  We experimentally investigate the viability of a variational quantum gate
optimization protocol informed by the underlying physical Hamiltonian of
fixed-frequency transmon qubits. The utility of the scheme is demonstrated
through the successful experimental optimization of two and three qubit quantum
gates tailored on the native cross-resonance interaction. The limits of such a
strategy are investigated through the optimization of a gate based on
Floquet-engineered three-qubit interactions, however parameter drift is
identified as a key limiting factor preventing the implementation of such a
scheme which the variational optimization protocol is unable to overcome.
