  There is a recent surge of interest and insights regarding the interplay of
quantum optimal control and variational quantum algorithms. We study the
framework in the context of qudits which are, for instance, definable as
controllable electromagnetic modes of a superconducting cavity system coupled
to a transmon. By employing recent quantum optimal control approaches described
in (Petersson and Garcia, 2021), we showcase control of single-qudit operations
up to eight states, and two-qutrit operations, mapped respectively onto a
single mode and two modes of the resonator. We discuss the results of numerical
pulse engineering on the closed system for parametrized gates useful to
implement Quantum Approximate Optimization Algorithm (QAOA) for qudits. The
results show that high fidelity ($>$ 0.99) is achievable with sufficient
computational effort for most cases under study, and extensions to multiple
modes and open, noisy systems are possible. The tailored pulses can be stored
and used as calibrated primitives for a future compiler in circuit quantum
electrodynamics (cQED) systems.
