Quantum logic gates are the foundation of circuit-based quantum computation and quantum simulation. Multi-qubit quantum controlled gates are of vital importance when large-scale quantum circuits are concerned. Here, assisted by diamond nitrogen-vacancy (NV) centers inside single-sided optical cavities, we first introduce the single-photon quantum routing operation, with which a single-photon polarization qubit can be coherently routed to the superposition of two spatial modes according to the state of the other polarization qubit. On this basis, we propose two schemes for the implementation of universal three-qubit quantum Fredkin- and Toffoli-gate operations on single-photon polarization qubits. Moreover, we show that a general multi-qubit controlled-unitary gate can be realized using the same strategy. Compared to other NV-cavity-based schemes, our schemes utilize photonic polarization, spatial, and temporal degrees of freedom (DOFs) simultaneously, so that the quantum circuits are simplified and the required quantum resources are reduced, making our schemes more resource-efficient and highly scalable. Our calculations show that the schemes can posses near-deterministic efficiency and near-unity fidelity with current or near-future techniques, which guarantees the experimental feasibility. Our schemes can therefore provide a new recipe for the construction of multi-qubit quantum gates and for efficient processing of multi-photon states.
