  Superconducting spin qubits, also known as Andreev spin qubits, promise to
combine the benefits of superconducting qubits and spin qubits defined in
quantum dots. While most approaches to control these qubits rely on controlling
the spin degree of freedom via the supercurrent, superconducting spin qubits
can also be coupled to each other via the superconductor to implement two-qubit
quantum gates. We theoretically investigate the interaction between
superconducting spin qubits in the weak tunneling regime and concentrate on the
effect of spin-orbit interaction (SOI), which can be large in
semiconductor-based quantum dots and thereby offers an additional tuning
parameter for quantum gates. We find analytically that the effective
interaction between two superconducting spin qubits consists of Ising,
Heisenberg, and Dzyaloshinskii-Moriya interactions and can be tuned by the
superconducting phase difference, the tunnel barrier strength, or the SOI
parameters. The Josephson current becomes dependent on SOI and spin
orientations. We demonstrate that this interaction can be used for fast
controlled phase-flip gates with a fidelity >99.99%. We propose a scalable
network of superconducting spin qubits which is suitable for implementing the
surface code.
