re defined according to: (N10) The above exponential-operator definitions are relevant, since Pauli matrices satisfy the condition X2 = = 1. Substituting the definitions in Eqs. (N1) and (N8) into Eq. (N10) yields (as easily checked): Rx(y) (N11) Appendix N Figure N1 Qubit represented as point of coordinates (A, 4) on Bloch sphere. I shall now interpret the effect of the three above rotation operators on qubits. As we have seen from the main text, the general definition of a qubit lq) is (N12) The above definition shows that any qubit can be represented by a single point of angular coordinates (A, 4) on the surface of the Bloch sphere (with 0 = 0 IT and -> 2), which is show in Fig. N1 for convenience. It is left as an exercise to show that the three operators Rk(s) (k = x,y,z) rotate the qubit by an angle y about the axis k, in the counterclockwise direction. Consider next a more general form of the rotation operator. First, we define the Pauli vector in the Cartesian reference frame (x, y, z)) according to (N13) =(01,02,03). Given a unitary vector n = (nx, Ny,Nz), the scalar product corresponds to the matrix operator defined by U =n.o NxO1 NyO2 + NzO3. It is easily established that U is unitary or U2 I. Indeed, we have U2=UU (nxo1 + = () +nyNz(0203+0302)NzNx(0301+0103) (N14) Pauli matrices, rotations, and unitary operators In the above result, we have applied the property in Eq. (16.19) from the text, namely for i # j and our assumption that n is unitary (n2 +n, + n2 = 1). Since U2=1, theexponential-operator definition in Eq. (N9) is valid for the operator U, and we have, for any real O: exp(iU0)=Icos0+iUsin6 (N15) exp[i(n ) = = I cos O + i(n.o) sin O. We recognize in this result a more general expression from which the elementary rotation operators Rk(s) (k = X, y, z) defined in Eq. (N10) can be derived, i.e., by setting n = x, n or = IN with O = -r/2. The general expression corresponds to a qubit rotation of (counterclockwise) angle y about the axis defined by the unit vector n = (nx, Ny, nz). Since the result