The minimal canonical form of a tensor network

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Acuaviva Huertos, Arturo and Makam, Visu and Nieuwboer, Harold and Pérez García, David and Sittner, Friedrich and Walter, Michael and Witteveen, Freek (2022) The minimal canonical form of a tensor network. (Unpublished)

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Abstract

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.


Item Type:Article
Subjects:Sciences > Physics > Mathematical physics
Sciences > Mathematics > Operations research
ID Code:75768
Deposited On:12 Dec 2022 18:02
Last Modified:13 Dec 2022 08:12

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