Tensor Networks in a Nutshell

TL;DR

This paper provides an accessible introduction to tensor network methods, illustrating their use in quantum physics and combinatorial problems through graphical language and practical examples.

Contribution

It offers a quick, beginner-friendly tutorial on tensor networks, covering key concepts, graphical notation, matrix product states, and applications in counting problems.

Findings

Tensor networks efficiently approximate certain quantum states.

Graphical language simplifies reasoning about quantum circuits.

Tensor contractions can solve combinatorial counting problems.

Abstract

Tensor network methods are taking a central role in modern quantum physics and beyond. They can provide an efficient approximation to certain classes of quantum states, and the associated graphical language makes it easy to describe and pictorially reason about quantum circuits, channels, protocols, open systems and more. Our goal is to explain tensor networks and some associated methods as quickly and as painlessly as possible. Beginning with the key definitions, the graphical tensor network language is presented through examples. We then provide an introduction to matrix product states. We conclude the tutorial with tensor contractions evaluating combinatorial counting problems. The first one counts the number of solutions for Boolean formulae, whereas the second is Penrose's tensor contraction algorithm, returning the number of $3$-edge-colorings of $3$-regular planar graphs.