Long-range two-qubit gate between nuclear spins in diamond mediated via an optical cavity

Nitrogen-vacancy (NV) centers in diamond represent a promising possibility for a solid-state based realization of a qubit due to their excellent electron- and nuclear-spin coherence properties. Single-qubit gates for the nitrogen nuclear spin have been implemented [1]. Here, we extend an earlier proposal [2] for cavity-mediated coupling between NV electron spins and develop a scheme to implement a universal two-qubit gate between $^{14}$N or $^{15}$N nuclear spins. By virtually exciting a single NV center with an external laser field, a photon can be scattered into a surrounding cavity; we show that this process depends on the spin state of the nitrogen nucleus. For the two-qubit gate, we consider two NV centers coupled to a common cavity mode and each being excited individually. Virtual cavity excitation can then mediate an effective interaction between the NV nuclear spin qubits, generating a controlled-$\textit{Z}$ gate. Operation times for the gate implementation are found to be below 100 nanoseconds, which is orders of magnitude faster than the decoherence time of nuclear spin qubits in diamond.\\[4pt] [1] S.~Sangtawesin \textit{et al.}, Phys.~Rev.~Lett.~\textbf{113}, 020506 (2014).\newline [2] G.~Burkard and D.~D.~Awschalom, arXiv:1402.6351.