  Physical quantum systems are commonly composed of more than two levels and
offer the capacity to encode information in higher-dimensional spaces beyond
the qubit, starting with the three-level qutrit. Here, we encode neutral-atom
qutrits in an ensemble of ultracold $^{87}$Rb and demonstrate arbitrary
single-qutrit SU(3) gates. We generate a full set of gates using only two
resonant microwave tones, including synthesizing a gate that effects a direct
coupling between the two disconnected levels in the three-level
$\Lambda$-scheme. Using two different gate sets, we implement and characterize
the Walsh-Hadamard Fourier transform, and find similar final-state fidelity and
purity from both approaches. This work establishes the ultracold neutral-atom
qutrit as a promising platform for qutrit-based quantum information processing,
extensions to $d$-dimensional qudits, and explorations in multilevel quantum
state manipulations with nontrivial geometric phases.
