Majorana bound state parity exchanges in planar Josephson junctions

We describe a scheme to exchange fermion parity between two pairs of Majorana bound states mediated by coupling with a centralized quantum dot. We formulate such a scheme for Majorana bound states nucleated in the Josephson vortices formed in a fourfold crossroads junction of planar topological superconductors in the presence of a perpendicular magnetic field. This platform yields several advantages to the execution of our scheme as compared to similar ideas proposed in wire geometries, including control over the positions of the Majorana bound states (MBS) and hence a tunable coupling with the quantum dot. We show that moving the MBS along the junctions through voltage pulses can facilitate parity exchange via a two-step process, with intermediate projective measurements of the quantum dot charge. Thus we formulate a way to achieve single qubit operations for MBS in extended Josephson junctions through projective measurements of quantum dot charge. We also discuss the physical viability of our scheme with a particular focus on changes in quantum dot energy levels as a measurable indicator of the success of the scheme.