  It is desirable to implement an efficient quantum information process
demanding fewer quantum resources. We designed two compact quantum circuits for
determinately implementing four-qubit Toffoli and Fredkin gates on
single-photon systems in both the polarization and spatial degrees of freedom
(DoFs) via diamond nitrogen-vacancy (NV) centers in resonators. The gates are
heralded by the electron spins associated with the diamond NV centers. In
contrast to the ones with one DoF, our implementations reduce the quantum
resource and are robust against the decoherence. Evaluations of fidelities and
efficiencies of our gates show that our schemes may be implemented with current
technology.
