Abstract: A method for solving many classes of real-world problems with a quantum mechanical system by coaxing the system from a starting state to a desired state that is present in the quantum mechanical system with a probability greater than zero. Moving of the quantum mechanical system toward the desired states is effected by a repeated application of operator Q≡−IsU−1ItU, U is a unitary matrix, Is is a diagonal matrix that characterizes the starting state of the system with 1's in every position along the diagonal, except at the position corresponding to column s and row s, which represents the starting state, where it is −1. It is a similar diagonal matrix that characterizes the desired target state, t. The Q transformation is applied a prescribed number of times, followed by a single transfomation U, yielding the desired target state.