Effects of decoherence on diabatic errors in Majorana braiding

The braiding of two non-Abelian Majorana modes is important for realizing topological quantum computation. It can be achieved through tuning the coupling between the two Majorana modes to be exchanged and two ancillary Majorana modes. However, this coupling also makes the braiding subject to environment-induced decoherence. Here, we study the effects of decoherence on the diabatic errors in the braiding process for a set of time-dependent Hamiltonians with finite smoothness. To this end, we employ the master equation to calculate the diabatic excitation population for three kinds of decoherence processes. (1) Only pure dehasing: the scaling of the excitation population changed from $T^{-2k-2}$ to $T^{-1}$ ($k$ is the number of the Hamiltonian's time derivatives vanishing at the initial and final times) as the braiding duration $T$ exceeds a certain value. (2) Only relaxation: the scaling transforms from $T^{-2k-2}$ to $T^{-2}$ for $k=0$ and to $T^{-a}$ ($a>3$) for $k>0$. (3) Pure dephasing and relaxation: the original scaling switches to $T^{-1}$ firstly and then evolves to $T^{-2}$ in the adiabatic limit. Interestingly, the third scaling-varying style holds even when the expectation of pure dephasing rate is much smaller than that of the relaxation rate, which is attributed to the vanishing relaxation at the turning points of the braiding.