The ribbon category framework for topological quantum computing

TL;DR

This paper explains the mathematical framework of ribbon fusion categories, crucial for understanding topological quantum computing, focusing on braid representations and their origins from quantum groups, with accessible explanations for newcomers.

Contribution

It provides an accessible exposition of ribbon fusion categories and their role in topological quantum computing, linking category theory with quantum group representations.

Findings

Clarifies the structure of ribbon fusion categories

Details the braiding in RFCs relevant to quantum computing

Connects RFCs with quantum group representation theory

Abstract

This expository article supplies the mathematical background underpinning the braid representation calculator introduced in arXiv:2212.00831; those representations describe the sets of logic gates available to a topological quantum computer for processing encoded qubits. Assuming little background in category theory, we first recall the notion of a ribbon fusion category (RFC), collecting most of the necessary definitions. Then we discuss how certain RFCs arise from the representation theory of quantum groups. We explore the braiding in these categories in detail, since it is essential for the quantum computing application.