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. We find that in the absence of a bath, the error can be made exponentially small with increasing braiding time, only when the time variation in the Hamiltonian is completely smooth. 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 improves this scaling slightly to go as T-2. Thus, coupling of topological systems to Bosonic baths can lead to power law in braiding time diabatic errors that might limit the speed of topologically protected operations using Majorana modes.