Implementing a C$_k$NOT gate using Rydberg blockade

Multibit quantum gates are efficient primitives for quantum error correction and other algorithms. We report on an analysis of the fidelity and error scaling of a C$_k$NOT gate (CNOT gate with $k$ control bits) using Rydberg blockade. Provided all $k+1 $ bits lie within a blockade radius the gate can be implemented using a sequence of $2k+1$ Rydberg pulses. This is a more efficient scaling than would be obtained by decomposing the C$_k$NOT into one and two- qubit gates.