Abstract
This is a tutorial review of methods to braid 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''.
Highlights
This is a tutorial review of methods to braid non-Abelian anyons (Majorana zero-modes) in topological superconductors
Charge e/4 quasiparticles in the ν = 5/2 quantum Hall effect were the first candidates for non-Abelian anyons [4], followed by vortices in topological superconductors [5, 6]
The Majorana fermions that propagate along the edge of a topological superconductor [5] have conventional fermionic exchange statistics, while the non-Abelian anyons are midgap states (“zero-modes”) bound to a defect and are typically immobile
Summary
An operational description of braiding has the magical flavor of a cups and balls performance. The vortices share an unpaired electron or hole when the upper level is occupied, while all fermions are paired if the upper level is empty. The representation Pkl = 1 − 2ak†l akl in terms of fermionic creation and annihilation operators ak†l and akl is inconvenient because it does not distinguish the contributions from the individual vortices k, l that make up the pair. Each vortex is the origin of a 2π branch cut in the phase φ of the superconducting pair potential, corresponding to a π phase jump for fermion operators. The initial state |+〉|−〉 of even–odd fermion parity is converted into the odd–even state |−〉|+〉, meaning that a fermion has been exchanged between vortex pairs.
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