RT Journal Article
T1 The minimal canonical form of a tensor network
A1 Acuaviva Huertos, Arturo
A1 Makam, Visu
A1 Nieuwboer, Harold
A1 Pérez García, David
A1 Sittner, Friedrich
A1 Walter, Michael
A1 Witteveen, Freek
AB 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.
YR 2022
FD 2022
LK https://hdl.handle.net/20.500.14352/72734
UL https://hdl.handle.net/20.500.14352/72734
LA eng
NO Ministerio de Ciencia e Innovación (MICINN)
NO Comunidad de Madrid
DS Docta Complutense
RD 19 abr 2025