We study two-dimensional (2D) solitons in the mean-field models of ultracold
gases with long-range quadrupole-quadrupole interaction (QQI) between
particles. The condensate is loaded into a deep optical-lattice (OL) potential,
therefore the model is based on the 2D discrete nonlinear Schr\"{o}dinger
equation with contact onsite and long-range intersite interactions, which
represent the QQI. The quadrupoles are built as pairs of electric dipoles and
anti-dipoles orientated perpendicular to the 2D plane to which the gas is
confined. Because the quadrupoles interact with the local gradient of the
external field, they are polarized by inhomogeneous dc electric field that may
be supplied by a tapered capacitor. Shapes, stability, mobility, and collisions
of fundamental discrete solitons are studied by means of systematic
simulations. In particular, threshold values of the norm, necessary for the
existence of the solitons, are found, and anisotropy of their static and
dynamical properties is explored. As concerns the mobility and collisions, it
is the first analysis of such properties for discrete solitons on 2D lattices
with long-range intersite interactions of any type. Estimates demonstrate that
the setting can be realized under experimentally available conditions,
predicting solitons built of $\sim$ 10,000 particles.Comment: 13 pages, 11 figures, 97 references, Physical Review A, in pres