Timescales for charge transfer based operations on Majorana systems

In this Rapid Communication we analyze the efficiency of operations based on transferring charge from a quantum dot (QD) to two coupled topological superconductors, which can be used for performing non-Abelian operations on Majorana bound states (MBSs). We develop a method which allows us to describe the full time evolution of the system as the QD energy is manipulated. Using a full counting statistics analysis, we set bounds to the operation timescales. The lower bound depends on the superconducting phase difference due to a partial decoupling of the different MBS parity sectors, while the upper bound is set by the tunneling of quasiparticles to the MBSs. Using realistic parameters, we find the existence of a regime where the operation can be carried out with a fidelity close to unity. Finally, we propose the use of a two-operation protocol to quantify the effect of the dephasing and accumulated dynamical phases, demonstrating their absence for certain superconducting phase differences.