  Hybrid quantum gates hold great promise for quantum information processing
since they preserve the advantages of different quantum systems. Here we
present compact quantum circuits to deterministically implement controlled-NOT,
Toffoli, and Fredkin gates between a flying photon qubit and diamond
nitrogen-vacancy (NV) centers assisted by microcavities. The target qubits of
these universal quantum gates are encoded on the spins of the electrons
associated with the diamond NV centers and they have long coherence time for
storing information, and the control qubit is encoded on the polarizations of
the flying photon and can be easily manipulated. Our quantum circuits are
compact, economic, and simple. Moreover, they do not require additional qubits.
The complexity of our schemes for universal three-qubit gates is much reduced,
compared to the synthesis with two-qubit entangling gates. These schemes have
high fidelities and efficiencies, and they are feasible in experiment.
