In this paper, we investigate the problem of simulation of quantum-mechanical dynamical system with prescribed properties using of exactly solvable Hamiltonians. A method for solving time-dependent matrix Schrodinger equations in an explicit analytical form will be developed. It is based on exactly solvable time-independent problems, a certain choice of the initial conditions, and special time-dependent gauge transformations converting time-independent problems to time-dependent ones. An application of the exactly solvable time-dependent problems thus obtained to the construction of a universal set of gates for quantum computers is also presented.