  The Q1 lattice equation, a member in the Adler-Bobenko-Suris list of 3D
consistent lattices, is investigated. By using the multidimensional
consistency, a novel Lax pair for Q1 equation is given, which can be
nonlinearised to produce integrable symplectic maps. Consequently, a Riemann
theta function expression for the discrete potential is derived with the help
of the Baker-Akhiezer functions. This expression leads to the algebro-geometric
integration of the Q1 lattice equation, based on the commutativity of discrete
phase flows generated from the iteration of integrable symplectic maps.
