  We investigate the quantum irreversibility and quantum diffusion in a
non-Hermitian kicked rotor model for which the kicking strength is complex. Our
results show that the exponential decay of Loschmidt echo gradually disappears
with increasing the strength of the imaginary part of non-Hermitian driven
potential, demonstrating the suppress of the exponential instability by
non-Hermiticity. The quantum diffusion exhibits the dynamical localization in
momentum space, namely, the mean square of momentum increases to saturation
with time evolution, which decreases with the increase of the strength of the
imaginary part of the kicking. This clearly reveals the enhancement of
dynamical localization by non-Hermiticity. We find, both analytically and
numerically, that the quantum state are mainly populated on a very few
quasieigenstates with significantly large value of the imaginary part of
quasienergies. Interestingly, the average value of the inverse participation
ratio of quasieigenstates decreases with the increase of the strength of the
imaginary part of the kicking potential, which implies that the feature of
quasieigenstates determines the stability of wavepacket's dynamics and the
dynamical localization of energy diffusion.
