Cavity polariton blockade for non-local entangling gates with trapped atoms

We propose a scheme for realizing multi-qubit entangled W-state and non-local $CZ$ and $C_{2}Z$ gates via a cavity polariton blockade mechanism with a system of atomic qubits coupled to a common cavity mode. The polariton blockade is achieved by tuning the system, an $N-$qubit register, such that no two atoms are simultaneously excited to the qubit excited state, and there is an effective coupling only between the ground state and a singly-excited W state of the qubit register. The control step requires only an external drive of the cavity mode and a global qubit pulse and no individual qubit addressing. We analytically obtain the state preparation error for an $N-$qubit W state which scales as $\sqrt{(1-1\mathrm{/}N)}\mathrm{/}\sqrt{C}$ where $C$ is the single particle cooperativity. We additionally show the application of the polariton blockade mechanism in realizing a non-local $CZ$ and $C_{2}Z$ gate by using a different set of computational qubit states, and characterize the gate errors which scale as $\sim 1\mathrm{/}\sqrt{C}$.