The resource theory of tensor networks

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Abstract

Tensor networks provide succinct representations of quantum many-body states and are an important computational tool for strongly correlated quantum systems. Their expressive and computational power is characterized by an underlying entanglement structure, on a lattice or more generally a (hyper)graph, with virtual entangled pairs or multipartite entangled states associated to (hyper)edges. Changing this underlying entanglement structure into another can lead to both theoretical and computational benefits. We study a natural resource theory which generalizes the notion of bond dimension to entanglement structures using multipartite entanglement. It is a direct extension of resource theories of tensors studied in the context of multipartite entanglement and algebraic complexity theory, allowing for the application of the sophisticated methods developed in these fields to tensor networks. The resource theory of tensor networks concerns both the local entanglement structure of a quantum many-body state and the (algebraic) complexity of tensor network contractions using this entanglement structure. We show that there are transformations between entanglement structures which go beyond edge-by-edge conversions, highlighting efficiency gains of our resource theory that mirror those obtained in the search for better matrix multiplication algorithms. We also provide obstructions to the existence of such transformations by extending a variety of methods originally developed in algebraic complexity theory for obtaining complexity lower bounds. The resource theory of tensor networks allows one to compare different entanglement structures and could lead to more efficient tensor network representations and contraction algorithms.

OriginalsprogEngelsk
Artikelnummer1560
TidsskriftQuantum
Vol/bind8
Antal sider66
ISSN2521-327X
DOI
StatusUdgivet - 2024

Bibliografisk note

Funding Information:
We acknowledge financial support from the European Research Council (ERC Grant Agreement No. 818761), VILLUM FONDEN via the QMATH Centre of Excellence (Grant No.10059) and the Villum Young Investigator program (Grant No. 25452) and the Novo Nordisk Foundation (grant NNF20OC0059939 \u2018Quantum for Life\u2019). MC thanks the National Center for Competence in Research SwissMAP of the Swiss National Science Foundation and the Section of Mathematics at the University of Geneva for their hospitality. VL additionally acknowledges financial support from the European Union (ERC Grant No. 101040907). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

Publisher Copyright:
© 2024 Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften. All rights reserved.

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