# Parameterization of tensor network contraction

arXiv: https://arxiv.org/abs/1906.00013 · HTML: https://mlsanity.com/paper/1906.00013/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 May 2019.

## Built on by (most cited first, 3 of 3)

- [Near-Linear Time and Fixed-Parameter Tractable Algorithms for Tensor Decompositions](https://mlsanity.com/paper/2207.07417/) (2022)
- [Algorithmic algorithm development with LLMs: A Case Study on LLM-Usage for Contraction Order Optimization in Tensor Networks](https://mlsanity.com/paper/2606.01975/) (2026)
- [Learning to Rank Tensor Network Contraction Plans for GPU-Accelerated Quantum Circuit Simulation](https://mlsanity.com/paper/2608.05819/) (2026)

## Abstract (arXiv)

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.