  We employ quantum optimal control theory to realize quantum gates for two
protected superconducting circuits: the heavy-fluxonium qubit and the 0-$\pi$
qubit. Utilizing automatic differentiation facilitates the simultaneous
inclusion of multiple optimization targets, allowing one to obtain
high-fidelity gates with realistic pulse shapes. For both qubits, disjoint
support of low-lying wave functions prevents direct population transfer between
the computational-basis states. Instead, optimal control favors dynamics
involving higher-lying levels, effectively lifting the protection for a
fraction of the gate duration. For the 0-$\pi$ qubit, offset-charge dependence
of matrix elements among higher levels poses an additional challenge for gate
protocols. To mitigate this issue, we randomize the offset charge during the
optimization process, steering the system towards pulse shapes insensitive to
charge variations. Closed-system fidelities obtained are 99% or higher, and
show slight reductions in open-system simulations.

Quantum Physics;Mesoscale and Nanoscale Physics