Majorana phase gate based on the geometric phase

We study the dynamics of a single qubit encoded in two pairs of Majorana modes, whereby each pair is hosted on a trijunction described by the Kitaev model extended by many-body interactions. We demonstrated that the challenging phase gate may be efficiently implemented via braiding of partially overlapping modes. Although such a qubit acquires both geometric and dynamical phases during the braiding protocol, the latter phase may be eliminated if the Majorana modes are hosted by systems with appropriate particle-hole symmetry.