Braids, Motions and Topological Quantum Computing

TL;DR

This survey explores how the mathematical theory of braids underpins topological quantum computing, emphasizing the role of braid groups, their physical relevance, and generalizations to knot motions.

Contribution

It provides an overview of the importance of braid theory in topological quantum computing and discusses its physical applications and higher-dimensional generalizations.

Findings

Braid groups are fundamental in fault-tolerant quantum computation.

Braiding appears in various physical systems involving anyons.

Generalizations to knot motions extend the mathematical framework.

Abstract

The topological model for quantum computation is an inherently fault-tolerant model built on anyons in topological phases of matter. A key role is played by the braid group, and in this survey we focus on a selection of ways that the mathematical study of braids is crucial for the theory. We provide some brief historical context as well, emphasizing ways that braiding appears in physical contexts. We also briefly discuss the 3-dimensional generalization of braiding: motions of knots.