  We study quantum invariant Z(M) for cusped hyperbolic 3-manifold M. We
construct this invariant based on oriented ideal triangulation of M by
assigning to each tetrahedron the quantum dilogarithm function, which is
introduced by Faddeev in studies of the modular double of the quantum group.
Following Thurston and Neumann-Zagier, we deform a complete hyperbolic
structure of M, and correspondingly we define quantum invariant Z(M_u). This
quantum invariant is shown to give the Neumann--Zagier potential function in
the classical limit, and the A-polynomial can be derived from the potential
function. We explain our construction by taking examples of 3-manifolds such as
complements of hyperbolic knots and punctured torus bundle over the circle.
