  The cross-resonant gate is an entangling gate for fixed frequency
superconducting qubits introduced for untunable qubits. While being simple and
extensible, it suffers from long duration and limited fidelity. Using two
different optimal control algorithms, we probe the quantum speed limit for a
CNOT gate in this system. We show that the ability to approach this limit
depends strongly on the ansatz used to describe the optimal control pulse. A
piecewise constant ansatz with a single carrier leads to an experimentally
feasible pulse shape, shorter than the one currently used in experiments, but
that remains relatively far from the speed limit. On the other hand, an ansatz
based on the two dominant frequencies involved in the optimal control problem
allows to generate an optimal solution more than twice as fast, in under
$30$ns. This comes close to the theoretical quantum speed limit, which we
estimate at $15$ns for typical circuit-QED parameters, which is more than an
order of magnitude faster than current experimental microwave-driven
realizations, and more than twice as fast as tunable direct-coupling
experimental realizations.
