High-Dimensional Photonic Quantum Computing with a Measurement-Free Auxiliary System

Enhancing the capabilities of quantum computing relies heavily on harnessing the power of qudit-based high-dimensional quantum gates. In the study, single-qudit 4D $X$, $X^2$, and $X^{†}$ gates tailored for a two-photon system in polarization states are presented. Furthermore, a two-qudit $4\times 4$-dimensional controlled-not (CNOT) gate designed for a four-photon system is introduced. These high-dimensional gates can offer versatile and straightforward optical implementations, ensuring them to fulfill in a deterministic way. To facilitate these processes, an auxiliary system in the form of a $\Lambda$-type atom residing in a cavity is employed. Remarkably, the auxiliary system retains its original state after the operation process ends, so it is not required to measure and plays a pivotal role in promoting effective interactions among distinct photons in its extended coherence time. Importantly, the in-depth analysis of the fidelities and efficiencies of these quantum gates showcase remarkable outcomes, affirming the superiority of the proposed protocols. Therefore, these high-dimensional gates not only amplify quantum parallelism, but also bolster the speed of quantum computations, fortify resilience against errors, and foster scalability for executing intricate quantum operations.