  Just like decent classical difference-difference systems define symplectic
maps on suitable phase spaces, their counterparts with properly ordered
noncommutative entries come as Heisenberg equations of motion for corresponding
quantum discrete-discrete models. We observe how this idea applies to a
difference-difference counterpart of the Liouville equation. We produce
explicit forms of of its evolution operator for the two natural space-time
coordinate systems. We discover that discrete-discrete models inherit crucial
features of their continuous-time parents like locality and integrability while
the new-found algebraic transparency promises a useful progress in some
branches of Quantum Inverse Scattering Method.
