  We investigate, both analytically and numerically, the dynamics of quantum
chaos and quantum scrambling under many-body interaction effects via a
non-Hermitian Gross-Pitaevski map model, incorporating a periodically
modulated, complex strength of nonlinear interaction as delta kicks. We
establish a theoretical equivalence between a particular out-of-time ordered
correlator and the fidelity of a quantum state, which is proportional to the
Loschmidt echo. Both exhibit a double exponential growth in relation to time,
indicating the emergence of superexponential instability and
superexponentially-fast scrambling. The underlying mechanism is rooted in the
superexponentially-fast diffusion of energy, arising from the interplay between
the positive feedback of nonlinear interaction and the growth of the amplitude
of quantum state due to non-Hermiticity. Our findings suggest a kind of fastest
divergence of two nearby quantum states, which has implication in information
scrambling and brachistochrone evolution of quantum states.
