitrary structures and dimensions.
- Edge congestion is nearly equal to the branchwidth of the line graph.

## Abstract

We present a conceptually clear and algorithmically useful framework for parameterizing the costs of tensor network contraction. Our framework is completely general, applying to tensor networks with arbitrary bond dimensions, open legs, and hyperedges. The fundamental objects of our framework are rooted and unrooted contraction trees, which represent classes of contraction orders. Properties of a contraction tree correspond directly and precisely to the time and space costs of tensor network contraction. The properties of rooted contraction trees give the costs of parallelized contraction algorithms. We show how contraction trees relate to existing tree-like objects in the graph theory literature, bringing to bear a wide range of graph algorithms and tools to tensor network contraction. Independent of tensor networks, we show that the edge congestion of a graph is almost equal to the branchwidth of its line graph.