Leveraging the higher harmonics content of the Josephson potential in a superconducting circuit offers a promising route in the search for new qubits with increased protection against decoherence. In this work, we demonstrate how the flux tunability of a hybrid semiconductor-superconductor Josephson junction based on a single full-shell nanowire enables this possibility. Near one flux quantum, $\mathrm{\Phi }\approx {\mathrm{\Phi }}_0=h/2e$, we find that the qubit system can be tuned from a gatemon regime to a parity-protected regime with qubit eigenstates localized in phase space in the ${\phi }_0=0$ and $\pi$ minima of the Josephson potential $(cos2{\phi }_0)$. Estimates of qubit coherence and relaxation times due to different noise sources are presented.

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• Mesoscopics
• Quantum control
• Superconducting qubits
• Josephson junctions
• Bogoliubov-de Gennes equations