Lecture Notes of Tensor Network Contractions

TL;DR

This paper provides a comprehensive overview of tensor network contraction algorithms, their applications in quantum many-body simulations, and the underlying mathematical structures, highlighting recent developments and systematic understanding.

Contribution

It systematically reviews tensor network contraction methods, their applications, and the connections between different algorithms from a tensor algebra perspective.

Findings

Several paradigm algorithms are presented, including DMRG and TEBD.

Tensor network methods effectively simulate quantum many-body systems.

A unified view of TN algorithms from multilinear algebra is proposed.

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.