Long-range exchange interaction between spin qubits mediated by a superconducting link at finite magnetic field

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. Here, we study a setup where an extension of the tunneling range is obtained by using a superconductor as a quantum mediator. We analyze the impact of spin-orbit (SO) coupling, external magnetic fields, and the geometry of the superconductor. We show that while SO scattering in the superconducting bulk and the addition of an external magnetic field decrease the strength of the exchange interaction, 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 (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.