Diabatic errors in Majorana braiding with bosonic bath

Majorana mode based topological qubits are potentially subject to diabatic errors that in principle can limit the utility of topological quantum computation. Using a combination of analytical and numerical methods we study the diabatic errors in Majorana-based topological Y-junction that are coupled to a Bosonic bath in the Markovian approximation. From the study we find analytically that in the absence of a bath, the error rate can be made exponentially small in the braiding time only for completely smooth pulse shapes. Thus, pristine topological systems can reach exponentially small errors even for finite braid times. The presence of a dominantly dissipative Markovian bath is found to eliminate this exponential scaling of error to a power-law scaling as $T^{-1}$ with $T$ being the braiding time. However, the inclusion of relaxation imroves this scaling slightly to go as $T^{-2}$. Thus, coupling of topological systems to Bosonic baths can lead to powerlaw in braiding time diabatic errors that might limit the speed of topologically protected operations.