Tensor Network Contractions Methods and Applications to Quantum Many-Body Systems
Tensor network is a fundamental mathematical tool with a huge range of applications in physics, such as condensed matter physics, statistic physics, high energy physics, and quantum information sciences. This open access book aims to explain the tensor network contraction approaches in a systematic...
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| Format: | E-Book Buch |
| Sprache: | Englisch |
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Springer Nature
2020
Springer Springer International Publishing AG |
| Ausgabe: | 1 |
| Schriftenreihe: | Lecture Notes in Physics |
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| ISBN: | 9783030344894, 3030344894, 3030344886, 9783030344887 |
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| Abstract | Tensor network is a fundamental mathematical tool with a huge range of applications in physics, such as condensed matter physics, statistic physics, high energy physics, and quantum information sciences. This open access book aims to explain the tensor network contraction approaches in a systematic way, from the basic definitions to the important applications. This book is also useful to those who apply tensor networks in areas beyond physics, such as machine learning and the big-data analysis. Tensor network originates from the numerical renormalization group approach proposed by K. G. Wilson in 1975. Through a rapid development in the last two decades, tensor network has become a powerful numerical tool that can efficiently simulate a wide range of scientific problems, with particular success in quantum many-body physics. Varieties of tensor network algorithms have been proposed for different problems. However, the connections among different algorithms are not well discussed or reviewed. To fill this gap, this book explains the fundamental concepts and basic ideas that connect and/or unify different strategies of the tensor network contraction algorithms. In addition, some of the recent progresses in dealing with tensor decomposition techniques and quantum simulations are also represented in this book to help the readers to better understand tensor network. This open access book is intended for graduated students, but can also be used as a professional book for researchers in the related fields. To understand most of the contents in the book, only basic knowledge of quantum mechanics and linear algebra is required. In order to fully understand some advanced parts, the reader will need to be familiar with notion of condensed matter physics and quantum information, that however are not necessary to understand the main parts of the book. This book is a good source for non-specialists on quantum physics to understand tensor network algorithms and the related mathematics. |
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| AbstractList | Tensor network is a fundamental mathematical tool with a huge range of applications in physics, such as condensed matter physics, statistic physics, high energy physics, and quantum information sciences. This open access book aims to explain the tensor network contraction approaches in a systematic way, from the basic definitions to the important applications. This book is also useful to those who apply tensor networks in areas beyond physics, such as machine learning and the big-data analysis. Tensor network originates from the numerical renormalization group approach proposed by K. G. Wilson in 1975. Through a rapid development in the last two decades, tensor network has become a powerful numerical tool that can efficiently simulate a wide range of scientific problems, with particular success in quantum many-body physics. Varieties of tensor network algorithms have been proposed for different problems. However, the connections among different algorithms are not well discussed or reviewed. To fill this gap, this book explains the fundamental concepts and basic ideas that connect and/or unify different strategies of the tensor network contraction algorithms. In addition, some of the recent progresses in dealing with tensor decomposition techniques and quantum simulations are also represented in this book to help the readers to better understand tensor network. This open access book is intended for graduated students, but can also be used as a professional book for researchers in the related fields. To understand most of the contents in the book, only basic knowledge of quantum mechanics and linear algebra is required. In order to fully understand some advanced parts, the reader will need to be familiar with notion of condensed matter physics and quantum information, that however are not necessary to understand the main parts of the book. This book is a good source for non-specialists on quantum physics to understand tensor network algorithms and the related mathematics. Tensor network is a fundamental mathematical tool with a huge range of applications in physics, such as condensed matter physics, statistic physics, high energy physics, and quantum information sciences. |
