Teaching plan for the course unit



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General information


Course unit name: Tensor Networks

Course unit code: 574645

Academic year: 2021-2022

Coordinator: Bruno Julia Diaz

Department: Department of Quantum Physics and Astrophysics

Credits: 3

Single program: S



Estimated learning time

Total number of hours 75


Face-to-face and/or online activities



-  Lecture

Face-to-face and online




-  Lecture with practical component

Face-to-face and online



Independent learning






  1.  Students are expected to have a solid background in quantum mechanics and statistical mechanics and master one programming language.



Competences to be gained during study


  1. Ability to understand Tensor Network approaches, use the most common tensor network packages and produce their own codes. 





Learning objectives


Referring to knowledge

  1. To develop a broad and unified perspective on tensor networks. 

  2. To become familiar with the basic tensor network structures like Matrix product states, Tree Tensor networks and Projected entangled pair states. 

  3. To understand the fundamentals numerical algorithms used in the context of tensor networks. 



Teaching blocks


1. Generalities


  1. An introduction to the tensor network formalism and graphical notation,

  2. Introduction to the tensor network zoo, MPS, PEPS, TTN, MERAs ...

2. Analytical foundation of tensor networks


  1. Matrix product states and their representation 

  2. Canonical form and injectivity

  3. Entanglement  in many-body quantum systems 

  4. The geometry of tensor networks

3. Tensor networks approaches to statistical mechanics


  1. Encoding partition functions

  2. Calculating entropies

4. Back to quantum systems


  1. Transfer matrix and the quantum Hamiltonian

  2. The renormalization group 

  3. DMRG as a RG algorithm

5. Numerical simulation the existing software


  1. An overview of the steps in numerical algorithms

  2. Tensor contractions, their cost and strategies

  3. Overview of the main software packages

6. Numerical simulations building new algorithms


  1. Guiding principles

  2. Software libraries helping designing new algorithms

7. Symmetries in tensor Networks


  1. The physical relevance of symmetry

  2. Symmetries in tensor networks

  3. Implementing symmetries in practice



Official assessment of learning outcomes


  1. Continuous assessment, based on working on a final project. The project will contain several sections that will be evaluated during the course.  Some sections will be common to all projects (the basic ones). 

  2. Exercises will be also provided, and their solution will contribute to the final mark. 



Reading and study resources

Consulteu la disponibilitat a CERCABIB


Tensor Network Contractions Methods and Applications to Quantum Many-Body Systems Authors: Ran, S.-J., Tirrito, E., Peng, C., Chen, X., Tagliacozzo, L., Su, G., Lewenstein, M.  Springer Lecture notes in Physics  Enllaç

Video, DVD and film

Topological Matter School. TMS18.L25. Frank Pollmann. Tensor Networks and Matrix Product States (I).; 2018. Accessed January 25, 2021.

ICTP Condensed Matter and Statistical Physics. Norbert Schuch Tensor Networks - Lecture 1.; 2017. Accessed January 25, 2021.  Enllaç

Cornell Laboratory of Atomic and Solid State Physics. Garnet Chan “Matrix Product States, DMRG, and Tensor Networks” (Part 1 of 2).; 2015. Accessed January 25, 2021.  Enllaç

ICAM - I2CAM. Tensor Network States & Entanglement Renormalization I - Verstraete.; 2016. Accessed January 25, 2021.

Mini Crash Course: Tensor Networks. Guifre Vidal Accessed February 19, 2020.




J. C. Bridgeman and C. T. Chubb, Hand-Waving and Interpretive Dance: An Introductory Course on Tensor Networks, J. Phys. A: Math. Theor. 50, 223001 (2017).

Haegeman and F. Verstraete, Diagonalizing Transfer Matrices and Matrix Product Operators: A Medley of Exact and Computational Methods, ArXiv:1611.08519 [Cond-Mat, Physics:Math-Ph, Physics:Quant-Ph] (2016).

R. Orus, A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States, Annals of Physics 349, 117 (2014).