Abstract

Tensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum information science, statistical physics, and so on. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarse-graining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multi-linear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches.

Highlights

  • Comes mathematics, a new world we made by numbers and symbols, where the nature is reproduced by laws and theorems in an extremely simple, beautiful, and unprecedentedly accurate manner

  • In the following as two straightforward examples, we show that projected entangled pair state (PEPS) can represents non-trivial physical states including nearest-neighbor resonating valence bond (RVB) and Z2 spin liquid states

  • We mainly focused on the infinite PEPS (iPEPS) algorithm that simulates the ground states of 2D lattice models

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Summary

Introduction

Abstract One characteristic that defines us, human beings, is the curiosity of the unknown. We have been trying to use any methods that human brains can comprehend to explore the nature: to mimic, to understand, and to utilize in a controlled and repeatable way. Comes mathematics, a new world we made by numbers and symbols, where the nature is reproduced by laws and theorems in an extremely simple, beautiful, and unprecedentedly accurate manner. With the explosive development of digital sciences, computer was created. It provided us the third way to investigate the nature, a digital world whose laws can be ruled by ourselves with codes and algorithms to numerically mimic the real universe. We briefly review the history of tensor network algorithms and the related progresses made recently.

Numeric Renormalization Group in One Dimension
Tensor Network States in Two
Tensor Network States in Two Dimensions
Tensor Renormalization Group and Tensor Network Algorithms
Organization of Lecture
Organization of Lecture Notes
Scalar, Vector, Matrix, and Tensor
A Simple Example of Two Spins and Schmidt Decomposition
Matrix Product State
Affleck–Kennedy–Lieb–Tasaki State
Tensor Network
Tree Tensor Network State (TTNS) and Projected Entangled Pair State (PEPS)
PEPS Can Represent Non-trivial Many-Body States
Tensor Network Operators
Tensor Network for Quantum Circuits
Definition of Exactly Contractible Tensor Network States
MPS Wave-Functions
Tree Tensor Network Wave-Functions
MERA Wave-Functions
Sequentially Generated PEPS Wave-Functions
Exactly Contractible Tensor Networks
Gauge Degrees of Freedom
Tensor Network and Quantum Entanglement
Classical Partition Functions
Quantum Observables
Ground-State and Finite-Temperature Simulations
Tensor Renormalization Group
Corner Transfer Matrix Renormalization Group
Time-Evolving Block Decimation
Transverse Contraction and Folding Trick
Relations to Exactly Contractible Tensor Networks and Entanglement Renormalization
A Shot Summary
Variational Approaches of Projected-Entangled Pair State
Imaginary-Time Evolution Methods
Full, Simple, and Cluster Update Schemes
Summary of the Tensor Network Algorithms in Higher Dimensions
A Simple Example of Solving Tensor Network Contraction by Eigenvalue Decomposition
Canonicalization of Matrix Product State
Canonical Form and Globally Optimal Truncations of MPS
Canonicalization Algorithm and Some Related Topics
Super-Orthogonalization and Tucker Decomposition
Super-Orthogonalization
Super-Orthogonalization Algorithm
Super-Orthogonalization and Dimension Reduction by Tucker Decomposition
Super-Orthogonalization Works Well for Truncating the PEPS on Regular Lattice
Rank-1 Decomposition and Algorithm
Rank-1 Decomposition, Super-Orthogonalization, and Zero-Loop Approximation
Error of Zero-Loop Approximation and Tree-Expansion Theory Based on Rank-Decomposition
Revisiting iDMRG, iTEBD, and CTMRG: A Unified Description with Tensor Ring Decomposition
Extracting the Information of Tensor Networks From Eigenvalue Equations
Motivation and General Ideas
Simulating One-Dimensional Quantum Lattice Models
Simulating Higher-Dimensional Quantum Systems
Summary
Methods
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