Erasure conversion in Majorana qubits via local quasiparticle detection

Abstract

Quasiparticle poisoning errors in Majorana-based qubits are not suppressed by the underlying topological properties, which undermines the usefulness of this proposed platform. This work tackles the errors originating from intrinsically excited quasiparticles by developing an erasure conversion scheme based on local quasiparticle detection. To model such measurements, we begin by constructing the quasiparticle position operator for the Kitaev chain. A measurement probe coupling to this operator is shown to allow projective measurements in the Wannier quasiparticle basis. Detection of quasiparticles in a region of width $d$ adjacent to each Majorana zero-energy mode allows implementation of an error-detecting Majorana stabilizer code $\mathcal{C}_d$ based on microscopic fermionic (non-topological) physical degrees of freedom. The implementation of $\mathcal{C}_d$ converts a large fraction of Pauli errors to erasure errors, thus achieving `erasure conversion' in Majorana qubits. We show that the fraction of Pauli errors escaping conversion to erasure errors is exponentially small in $d$, a result tied to the exponential localization of Wannier functions which we prove rigorously. The suppression in Pauli error rate comes at the cost of the erasure rate increasing sublinearly with $d$, but this can be readily compensated for by a suitable outer code, with the net effect being a higher threshold rate of quasiparticle poisoning. The framework developed here serves as a basis for understanding how realistic measurements, such as conductance measurements, could be utilized for achieving fault tolerance in these systems.