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 Reviews and Lectures

• “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 algorithms for one- and two-dimensional quantum systems.

• “Tensor networks for complex quantum systems”
  Author: Roman Orus
  Pre-print: 1812.04011
  Journal: Nature Reviews Physics volume 1, pages 538–550 (2019)
  Notes: Accessible review of tensor networks, focusing on more recent developments and the growing role of tensor networks in other areas beyond their origin in condensed matter quantum physics.

• “Lectures on Matrix Product States and Density Matrix Renormalization Group (DMRG)”
  Author: Ulrich Schollwoeck
  Link to Videos
  Notes: Accessible, introductory lectures on matrix product states and associated concepts (SVD, area law of entanglement) and the DMRG algorithm.

• “Matrix product states, DMRG, and tensor networks”
  Author: Garnet Chan
  Video Link 1
  Video Link 2
  Notes: Accessible and gradual lectures introducing core tensor network concepts and algorithms in physics and chemistry.

• “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.

More Technical Review Articles, Books, etc.

• “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.

• “A Practical Guide to the Numerical Implementation of Tensor Networks I: Contractions, Decompositions and Gauge Freedom”
  Author: Glen Evenbly
  Pre-print: 2202.02138
  Notes: Presents an overview of the key ideas and skills necessary to begin implementing tensor network methods numerically, which is intended to facilitate the practical application of tensor network methods for researchers that are already versed with their theoretical foundations. These skills include an introduction to the contraction of tensor networks, to optimal tensor decompositions, and to the manipulation of gauge degrees of freedom in tensor networks.

• “Tensor Network Contractions: Methods and Applications to Quantum Many-Body Systems”
  Authors: Shi-Ju Ran, Emanuele Tirrito, Cheng Peng, Xi Chen, Luca Tagliacozzo, Gang Su, Maciej Lewenstein
  Link: Springer Lecture Notes in Physics, Open Access Book
  Notes: An open-access book reviewing basics of tensor networks in diagram notation, then introducing advanced techniques for tasks such as contracting two-dimensional tensor networks.

• “Efficient Numerical Simulations Using Matrix-Product States”
  Authors: Frank Pollmann
  Link to Notes
  Notes: An introduction to matrix product state methods in physics building up to Trotter gate (TEBD) algorithms for time evolution and the infinite DMRG algorithm.

• “Tangent-space methods for uniform matrix product states”
  Authors: Laurens Vanderstraeten, Jutho Haegeman, Frank Verstraete
  Pre-print: 1810.07006
  Journal Article: SciPost Phys. Lect. Notes 7 (2019)
  Notes: Overview of the concept of tangent spaces to MPS, methods for optimizing uniform (infinite) MPS, and finding excited states (sub-dominant eigenvectors) using tangent space methods.

• “Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems”
  Authors: J. Ignacio Cira, David Perez-Garcia, Norbert Schuch, Frank Verstraete
  Pre-print: 2011.12127
  Notes: An authoritative introduction to tensor networks from a quantum physics perspective with a rigorous discussion of concepts such as scaling of entanglement, symmetries and group theoretic properties, parent Hamiltonians, and canonical forms of tensor networks.

• “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.
  

• “Matrix Product Operators, Matrix Product States, and ab initio Density Matrix Renormalization Group algorithms”
  Authors: Garnet Kin-Lic Chan, Anna Keselman, Naoki Nakatani, Zhendong Li, Steven R. White
  Pre-print: 1605.02611
  Notes: An extensive and readable discussion of DMRG in the matrix product state formalism as applied to Hamiltonians with long-range interactions such as in quantum chemistry. Discusses the structure of matrix product operators (MPO) in particular and gives a novel way to construct compressed MPOs for complicated Hamiltonians.

• “Time-evolution methods for matrix-product states”
  Author: Sebastian Paeckel, Thomas Köhler, Andreas Swoboda, Salvatore R. Manmana, Ulrich Schollwöck, Claudius Hubig
  Pre-print: 1901.05824
  Journal: Annals of Physics 411, 167998 (2019) (open access)
  Notes: Explains five current time-evolution methods for matrix-product states in extensive detail and tests them on four relevant problem settings. The content of this review is also the basis of the articles on time-evolution methods on this website.

Selected Original Literature

• “O(d log(n))-Quantics Approximation of Nd tensors in High-Dimensional Numerical Modeling”
  Author: Boris N. Khoromskij
  Journal: Constr Approx 34, 257–280 (2011)
  Notes: Introduces the quantics, or quantized tensor train (QTT) format for representing functions as tensor trains or matrix product states.

• “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 Operator Representations”
  Authors: B. Pirvu and V. Murg and J.I. Cirac and F. Verstraete
  Pre-print: arxiv:0804.3976
  Journal: New J. Phys. 12 025012 (2010)
  Notes: Discusses various exact or controlled constructions of MPOs and PEPS, including long-range correlations or interactions.

• “Tensor Decompositions and Applications”
  Author: T. G. Kolda, B. W. Bader
  Journal: SIAM Review, 51(3), 455-500 (2009)
  Notes: Review article introducing tensor basics and tensor decompositions.

• “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.

• “Algorithms for Tensor Network Renormalization”
  Authors: G. Evenbly
  Pre-print: 1509.07484
  Journal: Phys. Rev. B 95, 045117
  Notes: Tutorial article with details about optimization steps for isometric tensors and steps of the tensor network renormalization (TNR) algorithm.

• “Tensor Network States and Geometry”
  Authors: G. Evenbly, G. Vidal
  Pre-print: 1106.1082
  Journal: J Stat Phys (2011) 145:891-918
  Notes: Discusses scaling of entanglement in different families of tensor networks and how it relates to their emergent, or holographic, geometries. Contains useful facts about how entanglement scales for various classes of quantum states in different spatial dimensions.

• “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.

• “From density-matrix renormalization group to matrix product states”
  Authors: Ian McCulloch
  Pre-print: cond‑mat/0701428
  Journal: J. Stat. Mech.: Theory and Experiment 10, P10014 (2007)
  Notes: Describes the “calculus” of matrix product states, the structure of matrix product states and opearators, and how the DMRG algorithm can be formulated in terms of tensor networks.

• “Classical Simulation of Infinite-Size Quantum Lattice Systems in One Spatial Dimension”
  Authors: Guifre Vidal
  Pre-print: cond‑mat/0605597
  Journal: Phys. Rev. Lett. 98, 070201 (2007)
  Notes: Extends the TEBD algorithm to infinite (translationally-invariant) systems.

• “Efficient Classical Simulation of Slightly Entangled Quantum Computations”
  Authors: Guifre Vidal
  Pre-print: quant‑ph/0301063
  Journal: Phys. Rev. Lett. 91, 147902 (2003)
  Notes: Proposes the “time-evolving block decimation” (TEBD) algorithm for applying a quantum circuit to a matrix product state tensor network, allowing a wide class of circuits to be classically simulated.

• “Thermodynamic Limit of Density Matrix Renormalization”
  Authors: Stellan Ostlund and Stefan Rommer
  Pre-print: cond‑mat/9503107
  Journal: Phys. Rev. Lett. 75, 3537 (1995)
  Notes: Explains how the format of the wavefunction found by the density matrix renormalization group (DMRG) algorithm can be understood as a matrix product state (MPS) tensor network.

Edit This Page