Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks

TL;DR

This paper introduces the fundamentals of tensor networks and algorithms for studying quantum many-body systems, providing an accessible lecture series to help students understand complex quantum states and phases.

Contribution

It offers a comprehensive, introductory course on tensor networks, covering notation, properties, algorithms, and applications in quantum information and condensed matter physics.

Findings

Explains tensor network notation and properties

Classifies quantum phases using tensor networks

Provides algorithms for matrix product states

Abstract

The curse of dimensionality associated with the Hilbert space of spin systems provides a significant obstruction to the study of condensed matter systems. Tensor networks have proven an important tool in attempting to overcome this difficulty in both the numerical and analytic regimes. These notes form the basis for a seven lecture course, introducing the basics of a range of common tensor networks and algorithms. In particular, we cover: introductory tensor network notation, applications to quantum information, basic properties of matrix product states, a classification of quantum phases using tensor networks, algorithms for finding matrix product states, basic properties of projected entangled pair states, and multiscale entanglement renormalisation ansatz states. The lectures are intended to be generally accessible, although the relevance of many of the examples may be lost on students without a background in many-body physics/quantum information. For each lecture, several problems are given, with worked solutions in an ancillary file.