TY - GEN

T1 - The minimal canonical form of a tensor network

AU - Acuaviva, Arturo

AU - Makam, Visu

AU - Nieuwboer, Harold

AU - Perez-Garcia, David

AU - Sittner, Friedrich

AU - Walter, Michael

AU - Witteveen, Freek

N1 - Publisher Copyright: © 2023 IEEE.

PY - 2023

Y1 - 2023

N2 - 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 equality of 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.

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 equality of 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.

KW - invariant theory

KW - non-commutative optimization

KW - tensor networks

U2 - 10.1109/FOCS57990.2023.00027

DO - 10.1109/FOCS57990.2023.00027

M3 - Article in proceedings

AN - SCOPUS:85182394423

SP - 328

EP - 362

BT - Proceedings - 2023 IEEE 64th Annual Symposium on Foundations of Computer Science, FOCS 2023

PB - IEEE

T2 - 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023

Y2 - 6 November 2023 through 9 November 2023

ER -