  High-dimensional quantum systems have been used to reveal interesting
fundamental physics and to improve information capacity and noise resilience in
quantum information processing. However, it remains a significant challenge to
realize universal two-photon quantum gates in high dimensions with high success
probability. Here, by considering an ion-cavity QED system, we theoretically
propose, to the best of our knowledge, the first high-dimensional,
deterministic and universal two-photon quantum gate. By using an optical cavity
embedded with a single trapped 40Ca+ ion, we achieve a high average fidelity
larger than 98% for a quantum controlled phase-flip gate in four-dimensional
space, spanned by photonic spin angular momenta and orbital angular momenta.
Our proposed system can be an essential building block for high-dimensional
quantum information processing, and also provides a platform for studying
high-dimensional cavity QED.
