Bang-bang control design for quantum state transfer based on hyperspherical coordinates and optimal time-energy control

We present a constructive control scheme for solving quantum state engineering problems based on the parameterization 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 generalized Euler angles in terms of the hyperspherical coordinates of the initial and final states. The factorization, when applicable, has added benefits that phase rotations can be combined and performed concurrently. The control procedure can be realized using simple bang-bang or square-wavefunction controls. Optimal time-energy control is considered to find the optimal control amplitude. The extension of the scheme to implement arbitrary unitary operators is also discussed.