  Contemporary quantum computers encode and process quantum information in
binary qubits (d = 2). However, many architectures include higher energy levels
that are left as unused computational resources. We demonstrate a
superconducting ququart (d = 4) processor and combine quantum optimal control
with efficient gate decompositions to implement high-fidelity ququart gates. We
distinguish between viewing the ququart as a generalized four-level qubit and
an encoded pair of qubits, and characterize the resulting gates in each case.
In randomized benchmarking experiments we observe gate fidelities greater 95%
and identify coherence as the primary limiting factor. Our results validate
ququarts as a viable tool for quantum information processing.
