 of any 2 X 2 unitary transformation A is to move a qubit on the surface of the Bloch sphere, there exists a unique rotation associated with such a move (like taking a direct flight from city to city on the Earth). This unique rotation is defined through the operator exp[i(n . )A] = A (within an unobservable phase factor). We can make this operator more explicit, by developing the definition in Eq. (N15), according (N16) (N17) where Mi are complex numbers defined by = cos 0, 11 = inx sin 0, M2 = iny sin 0, = in sin A, (N18) which, in particular, satisfy Eilmil2= = 1. To summarize this result, any 2 x 2 unitary transformation, or rotation on the Bloch sphere, can be expressed as a linear com- plex expansion of Pauli matrices. It is left as an elementary exercise to determine the parameters n, 0 associated with any unitary transformation A. Euler's theorem This theorem states that every 2 x 2 unitary matrix U can be expressed from the two rotation operators Ry, R2 and a set of four real numbers a, B, y, 8, according to the product: (N19) Appendix N To show this, we first substitute the corresponding definitions of Ry, R2 from Eq. (N11) to obtain: (N20) It is left as an exercise to prove that the unitary condition U+U = 1 is both necessary and sufficient for the above matrix coefficients to represent any unitary matrix U. Euler's theorem, as defined in Eq. (N19), is referred to as Z-Y decomposition of rotations. In fact Z-X decomposition, according to U = is also possible, as well as any decomposition involving two different Pauli operators. Most generally, given any two nonparallel unitary vectors n, m (N21) where R(0) )=exp[i(p.)A] is the rotation operator of angle 0 about the axis defined by the unitary vector p. Decomposition of 2 X 2 unitary matrices A general property stemming from Euler's theorem is that any 2 X 2 unitary matrix can be decomposed according to (N22) where A, B, C are 2 x 2 unitary matrices satisfying ABC = I and X = 01. To prove this property, we must find at least one set of three matrices A,