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There is a recent surge of interest and insights regarding the interplay ofquantum optimal control and variational quantum algorithms. We study theframework in the context of qudits which are, for instance, definable ascontrollable electromagnetic modes of a superconducting cavity system coupledto a transmon. By employing recent quantum optimal control approaches describedin (Petersson and Garcia, 2021), we showcase control of single-qudit operationsup to eight states, and two-qutrit operations, mapped respectively onto asingle mode and two modes of the resonator. We discuss the results of numericalpulse engineering on the closed system for parametrized gates useful toimplement Quantum Approximate Optimization Algorithm (QAOA) for qudits. Theresults show that high fidelity ($>$ 0.99) is achievable with sufficientcomputational effort for most cases under study, and extensions to multiplemodes and open, noisy systems are possible. The tailored pulses can be storedand used as calibrated primitives for a future compiler in circuit quantumelectrodynamics (cQED) systems.

Numerical Gate Synthesis for Quantum Heuristics on Bosonic Quantum Processors