The equivalence between the two-dimensional Ising model and the one-dimensional quantum XY model is generalized to the cases with alternating/random interactions and with periodic/free boundary conditions. It is proved that the eigenstate for the maximum eigenvalue of the transfer matrix of the two-dimensional Ising model corresponds to the ground state of the Hamiltonian of the one-dimensional XY model, which certifies direct relations between the physical quantities of the two models. Copyright (C) EPLA, 2014