Review Articles and Learning Resources

Introductory Websites or Blogs

• Tensors.net
  Author: Glen Evenbly
  Notes: Extensive introductory tutorials about tensor networks including working sample codes in MATLAB, python, and Julia.

Introductory Review Articles

• “A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States”
  Author: Roman Orus
  Pre-print: 1306.2164
  Journal: Annals of Physics 349 (2014) 117-158
  Notes: Friendly introduction to tensor networks for readers with a quantum physics background. Gives some overviews of basics of tensor network algorithms as well as more detailed descriptions of aalgorithms for one- and two-dimensional quantum systems.

• “Tensor Networks for Big Data Analytics and Large-Scale Optimization Problems”
  Author: Andrzej Cichocki
  Pre-print: 1407.3124
  Notes: Good introduction to tensor networks for those more familiar with applied mathematics literature and notation.

• “Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks”
  Authors: Jacob C. Bridgeman, Christopher T. Chubb
  Pre-print: 1603.03039
  Journal: J. Phys. A: Math. Theor. 50 223001 (2017)
  Notes: For those with a quantum information background, this is a very friendly and readable introduction to tensor networks and tensor diagram notation.

Technical Introductions or Review Articles

• “Tensor Networks and Applications”
  Author: Miles Stoudenmire
  Description: This is a one-week course, mainly on matrix product state (MPS) tensor networks, aimed at graduate students familiar with many-body quantum mechanics.
  Slides (pdf):
  Lecture 1 — Entanglement in quantum many-body systems.
  Lecture 2 — Introduction to MPS; efficient computations with MPS; AKLT state example.
  Lecture 3 — Gauges or canonical forms of MPS; matrix product operators; DMRG algorithm.
  Lecture 4 — Tensor networks; scaling of entanglement for critical and 2D systems; quantum chemistry; finite temperature systems.
  Lecture 5 — Introduction to machine learning; selected uses of tensor networks in machine learning.
  

• “The density-matrix renormalization group in the age of matrix product states”
  Author: Ulrich Schollwoeck
  Pre-print: 1008.3477
  Journal: Annals of Physics 326, 96 (2011)
  Notes: A nearly comprehensive, and very detailed explanation of matrix product state and DMRG methods as of 2011. This article could serve as a very good introduction for those familiar with quantum mechanics and who are willing to work carefully through the many helpful steps and details offered.

Selected Original Literature

• “Tensor-Train Decomposition”
  Author: Ivan Oseledets
  Journal: SIAM J. Sci. Comput., 33(5), 2295 (2011)
  Notes: Very readable article introducing the idea of the tensor-train decomposition into the mathematics literature.

• “Matrix Product State Representations”
  Authors: D. Perez-Garcia, F. Verstraete, M.M. Wolf, J.I. Cirac
  Pre-print: quant‑ph/0608197
  Journal: Quantum Info. Comput. 7, 401–430 (2007)
  Notes: One of the earlier articles proposing the idea of canonical forms of matrix product states (MPS), algorithms to compute MPS, and other properties of MPS.

• “Algorithms for Entanglement Renormalization (v2)”
  Author: G. Vidal
  Pre-print: 0707.1454v2
  Notes: Version 2 of this article is very different from the final published version (see below). It contains lots of interesting results about optimization strategies for tensor networks, and proposals to compute layers of hierarchical tensor networks.

• “Algorithms for Entanglement Renormalization”
  Authors: G. Evenbly, G. Vidal
  Pre-print: 0707.1454
  Journal: Phys. Rev. B 79, 144108
  Notes: Readable article with many details about MERA tensor networks including strategies for optimizing them.

• “Finite automata for caching in matrix product algorithms”
  Authors: Gregory M. Crosswhite, Dave Bacon
  Pre-print: 0708.1221
  Journal: Phys. Rev. A 78, 012356 (2008)
  Notes: Provides a nice picture and detailed understanding of how MPS and MPO tensor networks represent certain high-dimensional tensors in terms of a finite-state automaton picture.

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