| Author | Tirrito, Emanuele Chen, Xi Ran, Shi-Ju Lewenstein, Maciej Peng, Cheng Tagliacozzo, Luca Su, Gang |
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| EISBN | 9783030344894 3030344894 |
| Edition | 1 1st Edition 2020 |
| Editor | Chen, Xi Tirrito, Emanuele Tagliacozzo, Luca Peng, Cheng |
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| Notes | Other authors: Emanuele Tirrito, Cheng Peng, Xi Chen, Luca Tagliacozzo, Gang Su, Maciej Lewenstein SpringerOpen Includes bibliographical references and index |
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| Snippet | Tensor network is a fundamental mathematical tool with a huge range of applications in physics, such as condensed matter physics, statistic physics, high... |
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| SubjectTerms | Artificial intelligence Computer science Computing and Information Technology Differential equations, Partial Elementary particles (Physics) Machine learning Mathematical physics Mathematics and Science Optical physics Physics Quantum field theory Quantum optics Quantum physics Quantum physics (quantum mechanics and quantum field theory) Statistical physics |
| Subtitle | Methods and Applications to Quantum Many-Body Systems |
| TableOfContents | 3.7 A Shot Summary -- References -- 4 Tensor Network Approaches for Higher-Dimensional Quantum Lattice Models -- 4.1 Variational Approaches of Projected-Entangled Pair State -- 4.2 Imaginary-Time Evolution Methods -- 4.3 Full, Simple, and Cluster Update Schemes -- 4.4 Summary of the Tensor Network Algorithms in HigherDimensions -- References -- 5 Tensor Network Contraction and Multi-Linear Algebra -- 5.1 A Simple Example of Solving Tensor Network Contraction by Eigenvalue Decomposition -- 5.1.1 Canonicalization of Matrix Product State -- 5.1.2 Canonical Form and Globally Optimal Truncations ofMPS -- 5.1.3 Canonicalization Algorithm and Some Related Topics -- 5.2 Super-Orthogonalization and Tucker Decomposition -- 5.2.1 Super-Orthogonalization -- 5.2.2 Super-Orthogonalization Algorithm -- 5.2.3 Super-Orthogonalization and Dimension Reduction by Tucker Decomposition -- 5.3 Zero-Loop Approximation on Regular Lattices and Rank-1 Decomposition -- 5.3.1 Super-Orthogonalization Works Well for Truncating the PEPS on Regular Lattice: Some Intuitive Discussions -- 5.3.2 Rank-1 Decomposition and Algorithm -- 5.3.3 Rank-1 Decomposition, Super-Orthogonalization, and Zero-Loop Approximation -- 5.3.4 Error of Zero-Loop Approximation and Tree-Expansion Theory Based on Rank-Decomposition -- 5.4 iDMRG, iTEBD, and CTMRG Revisited by Tensor Ring Decomposition -- 5.4.1 Revisiting iDMRG, iTEBD, and CTMRG: A Unified Description with Tensor Ring Decomposition -- 5.4.2 Extracting the Information of Tensor Networks From Eigenvalue Equations: Two Examples -- References -- 6 Quantum Entanglement Simulation Inspired by Tensor Network -- 6.1 Motivation and General Ideas -- 6.2 Simulating One-Dimensional Quantum Lattice Models -- 6.3 Simulating Higher-Dimensional Quantum Systems -- 6.4 Quantum Entanglement Simulation by Tensor Network:Summary -- References -- 7 Summary -- Index Intro -- Preface -- Acknowledgements -- Contents -- Acronyms -- 1 Introduction -- 1.1 Numeric Renormalization Group in One Dimension -- 1.2 Tensor Network States in Two Dimensions -- 1.3 Tensor Renormalization Group and Tensor Network Algorithms -- 1.4 Organization of Lecture Notes -- References -- 2 Tensor Network: Basic Definitions and Properties -- 2.1 Scalar, Vector, Matrix, and Tensor -- 2.2 Tensor Network and Tensor Network States -- 2.2.1 A Simple Example of Two Spins and Schmidt Decomposition -- 2.2.2 Matrix Product State -- 2.2.3 Affleck-Kennedy-Lieb-Tasaki State -- 2.2.4 Tree Tensor Network State (TTNS) and Projected Entangled Pair State (PEPS) -- 2.2.5 PEPS Can Represent Non-trivial Many-Body States: Examples -- 2.2.6 Tensor Network Operators -- 2.2.7 Tensor Network for Quantum Circuits -- 2.3 Tensor Networks that Can Be Contracted Exactly -- 2.3.1 Definition of Exactly Contractible Tensor Network States -- 2.3.2 MPS Wave-Functions -- 2.3.3 Tree Tensor Network Wave-Functions -- 2.3.4 MERA Wave-Functions -- 2.3.5 Sequentially Generated PEPS Wave-Functions -- 2.3.6 Exactly Contractible Tensor Networks -- 2.4 Some Discussions -- 2.4.1 General Form of Tensor Network -- 2.4.2 Gauge Degrees of Freedom -- 2.4.3 Tensor Network and Quantum Entanglement -- References -- 3 Two-Dimensional Tensor Networks and Contraction Algorithms -- 3.1 From Physical Problems to Two-Dimensional Tensor Networks -- 3.1.1 Classical Partition Functions -- 3.1.2 Quantum Observables -- 3.1.3 Ground-State and Finite-Temperature Simulations -- 3.2 Tensor Renormalization Group -- 3.3 Corner Transfer Matrix Renormalization Group -- 3.4 Time-Evolving Block Decimation: Linearized Contraction and Boundary-State Methods -- 3.5 Transverse Contraction and Folding Trick -- 3.6 Relations to Exactly Contractible Tensor Networks and Entanglement Renormalization |
| Title | Tensor Network Contractions |
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