A deterministic qudit-dependent high-dimensional photonic quantum gate with weak cross-kerr nonlinearity

High-dimensional quantum systems expand the Hilbert space, enabling the processing of more intricate information and the execution of wider quantum operations. The development of high-dimensional quantum systems relies significantly on the implementation of high-dimensional gates. In the paper, we propose a deterministic 4 × 4-dimensional controlled-not (CNOT) gate for a three-photon system, where encoding on the polarization-spatial state of a single photon acts as a 4-dimensional control qudit, meanwhile the polarized state of the remaining two photons serves as a 4-dimensional target qudit. The implementation of the gate leverages weak cross-kerr nonlinearities and X-homodyne detectors, which impose lower requirements on the interaction strength and demonstrate robustness against photon loss. In accordance with the measurement of the coherent state, the relevant classical feed-forward operations are employed to make high-dimensional CNOT gate effective. Additionally, the proposed scheme demonstrates a high successful probability as a result of the utilization of mature measurement methods and straightforward optical elements. Further, the fidelity of the CNOT gate is robust against photon loss and is approximate to 1 with the existing technology. Consequently, our deterministic protocol presents a promising approach for the practical realization of high-dimensional quantum computation for photon systems.