  We investigate the quantum diffusion of a periodically kicked particle
subjecting to both nonlinearity induced self-interactions and
$\mathcal{PT}$-symmetric potentials. We find that, due to the interplay between
the nonlinearity and non-Hermiticity, the expectation value of mean square of
momentum scales with time in a super-exponential form $\langle
p^2(t)\rangle\propto\exp[\beta\exp(\alpha t)]$, which is faster than any known
rates of quantum diffusion. In the $\mathcal{PT}$-symmetry-breaking phase, the
intensity of a state increases exponentially with time, leading to the
exponential growth of the interaction strength. The feedback of the
intensity-dependent nonlinearity further turns the interaction energy into the
kinetic energy, resulting in a super-exponential growth of the mean energy.
These theoretical predictions are in good agreement with numerical simulations
in a $\cal{PT}$-symmetric nonlinear kicked particle. Our discovery establishes
a new mechanism of diffusion in interacting and dissipative quantum systems.
Important implications and possible experimental observations are discussed.
