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Dimension of tensor network varieties

Alessandra Bernardi

Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38123 Povo (TN), Italy

E-mail Address: alessandra.bernardi@unitn.it

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Claudia De Lazzari

Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38123 Povo (TN), Italy

E-mail Address: claudia.delazzari@unitn.it

Corresponding author.

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Fulvio Gesmundo

Saarland University, Saarland Informatics Campus, 66123 Saarbrücken, Germany

E-mail Address: gesmundo@cs.uni-saarland.de

Current address: Saarland University, Saarbrücken, Germany.

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https://doi.org/10.1142/S0219199722500596Cited by:5 (Source: Crossref)

Abstract

The tensor network variety is a variety of tensors associated to a graph and a set of positive integer weights on its edges, called bond dimensions. We determine an upper bound on the dimension of the tensor network variety. A refined upper bound is given in cases relevant for applications such as varieties of matrix product states and projected entangled pairs states. We provide a range (the “supercritical range”) of the parameters where the upper bound is sharp.

Keywords:

AMSC: 15A69, 81P45

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