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“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.
“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.
“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.