Quantum circuits with multiterminal Josephson-Andreev junctions

We explore superconducting quantum circuits where several leads are simultaneously connected beyond the tunneling regime, such that the fermionic structure of Andreev bound states in the resulting multiterminal Josephson junction influences the states of the full circuit. Using a simple model of single channel contacts and a single level in the middle region, we discuss different circuit configurations where the leads are islands with finite capacitance and/or form loops with finite inductance. We find situations of practical interest where the circuits can be used to define noise protected qubits, which map to the bifluxon and $0{-}\pi$ qubits in the tunneling regime. We also point out the subtleties of the gauge choice for a proper description of these quantum circuits dynamics.