Braiding Majorana Fermions

TL;DR

This paper explores the mathematical structure of Majorana fermions using Clifford algebra, introduces the Clifford Braiding Theorem, and discusses their implications for topological quantum computing and braid group representations.

Contribution

It presents the Clifford Braiding Theorem, establishing a robust braid group representation derived from Majorana operators, linking physics, quantum information, and topology.

Findings

Majorana operators yield a robust braid group representation

Introduction of the Clifford Braiding Theorem

Analysis of Majorana fermions in quantum and topological contexts

Abstract

In this paper we study a Clifford algebra generalization of the quaternions and its relationship with braid group representations related to Majorana fermions. The Fibonacci model for topological quantum computing is based on the fusion rules for a Majorana fermion. Majorana fermions can be seen not only in the structure of collectivies of electrons, as in the quantum Hall effect, but also in the structure of single electrons both by experiments with electrons in nanowires and also by the decomposition of the operator algebra for a fermion into a Clifford algebra generated by two Majorana operators. The purpose of this paper is to discuss these braiding representations, important for relationships among physics, quantum information and topology. A new result in this paper is the Clifford Braiding Theorem. This theorem shows that the Majorana operators give rise to a particularly robust representation of the braid group that is then further represented to find the phases of the fermions under their exchanges in a plane space. The more robust representation in our braiding theorem will be the subject of further work.