(color online) Bloch sphere representation of a qubit. (a) State |ψ〉 is a superposition state of |0〉 and |1〉. (b) Single qubit gate operation shown in a Bloch sphere. Rotation from the initial state |ψ_i〉 to the final state |ψ_f〉 is finished by successive rotations around the z axis by ϕ_z and the y axis by ϕ_y.