# Mathematical open problems in projected entangled pair states

@article{Cirac2019MathematicalOP, title={Mathematical open problems in projected entangled pair states}, author={Juan Ignacio Cirac and Jos{\'e} Garre-Rubio and David P'erez-Garc'ia}, journal={Revista Matem{\'a}tica Complutense}, year={2019} }

Projected Entangled Pair States (PEPS) are used in practice as an efficient parametrization of the set of ground states of quantum many body systems. The aim of this paper is to present, for a broad mathematical audience, some mathematical questions about PEPS.

## 11 Citations

### A Tensor Version of the Quantum Wielandt Theorem

- MathematicsSIAM J. Matrix Anal. Appl.
- 2019

Boundedness results for the injectivity regions for projected entangled pair states are proved for a higher-dimensional generalization of the quantum Wielandt inequality.

### Simulation methods for open quantum many-body systems

- Physics
- 2019

This work reviews several approaches to simulate open many-body systems and point out the advances made in recent years towards the simulation of large system sizes.

### Symmetries in topological tensor network states: classification, construction and detection

- Mathematics
- 2019

This thesis contributes to the understanding of symmetry-enriched topological phases focusing on their descriptions in terms of tensor network states by proposing a family of gauge invariant quantities and their corresponding order parameters to detect the corresponding quantum phases, in particular their symmetry fractionalization patterns.

### Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice.

- PhysicsPhysical review letters
- 2020

A mathematical finite-size criterion is proved which gives an analytical, size-independent bound on the spectral gap if the gap of a particular cut-out subsystem of 36 spins exceeds a certain threshold value, and this is verified numerically by performing state-of-the-art DMRG calculations on the subsystem.

### Demonstrating the Affleck-Kennedy-Lieb-Tasaki Spectral Gap on 2D Degree-3 Lattices.

- PhysicsPhysical review letters
- 2020

A standard Lanczos method is used to establish the existence of a nonzero gap above its ground state in the two-dimensional spin-3/2 antiferromagnetic valence-bond model of Affleck, Kennedy, Lieb, and Tasaki (AKLT).

### Lower and Upper Bounds on the Pseudo-Dimension of Tensor Network Models

- Computer ScienceNeurIPS
- 2021

Upper and lower bounds on the VC-dimension and pseudo-dimension of a large class of TN models for classification, regression and completion are derived and a generalization bound is derived which can be applied to classification with low-rank matrices as well as linear classifiers based on any of the commonly used tensor decomposition models.

### Lower and Upper Bounds on the VC-Dimension of Tensor Network Models

- Computer ScienceArXiv
- 2021

Upper and lower bounds on the VC-dimension and pseudo-dimension of a large class of TN models for classification, regression and completion are derived and a generalization bound is derived which can be applied to classification with low-rank matrices as well as linear classifiers based on any of the commonly used tensor decomposition models.

### Spectral function of the $J_1-J_2$ Heisenberg model on the triangular lattice

- Physics
- 2022

Spectral probes, such as neutron scattering, are crucial for characterizing excitations in quantum many-body systems and the properties of quantum materials. Among the most elusive phases of matter…

### Classification of symmetry-protected topological phases in two-dimensional many-body localized systems

- Mathematics
- 2020

We use low-depth quantum circuits, a specific type of tensor networks, to classify two-dimensional symmetry-protected topological many-body localized phases. For (anti-)unitary on-site symmetries we…

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