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In this paper, we explore how to constructively manipulate qubits by rotatingBloch spheres. It is revealed that three-rotation and one-rotation Hamiltoniancontrols can be constructed to steer qubits when two tunable Hamiltoniancontrols are available. It is demonstrated in this research that local-wavefunction controls such as Bang-Bang, triangle-function and quadratic functioncontrols can be utilized to manipulate quantum states on the Bloch sphere. Anew kind of time-energy performance index is proposed to trade-off time andenergy resource cost, in which control magnitudes are optimized in terms ofthis kind of performance. It is further exemplified that this idea can begeneralized to manipulate encoded qubits on the controllable subspace.

Manipulating quantum information on the controllable systems or subspaces