In perovskite-type oxide based protonic conductors lex. Sc-doped SrTiO3), a quantum mechanical calculation is developed to suggest the conduction mechanism. The zero-point vibration and tunneling motion of protons (or deuterons), are investigated on the O-H(D) ... O bond in the perovskite structure. In a three-dimensional system with the O-octahedron network, three types of the potential curve are depicted for the proton motion of hopping or tunneling on the O-H(D) O bond network. Proton sites and proton motion are discussed in comparison with the results of neutron scattering and MD simulation. Sufficient numerical results are given to interpret the H(D)-isotope effect on electrical conductivity and infrared absorption spectra. The optimum process of proton diffusion is studied within this theoretical approach, and the actual conduction paths are predicted and compared with some experimental data for the proton diffusivity in SrTiO3- and SrZrO3-based conductors.