  The use of multilevel information carriers, also known as qudits, is a
promising path for exploring scalability of quantum computing devices. Here we
present a proof-of-principle realization of a quantum processor register that
uses optically-addressed $^{171}$Yb$^{+}$ ion qudits in a linear trap. The rich
level structure of $^{171}$Yb$^{+}$ ions allows using the Zeeman sublevels of
the quadrupole clock transition at 435.5 nm for efficient and robust qudit
encoding. We demonstrate the realization of the universal set of gates
consisting of single-qudit rotations and a two-qudit Molmer-Sorensen operation
with a two-ququart system, which is formally equivalent to a universal
gate-based four-qubit processor. Our results paves a way towards further
studies of more efficient implementations of quantum algorithms with
trapped-ion-based processors and, specifically, exploring properties of
$^{171}$Yb$^{+}$ ion qudits.
