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We present a model of two Anderson impurities coupled to and through asuperconducting island. The model parametrizes the strength of the couplingbetween impurity sites, allowing it to represent a variable distance betweenthe impurities. We systematically explore the effect of the model parameters inthe subspaces with total even and odd occupancy, identifying unique features ofthe charge stability diagrams that distinguish the two parities. For total oddelectron parity, we identify a device tuning, in which the splitting betweenthe two lowest energy states is highly insensitive to changes of the chemicalpotentials. We investigate the degree to which a qubit based on such two statesis tunable and discuss an optimal choice of parameters to maximizeinhomogeneous dephasing time. Finally, we point out that the proposed qubitlacks a transition dipole moment, and outline some of the consequences onviable driving and readout mechanisms. The prototype qubits could be realizedwith the existing capabilities of coupling quantum dots to hard-gappedsuperconductors.

Two Anderson impurities coupled through a superconducting island: charge stability diagrams and double impurity qubit