  The notion of universal quantum computation can be generalized to multi-level
qudits, which offer advantages in resource usage and algorithmic efficiencies.
Trapped ions, which are pristine and well-controlled quantum systems, offer an
ideal platform to develop qudit-based quantum information processing. Previous
work has not fully explored the practicality of implementing trapped-ion qudits
accounting for known experimental error sources. Here, we describe a universal
set of protocols for state preparation, single-qudit gates, a new
generalization of the M\o{}lmer-S\o{}rensen gate for two-qudit gates, and a
measurement scheme which utilizes shelving to a meta-stable state. We
numerically simulate known sources of error from previous trapped ion
experiments, and show that there are no fundamental limitations to achieving
fidelities above \(99\%\) for three-level qudits encoded in
\(^{137}\mathrm{Ba}^+\) ions. Our methods are extensible to higher-dimensional
qudits, and our measurement and single-qudit gate protocols can achieve
\(99\%\) fidelities for five-level qudits. We identify avenues to further
decrease errors in future work. Our results suggest that three-level trapped
ion qudits will be a useful technology for quantum information processing.
