The use of multilevel information carriers, also known as qudits, is a promising path for exploring scalability of quantum computing devices. In this work, we present a proof-of-principle realization of a quantum processor that uses optically-addressed 171 Yb + ion qudits in a linear trap. The rich level structure of 171 Yb + ions makes it possible to use 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 two-qudit entangling 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.