Dephasing of Charging-Energy-Protected Majorana Zero Mode Qubits

We analyze charging-energy-protected Majorana zero mode (MZM) qubits, focusing on the residual dephasing that is present when the distance between MZMs is insufficient for full topological protection. We argue that the leading source of dephasing is $1/f$ charge noise. This noise affects the qubit as a result of the hybridization energy and charge distribution associated with weakly-overlapping MZMs, which we calculate using a charge-conserving formalism. In contrast to mean-field descriptions, the latter emphasizes that a finite MZM overlap leads to the formation of a charge dipole rather than an absolute charge build-up. We estimate the coherence time to be of the order of a hundred nanoseconds for MZM qubits whose MZM separation is $L\sim 5\xi$ (with $\xi$ being the coherence length). The coherence time grows exponentially with MZM separation and eventually becomes temperature-limited, reaching values of the order of minutes for $L/\xi \sim 30$.