  Solid state spin qubits are promising candidates for the realization of a
quantum computer due to their long coherence times and easy electrical
manipulation. However, spin-spin interactions, which are needed for entangling
gates, have only limited range as they generally rely on tunneling between
neighboring quantum dots. This severely constrains scalability. Proposals to
extend the interaction range generally focus on coherent electron transport
between dots or on extending the coupling range. Here, we study a setup where
such an extension is obtained by using a superconductor as a quantum mediator.
Because of its gap, the superconductor effectively acts as a long tunnel
barrier. We analyze the impact of spin-orbit (SO) coupling, external magnetic
fields, and the geometry of the superconductor. We show that while spin
non-conserving tunneling between the dots and the superconductor due to SO
coupling does not affect the exchange interaction, strong SO scattering in the
superconducting bulk is detrimental. Moreover, we find that the addition of an
external magnetic field decreases the strength of the exchange interaction.
Fortunately, the geometry of the superconducting link offers a lot of room to
optimize the interaction range, with gains of over an order of magnitude from a
2D film to a quasi-1D strip. We estimate that for superconductors with weak SO
coupling (\textit{e.g.}, aluminum) exchange rates of up to 100\,MHz over a
micron-scale range can be achieved with this setup in the presence of magnetic
fields of the order of 100\,mT.
