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

High fidelity quantum information processing requires a combination of fast\ngates and long-lived quantum memories. In this work, we propose a hybrid\narchitecture, where a parity-protected superconducting qubit is directly\ncoupled to a Majorana qubit, which plays the role of a quantum memory. The\nsuperconducting qubit is based upon a $\\pi$-periodic Josephson junction\nrealized with gate-tunable semiconducting wires, where the tunneling of\nindividual Cooper pairs is suppressed. One of the wires additionally contains\nfour Majorana zero modes that define a qubit. We demonstrate that this enables\nthe implementation of a SWAP gate, allowing for the transduction of quantum\ninformation between the topological and conventional qubit. This architecture\ncombines fast gates, which can be realized with the superconducting qubit, with\na topologically protected Majorana memory.\n