SWAP gate between a Majorana qubit and a parity-protected superconducting qubit

In a noisy quantum computing environment, fast gates are necessary to overcome limitations from decoherence. At the same time, the storage of quantum information requires long-lived disconnected subspaces, where information can be stored before and after actual quantum computation.  We propose transduction of information by coupling a Majorana qubit to a parity-protected superconducting qubit based on π-periodic Josephson junctions, where tunneling of pairs of Cooper pairs preserves the Cooper pair parity. In the regime EC ≤ E_J the charging energy can resolve one Cooper pair in the superconducting island and thus distinguish the two states of the Majorana qubit. We show how this enables the implementation of a C-phase gate, and through proper driving of the Majorana qubit we can obtain a SWAP gate between the two. The point of the coupled circuit is that fast gates can be performed with the superconducting qubit and the information can be swapped and stored in the Majorana qubit.