  We report a new result concerning the dynamics of an initially localized wave
packet in quantum nonlinear Schr\"odinger lattices with a disordered potential.
A class of nonlinear lattices with subquadratic power nonlinearity is
considered. We show that there exists a parameter range for which an initially
localized wave packet can spread along the lattice to unlimited distances, but
the phenomenon is purely quantum and is hindered in the corresponding classical
lattices. The mechanism for this spreading assumes that the components of the
wave field may form coupled states by tunneling under the topological barriers
caused by multiple discontinuities in the operator space. Then these coupled
states thought of as quasiparticle states can propagate to long distances on
L\'evy flights with a distribution of waiting times. The overall process is
subdiffusive and occurs as a competition between long-distance jumps of the
quasiparticle states, on the one hand, and long-time trapping phenomena
mediated by clustering of unstable modes in wave number space, on the other
hand. The kinetic description of the transport, discussed in this work, is
based on fractional-derivative equations allowing for both non-Markovianity of
the spreading process as a result of attractive interaction among the unstable
modes and the effect of long-range correlations in wave number space tending to
introduce fast channels for the transport, the so-called stripes. We argue that
the notion of stripes is key to understand the topological constraints behind
the quantum spreading, and we involve the idea of stripy ordering to obtain
self-consistently the parameters of the associated waiting-time and jump-length
distributions. Finally, we predict the asymptotic laws for quantum transport
and show that the relevant parameter determining these laws is the exponent of
the power-law defining the type of the nonlinearity.
