This is a tutorial review of methods to braid the world lines of non-Abelian

anyons (Majorana zero-modes) in topological superconductors. That "Holy Grail"

of topological quantum information processing has not yet been reached in the

laboratory, but there now exists a variety of platforms in which one can search

for the Majorana braiding statistics. After an introduction to the basic

concepts of braiding we discuss how one might be able to braid immobile

Majorana zero-modes, bound to the end points of a nanowire, by performing the

exchange in parameter space, rather than in real space. We explain how Coulomb

interaction can be used to both control and read out the braiding operation,

even though Majorana zero-modes are charge neutral. We ask whether the fusion

rule might provide for an easier pathway towards the demonstration of

non-Abelian statistics. In the final part we discuss an approach to braiding in

real space, rather than parameter space, using vortices injected into a chiral

Majorana edge mode as "flying qubits".

Contents:

I. Introduction

II. Basic Concepts (The magic of braiding; Non-Abelian statistics; Fusion

rules; Clifford gates; Topological protection)

III. Braiding of Majorana zero-modes in nanowires (The three-point turn;

Non-Abelian Berry phase; Coulomb-assisted braiding; Anyon teleportation)

IV. Read-out of Majorana qubits (Majorana interferometry; Inductive coupling

to a flux qubit; Microwave coupling to a transmon qubit; Capacitive coupling to

a quantum dot; Random Access Majorana Memory)

V. Fusion of Majorana zero-modes in nanowires (Linear junction or

tri-junction; If we can fuse, do we need to braid?)

VI. How to braid Majorana edge modes (Chiral edge modes in a superconductor;

Edge vortex injection; Construction of the vortex operator; Edge vortex

braiding)

VII. Outlook on the experimental progress (to be added later)