  We investigate the dynamics of the out-of-time-ordered correlators (OTOCs)
via a non-Hermitian extension of the quantum kicked rotor model, where the
kicking potential satisfies $\mathcal{PT}$-symmetry. The spontaneous
$\cal{PT}$-symmetry breaking emerges when the strength of the imaginary part of
the kicking potential exceeds a threshold value. We find, both analytically and
numerically, that in the broken phase of $\cal{PT}$ symmetry, the OTOCs rapidly
saturate with time evolution. Interestingly, the late-time saturation value
scales as a pow-law in the system size. The mechanism of such scaling law
results from the interplay between the effects of nonlocal operator in OTOCs
and the time reversal induced by non-Hermitian driven potential.
