Efficient implementation of a quantum algorithm with a trapped ion qudit

Demonstration of quantum advantage remains challenging due to the increased overhead of controlling large quantum systems. While significant effort has been devoted to qubit-based devices, qudits (d-level systems) offer potential advantages in both hardware efficiency and algorithmic performance. In this paper, we demonstrate multi-tone control of a single trapped ion qudit of up to eight levels, as well as the implementation of Grover's search algorithm on a qudit with dimensions five and eight, achieving operation fidelity of 96.8(3)% and 69(6)%, respectively, which correspond to 99.9(1)% and 97.1(3) % squared statistical overlap, respectively, with the expected result for a single iteration of the Grover search algorithm. The performance is competitive when compared to qubit-based systems; moreover, the sequence requires only $O\left(d\right)$ single-qudit gates and no entangling gates. This work highlights the potential of using qudits for efficient implementations of quantum algorithms.