Title: Tensor Networks in a Nutshell,
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.