Microsoft has invested huge resources into engineering topological qubits. Their approach is based on topological Majorana states, which occur at the edges of a topological superconducting chain or at interfaces between such chains. For those who see these words for the first time, a quick mental representation is supplied by a ribbon, which can be twisted a number of times, either clockwise or counter-clockwise. As long as the ends of the ribbon are held fixed, the information stored in the those windings remains safe against external factors. Unfortunately, this is a classical picture that needs to be lifted to the quantum realm. Nevertheless, these ribbons are 1-dimensional and anyons usually occur in 2-dimensions. The trick is to use a network of ribbons, which can be connected and disconnected such that the windings can be passed from one ribbon to another. In fact, there is a trick called the T-junction process, which uses the coupling of three wires to reverse the order of the Majorana anyons in a wire. See
J. Alicea, Y. Oreg, G. Refael, F. v. Oppen and M. P. A. Fisher, Non-Abelian statistics and topological quantum information processing in 1D wire networks, Nature Physics 7, 412, (2011).
These exchanges of anyons supply the elementary unitary operations one uses in topological quantum computation. In any case, some news about Microsoft's efforts are highlighted here:
The sad reality is that: ``One drawback of Microsoft's topological qubit, though, is that they're not available yet." The word is that the effort towards a topological quantum computer has reached the point of make-or-break and we will definitely hear more news soon from Microsoft.
Lastly, a comment about graphene and 2-systems. Evidence of non-abelian anyon presence is routinely observed in the dynamics of electrons trapped at the interface between semiconductor films (quantum wells) when subjected to strong magnetic fields. However, manipulating these anyons is an entirely different task. While similar evidence was observed in graphene when subjected to strong magnetic fields,
graphene is not actively researched for the purpose of topological quantum computation.