The minimal canonical form of a tensor network

Tensor networks have a gauge degree of freedom on the virtual degrees of freedom that are contracted. A canonical form is a choice of fixing this degree of freedom. For matrix product states, choosing a canonical form is a powerful tool, both for theoretical and numerical purposes. On the other hand...

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Autores: Acuaviva Huertos, Arturo, Makam, Visu, Nieuwboer, Harold, Pérez García, David, Sittner, Friedrich, Walter, Michael, Witteveen, Freek
Tipo de recurso: artículo
Fecha de publicación:2022
País:España
Institución:Universidad Complutense de Madrid (UCM)
Repositorio:Docta Complutense
Idioma:inglés
OAI Identifier:oai:docta.ucm.es:20.500.14352/72734
Acceso en línea:https://hdl.handle.net/20.500.14352/72734
Access Level:acceso abierto
Palabra clave:Física matemática
Investigación operativa (Matemáticas)
1207 Investigación Operativa
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spelling The minimal canonical form of a tensor networkAcuaviva Huertos, ArturoMakam, VisuNieuwboer, HaroldPérez García, DavidSittner, FriedrichWalter, MichaelWitteveen, FreekFísica matemáticaInvestigación operativa (Matemáticas)1207 Investigación OperativaTensor networks have a gauge degree of freedom on the virtual degrees of freedom that are contracted. A canonical form is a choice of fixing this degree of freedom. For matrix product states, choosing a canonical form is a powerful tool, both for theoretical and numerical purposes. On the other hand, for tensor networks in dimension two or greater there is only limited understanding of the gauge symmetry. Here we introduce a new canonical form, the minimal canonical form, which applies to projected entangled pair states (PEPS) in any dimension, and prove a corresponding fundamental theorem. Already for matrix product states this gives a new canonical form, while in higher dimensions it is the first rigorous definition of a canonical form valid for any choice of tensor. We show that two tensors have the same minimal canonical forms if and only if they are gauge equivalent up to taking limits; moreover, this is the case if and only if they give the same quantum state for any geometry. In particular, this implies that the latter problem is decidable - in contrast to the well-known undecidability for PEPS on grids. We also provide rigorous algorithms for computing minimal canonical forms. To achieve this we draw on geometric invariant theory and recent progress in theoretical computer science in non-commutative group optimization.Universidad Complutense de Madrid20222022-01-0120222022-01-01journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14352/72734reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/727342026-06-02T12:44:21Z
dc.title.none.fl_str_mv The minimal canonical form of a tensor network
title The minimal canonical form of a tensor network
spellingShingle The minimal canonical form of a tensor network
Acuaviva Huertos, Arturo
Física matemática
Investigación operativa (Matemáticas)
1207 Investigación Operativa
title_short The minimal canonical form of a tensor network
title_full The minimal canonical form of a tensor network
title_fullStr The minimal canonical form of a tensor network
title_full_unstemmed The minimal canonical form of a tensor network
title_sort The minimal canonical form of a tensor network
dc.creator.none.fl_str_mv Acuaviva Huertos, Arturo
Makam, Visu
Nieuwboer, Harold
Pérez García, David
Sittner, Friedrich
Walter, Michael
Witteveen, Freek
author Acuaviva Huertos, Arturo
author_facet Acuaviva Huertos, Arturo
Makam, Visu
Nieuwboer, Harold
Pérez García, David
Sittner, Friedrich
Walter, Michael
Witteveen, Freek
author_role author
author2 Makam, Visu
Nieuwboer, Harold
Pérez García, David
Sittner, Friedrich
Walter, Michael
Witteveen, Freek
author2_role author
author
author
author
author
author
dc.contributor.none.fl_str_mv Universidad Complutense de Madrid
dc.subject.none.fl_str_mv Física matemática
Investigación operativa (Matemáticas)
1207 Investigación Operativa
topic Física matemática
Investigación operativa (Matemáticas)
1207 Investigación Operativa
description Tensor networks have a gauge degree of freedom on the virtual degrees of freedom that are contracted. A canonical form is a choice of fixing this degree of freedom. For matrix product states, choosing a canonical form is a powerful tool, both for theoretical and numerical purposes. On the other hand, for tensor networks in dimension two or greater there is only limited understanding of the gauge symmetry. Here we introduce a new canonical form, the minimal canonical form, which applies to projected entangled pair states (PEPS) in any dimension, and prove a corresponding fundamental theorem. Already for matrix product states this gives a new canonical form, while in higher dimensions it is the first rigorous definition of a canonical form valid for any choice of tensor. We show that two tensors have the same minimal canonical forms if and only if they are gauge equivalent up to taking limits; moreover, this is the case if and only if they give the same quantum state for any geometry. In particular, this implies that the latter problem is decidable - in contrast to the well-known undecidability for PEPS on grids. We also provide rigorous algorithms for computing minimal canonical forms. To achieve this we draw on geometric invariant theory and recent progress in theoretical computer science in non-commutative group optimization.
publishDate 2022
dc.date.none.fl_str_mv 2022
2022-01-01
2022
2022-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14352/72734
url https://hdl.handle.net/20.500.14352/72734
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Docta Complutense
instname:Universidad Complutense de Madrid (UCM)
instname_str Universidad Complutense de Madrid (UCM)
reponame_str Docta Complutense
collection Docta Complutense
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300724

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