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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 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.
“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.
“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.