  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 $\sim N/\sqrt{C}$, with $C$ the cooperativity and $N$ 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.

Quantum Physics;Other Condensed Matter