Citation: Alicea, Jason, et al. "Non-Abelian statistics and topological quantum information processing in 1D wire networks." Nature Physics 7.5 (2011): 412-417.
Web: https://arxiv.org/abs/1006.4395
Tags: Physical, Ising-computer, Majorana-fermions, Spin-chains
In this paper, the authors demonstrate how to braid Majorana fermions in 1D wire networks. Importantly: you never have to couple the topological behavior of the wire to any bulk topological degrees of freedom, and you never need to physically more the wires. The protocol looks something like the classic move where you use a T-junction to make room for braiding. The move is simple, but you then have to argue from first-principles that the logical operation you get is actually braiding. To show that this is nontrivial, the argument that vortices in p+ip superconductors have Majorana braiding is a celebrated result:
> Ivanov, Dmitri A. "Non-Abelian statistics of half-quantum vortices in p-wave superconductors." Physical review letters 86.2 (2001): 268.
This paper gives a nice little argument. Alternatively, you can use an argument I learned from Alexei recently. You can track the logical operators in the 2D Fock space through the whole process, and watch rather explicitly how the logical operations change. From this tracking process you find that the braiding operation is actually a standard Majorana braiding operator. This serves as a basis for the braiding protocols to be used in topological quantum computers.
Note that the wires never move. The thing that moves is the interface between the trivial and non-trivial phases, which can be moved by tuning a space-dependent potential.