  To implement useful quantum algorithms which demonstrate quantum advantage,
we must scale currently demonstrated quantum computers up significantly.
Leading platforms such as trapped ions face physical challenges in including
more information carriers. A less explored avenue for scaling up the
computational space involves utilizing the rich energy level structure of a
trapped ion to encode multi-level qudits rather than two-level qubits. Here we
show control and single-shot readout of qudits with up to 13 computational
states, using protocols which can be extended directly to manipulate qudits of
up to 25 levels in our chosen information host, $^{137}\text{Ba}^{+}$. This
represents more than twice as many computational states per qudit compared with
prior work in trapped ions. In addition to the preparation and readout
protocols we demonstrate, universal quantum computation requires other quantum
logic primitives such as entangling gates. These primitives have been
demonstrated for lower qudit dimensions and can be directly generalized to the
higher dimensions we employ. Hence, our advance opens an avenue towards using
high-dimensional qudits for large-scale quantum computation. We anticipate
efficiently utilizing available energy states in a trapped ion to play a
significant and complementary role in tackling the challenge in scaling up the
computational space of a trapped ion quantum computer. A qudit architecture
also offers other practical benefits, which include affording relaxed fault
tolerance thresholds for quantum error correction, providing an avenue for
efficient quantum simulation of higher spin systems, and more efficient qubit
gates.
