Tensor Networks in a Nutshell
Jacob Biamonte, Ville Bergholm
TL;DR
Tensor Networks in a Nutshell provides a compact, diagrammatic introduction to tensor networks and their use in quantum physics and combinatorial counting. It builds from fundamental Penrose-style graphical notation to practical constructions such as diagrammatic SVD and matrix product states, highlighting how entanglement and low-rank structure enable efficient representations and computations. The tutorial then demonstrates counting problems—like Boolean satisfiability and graph colorings—as tensor contractions, illustrating the broad applicability of tensor networks beyond physics. It closes with an overview of core architectures and frontier ideas, offering a practical toolkit for reasoning about quantum states and counting tasks using a unified graphical calculus.
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.