# Benchmarking treewidth as a practical component of tensor network simulations

## 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 futuremore »

- Authors:

- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- North Carolina State Univ., Raleigh, NC (United States)

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1490605

- Grant/Contract Number:
- AC05-00OR22725

- Resource Type:
- Accepted Manuscript

- Journal Name:
- PLoS ONE

- Additional Journal Information:
- Journal Volume: 13; Journal Issue: 12; Journal ID: ISSN 1932-6203

- Publisher:
- Public Library of Science

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Dumitrescu, Eugene F., Fisher, Allison L., Goodrich, Timothy D., Humble, Travis S., Sullivan, Blair D., and Wright, Andrew L. Benchmarking treewidth as a practical component of tensor network simulations. United States: N. p., 2018.
Web. doi:10.1371/journal.pone.0207827.
```

```
Dumitrescu, Eugene F., Fisher, Allison L., Goodrich, Timothy D., Humble, Travis S., Sullivan, Blair D., & Wright, Andrew L. Benchmarking treewidth as a practical component of tensor network simulations. United States. doi:10.1371/journal.pone.0207827.
```

```
Dumitrescu, Eugene F., Fisher, Allison L., Goodrich, Timothy D., Humble, Travis S., Sullivan, Blair D., and Wright, Andrew L. Tue .
"Benchmarking treewidth as a practical component of tensor network simulations". United States. doi:10.1371/journal.pone.0207827. https://www.osti.gov/servlets/purl/1490605.
```

```
@article{osti_1490605,
```

title = {Benchmarking treewidth as a practical component of tensor network simulations},

author = {Dumitrescu, Eugene F. and Fisher, Allison L. and Goodrich, Timothy D. and Humble, Travis S. and Sullivan, Blair D. and Wright, Andrew L.},

abstractNote = {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.},

doi = {10.1371/journal.pone.0207827},

journal = {PLoS ONE},

number = 12,

volume = 13,

place = {United States},

year = {2018},

month = {12}

}

*Citation information provided by*

Web of Science

Web of Science

#### Figures / Tables:

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Figures / Tables found in this record:

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*