  We present a constructive control scheme for solving quantum state
engineering problems based on a parametrization of the state vector in terms of
complex hyperspherical coordinates. Unlike many control schemes based on
factorization of unitary operators the scheme gives explicit expressions for
all the Euler angles in terms of the hyperspherical coordinates of the initial
and final state. The factorization, when applicable, has added benefits that
phase rotations can be combined and performed concurrently. The control
procedure can be realized using a simple bang-bang or square-wave-function
controls. Optimal time-energy control is considered to find the optimal control
amplitudes. The extension of the scheme to implement unitary operators is also
discussed.
