From Rotations and Groups to Tensor Networks

E. Miles Stoudenmire, PhD
Research Scientist, Center for Computational Quantum Physics
Flatiron Institute, New York, NY 10010

Published: Spring 2023, Issue 2, pp. 29 – 83

Abstract: Even the simplest mathematical concepts can have surprisingly deep structure and hidden connections to other topics. In this article we explore rotations of two-dimensional objects. After viewing rotations first concretely as small matrices, then more abstractly as members of a “group”, we consider how rotations transform simple functions ($x,y,x^2 − y^2$ etc.) and their products. Transforming products of three or more functions is best captured by introducing multi-dimensional generalizations of vectors and matrices known as tensors and chaining these tensors together. The resulting object known as a tensor train (or matrix product state) is one of the most powerful computational tools for applications such as modeling fluid flows, understanding properties of metals and crystals, predicting financial markets, and simulating quantum computers.