  As the building blocks of topological quantum computation, Majorana zero
modes (MZMs) have attracted tremendous attention in recent years. Scalable
mesoscopic island designs with MZMs show great potential in quantum information
processing. However, these systems are susceptible to quasi-particle poisoning
which would induce various parity-breaking errors. To solve this problem, we
modify the mesoscopic islands with gate-tunable valves and non-topological
backbones. We study the lifetime of the Majorana qubits on these modified
islands which are coupled to local bosonic and fermionic thermal baths. We
consider both the parity-breaking and parity-preserving errors, and propose a
parity correction scheme. By using Jordan-Wigner transformation, we analyze the
probability of logical X and Y errors. The open quantum system is described by
the Pauli master equation, and standard Monte Carlo simulations are applied to
observe the behavior of the system when the parity correction proposal is
implemented. The results demonstrate that (1) our parity correction proposal is
effective to most of the parity-breaking errors; (2) the lifetime of the qubit
benefits from larger island size before it meets the threshold; (3) small
chemical potential $ \mu $ on the non-topological backbones and fine tuned
paring potential $ \Delta $ of the topological bulk segment are required for
high probability of correctness. Our results provide an effective error
correction scheme for the parity-breaking errors.
