We study the approximation of the imbedding of functions from anisotropic and generalized Sobolev classes into Lq([0, 1]d) space in the quantum model of computation. Based on the quantum algorithms for approximation of finite imbedding from LNp to LNq , we develop quantum algorithms for approximating the imbedding from anisotropic Sobolev classes B(Wrp ([0, 1]d)) to Lq([0, 1]d) space for all 1 6 q; p 6 1 and prove their optimality. Our results show that for p < q the quantum model of computation can bring a speedup roughly up to a squaring of the rate in the classical deterministic and randomized settings.