Rotation operators

The rotation operators are defined as: $\ R_x\left ( \theta \right ) = \begin{pmatrix} \cos \left ( \frac{\theta}{2} \right ) & -i \sin \left ( \frac{\theta}{2} \right) \\ -i \sin \left ( \frac{\theta}{2} \right) & \cos \left ( \frac{\theta}{2} \right ) \end{pmatrix} \\$ $\ R_y\left ( \theta \right ) = \begin{pmatrix} \cos \left ( \frac{\theta}{2} \right ) & - \sin \left ( \frac{\theta}{2} \right) \\ \sin \left ( \frac{\theta}{2} \right) & \cos \left ( \frac{\theta}{2} \right ) \end{pmatrix} \\$ $\ R_z\left ( \theta \right ) = \begin{pmatrix} e^{-i \frac{\theta}{2}} & 0 \\ 0 & e^{i \frac{\theta}{2}} \end{pmatrix} \\$

The rotation operators are generated by exponentiation of the Pauli matrices according to $\ exp{(i A x)} = \cos\left ( x \right )I+i\sin\left ( x \right )A \\$ where A is one of the three Pauli Matrices.

Note that the Rz rotation operator can also be expressed as $\begin{pmatrix} e^{i \frac{\theta}{2}} & 0 \\ 0 & e^{i \frac{\theta}{2}} \end{pmatrix} \begin{pmatrix} e^{-i \frac{\theta}{2}} & 0 \\ 0 & e^{i \frac{\theta}{2}} \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & e^{i \theta} \end{pmatrix} \\$ which differs from the definition above by a global phase only.