Active correction of fermionic parity-preserving errors in an individual Majorana qubit*

Quantum information stored in the Majorana modes of a topological superconductor is only partially protected against fermionic parity-preserving local errors, due to finite-size effects. We address the problem whether total protection against such errors can be achieved by employing active error-correction techniques on an individual Majorana qubit. Majorana qubits in the tetron architecture are modeled by a pair of Kitaev chains. We show numerically that, for any parameter values in the topological phase, the two degenerate ground states that act as the physical qubit levels form an approximate error-correcting code, which can correct fermionic parity-preserving local errors on any one of the chains. We further propose a method to construct quasi-local syndrome measurements and correction procedures, and demonstrate our method for a few generic parameter values. Our results are achieved by re-expressing the spectrally flattened Hamiltonian of the Kitaev chain as a sum of commuting quasi-local terms. Our results indicate that each individual topological qubit can be made fully robust against fermionic parity-preserving local errors, thereby reducing the number of physical qubits required for fault-tolerant computation in comparison to the existing schemes.