We present two protocols for implementing deterministic non-local multi-qubit quantum gates on qubits coupled to a common cavity mode. The protocols rely only on a classical drive of the cavity modes, while no external drive of the qubits is required. In the first protocol, the state of the cavity follows a closed trajectory in phase space and accumulates a geometric phase depending on the state of the qubits. The second protocol uses an adiabatic evolution of the combined qubit-cavity system to accumulate a dynamical phase. Repeated applications of this protocol allow for the realization of phase gates with arbitrary phases, e.g. phase-rotation gates and multi-controlled-Z gates. For both protocols, we provide analytic solutions for the error rates. Our protocols are applicable to a variety of systems and can be generalized by replacing the cavity by a different bosonic mode, such as a phononic mode. We provide estimates of gate fidelities and durations for atomic and molecular qubits coupled to optical or microwave cavities, and suggest applications for quantum error correction.
S. Jandura, V. Srivastava, G. Brennen, G. Pupillo, arXiv:2303.13127