An utterly new application of superconducting device, the application to quantum computation, is demonstrated. In a single-Cooper-pair box, the number of Cooper pairs in the box is quantized and they form a single macroscopic quantum charge-number state. Two neighboring charge-number states that differ only by the occupation of one Cooper pair, form an artificial two-level system and they can be coherently coupled. Qubit operations were demonstrated in two different control techniques, dc electric-field gate bias and ac field bias. The dc method was unique compared with the commonly used Rabi-oscillation-type operation. Here the system was biased at the degenerate point of the two states so that the dynamical phase does not develop during the operation. This was the first time that the quantum coherent oscillation was observed in a solid-state device whose quantum states involved a macroscopic number of quantum particles. Multi-pulse experiments were also carried out. Phase control of the quantum state was demonstrated under a two-pulse operation. A three-pulse technique was implemented and it was shown to be effective in canceling the fluctuation of the bias point, and recover the coherent oscillation signal otherwise destroyed by such fluctuation.