Quantum Contextuality and Entanglement-Free Grover Search in a Trapped-Ion Optical Qudit

Quantum computational advantage is generally attributed to coherent interference and other non-classical resources, yet their respective roles remain difficult to disentangle in experimental platforms where multipartite entanglement is inherently present. High-dimensional quantum systems provide an attractive route for investigating these resources while simultaneously reducing hardware overhead for quantum information processing. Here we realize a programmable four-dimensional optical qudit encoded in a single trapped $^{138}\mathrm{Ba}^{+}$ ion and demonstrate universal coherent control through phase-programmable optical rotations. Using this platform, we implement an entanglement-free realization of Grover's quantum search algorithm, achieving target-state identification probabilities of up to $94.5\pm2.0\%$. Within the same processor, we further demonstrate state-dependent quantum contextuality through a Clauser--Horne--Shimony--Holt (CHSH)-type noncontextuality inequality, obtaining a maximum violation of $S = 2.816 \pm 0.082$, in close agreement with the Tsirelson bound. By integrating programmable quantum computation and contextuality measurements within a single multilevel trapped-ion platform, our work establishes a versatile architecture for investigating the relationship between coherent interference and contextuality in quantum information processing and provides a scalable route toward high-dimensional quantum technologies.