From Quantum Groups to Unitary Modular Tensor Categories

TL;DR

This paper surveys the current methods and progress in deriving unitary modular tensor categories from quantum groups, focusing on explicit computations relevant to quantum computing applications.

Contribution

It provides a comprehensive overview of approaches to construct unitary modular tensor categories from quantum groups, highlighting explicit computational techniques and recent developments.

Findings

Quantum groups are a key source of modular tensor categories.

Explicit computational methods are advancing the understanding of unitarity.

Progress is being made towards physically feasible models for quantum computing.

Abstract

Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are "physically feasible." We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasizing explicit computations.