Entanglement of encoded spin-qubits via curvature couplings to a superconducting cavity

We propose entangling operation and preparation procedures based on curvature couplings of encoded spin qubits to a superconducting cavity, exploring the non-linear qubit response to a voltage variation. For two-qubit (n-qubit) entangling gate we explore acquired geometric phases via a time-modulated longitudinal σ_z-coupling, offering gate times of 10s of ns. No dipole moment is necessary: the qubit transverse σ_x-coupling to the resonator is zero at full sweet spot. This approach allows always-on, exchange-only qubits, for example, to stay on their ``sweet spots'' during gate operations, minimizing the charge noise and eliminating an always-on static longitudinal qubit-qubit coupling. We calculate gate errors due to the diffusion noise and damping of the resonator, the qubit charge dephasing, and a static spin-dependent resonator frequency shift (via a ``dispersive-like'' curvature coupling). Using spin-echo-like error suppression at optimal regimes, gate infidelities 10^-2-10^-3 can be achieved. For entangling preparation, one uses designated resonators to perform joint n-qubit quantum measurements with entangling times of 10s of ns, exploring both longitudinal and dispersive-like curvature couplings. The proposed schemes seem suitable for remote spin-to-spin entanglement.