  Quantum state on Bloch sphere for superconducting charge qubit, phase qubit
and flux qubit for all time in absence of external drive is stable to initial
state. By driving the qubits, approximation of charge and flux Hamiltonian lead
to quantum state rotation in Bloch sphere around an axis completely differ from
rotation vector of exact Hamiltonian. The trajectory of quantum state for phase
qubit for approximated and exact Hamiltonian is the same but the expectation of
quantum observable has considerable errors as two other qubits. microwave drive
control is designed for approximated Hamiltonian and exerted on actual systems
and shows completely different trajectory with respect to desired trajectory.
Finally a nonlinear control with external $\mu$V voltage control and nA current
control is designed for general qubit which completely stabilizes quantum state
toward a desired state.
