Nonlocal multiqubit quantum gates via a driven cavity

We present two protocols for realizing deterministic nonlocal multiqubit quantum gates on qubits coupled to a common cavity mode. The protocols rely only on a classical drive of the cavity mode, while no external drive of the qubits is required. Applied to just two qubits, both protocols provide a universal gate set for quantum computing, together with single-qubit gates. 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. This geometric phase gate can be used together with global single-qubit gates to generate high-fidelity Greenberger--Horne--Zeilinger (GHZ) states. 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 a family 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 $\ensuremath{\sim}N/\sqrt{C}$ in the presence of relevant losses, 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 as well as superconducting fluxonium qubits coupled to optical or microwave cavities, and we outline implications for quantum error correction.