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Qudit, a high-dimensional quantum system, provides a larger Hilbert space toprocess the quantum information and has shown remarkable advantages over thequbit counterparts. It is a great challenge to realize the high fidelityuniversal quantum gates with qudits. Here we theoretically propose andexperimentally demonstrate a set of universal quantum gates for a singleoptical qudit with four dimensions (including the generalized Pauli $X_4$ gate,Pauli $Z_4$ gate, and all of their integer powers), which are encoded in thepolarization-spatial degree of freedom without multiple unstable cascadedinterferometers. Furthermore, we also realize the controlled-$X_4$ gate and allof its integer powers. We have achieved both the ultra-high average gatefidelity $99.73\%$ and efficiency $99.47\%$, which are above the the errorthreshold for fault-tolerant quantum computation. Our work paves a way for thelarge-scale high-dimensional fault-tolerant quantum computation with apolynomial resource cost.

Experimental realization of universal high-dimensional quantum gates with ultra-high fidelity and efficiency