Abstract
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, which scale as , with the cooperativity and the qubit number. 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.