Majorana based parity exchange and measurement via quantum dot coupling in planar topological Josephson junctions

Majorana bound states are nucleated in planar topological Josephson junctions when the junction region is permeated by a uniform magnetic flux. Such junctions offer an advantage over traditional wire geometries in that the bound Majorana can be moved along the junction by means of strategically applied biasing voltage or current pulses. In this work, we consider a crossroad of four such planar junctions hosting Majorana bound states that can be coupled to a centrally placed quantum dot. We formulate a scheme to detect the parity of pairs of Majorana shuttled towards the quantum dot as well as to exchange parity between such pairs. We comment on how such schemes can be extended to obtain braiding of Majorana bound states in planar junctions through a series of projective measurements.