Benchmarking treewidth as a practical component of tensor-network--based quantum simulation

TL;DR

This paper demonstrates that treewidth-based algorithms are practically effective for tensor network contraction in quantum simulations, outperforming domain-specific methods in various regimes, and provides an accessible software framework for further research.

Contribution

It shows the practical relevance of treewidth-based methods for tensor network contraction in quantum simulation and offers an open-source framework for experimentation.

Findings

Treewidth methods outperform domain-specific algorithms in certain regimes.

Optimal contraction depends on network density and complexity.

Open source software facilitates future research and benchmarking.

Abstract

Tensor networks are powerful factorization techniques which reduce resource requirements for numerically simulating principal quantum many-body systems and algorithms. The computational complexity of a tensor network simulation depends on the tensor ranks and the order in which they are contracted. Unfortunately, computing optimal contraction sequences (orderings) in general is known to be a computationally difficult (NP-complete) task. In 2005, Markov and Shi showed that optimal contraction sequences correspond to optimal (minimum width) tree decompositions of a tensor network's line graph, relating the contraction sequence problem to a rich literature in structural graph theory. While treewidth-based methods have largely been ignored in favor of dataset-specific algorithms in the prior tensor networks literature, we demonstrate their practical relevance for problems arising from two distinct methods used in quantum simulation: multi-scale entanglement renormalization ansatz (MERA) datasets and quantum circuits generated by the quantum approximate optimization algorithm (QAOA). We exhibit multiple regimes where treewidth-based algorithms outperform domain-specific algorithms, while demonstrating that the optimal choice of algorithm has a complex dependence on the network density, expected contraction complexity, and user run time requirements. We further provide an open source software framework designed with an emphasis on accessibility and extendability, enabling replicable experimental evaluations and future exploration of competing methods by practitioners.