  We present an iterative scheme to estimate the minimal duration in which a
quantum gate can be realized while satisfying hardware constraints on the
control pulse amplitudes. The scheme performs a sequence of unconstrained
numerical optimal control cycles that each minimize the gate fidelity for a
given gate duration alongside an additional penalty term for the control pulse
amplitudes. After each cycle, the gate duration is adjusted based on the
inverse of the resulting maximum control pulse amplitudes, by re-scaling the
dynamics to a new duration where control pulses satisfy the amplitude
constraints. Those scaled controls then serve as an initial guess for the next
unconstrained optimal control cycle, using the adjusted gate duration. We
provide multiple numerical examples that each demonstrate fast convergence of
the scheme towards a gate duration that is close to the quantum speed limit,
given the control pulse amplitude bound. The proposed technique is agnostic to
the underlying system and control Hamiltonian models, as well as the target
unitary gate operation, making the time-scaling iteration an easy to implement
and practically useful scheme for reducing the durations of quantum gate
operations.
