Ex22

$\def\abs#1{|#1}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 5.22}$

a) Show that the gates $H$, $P_{\frac{\pi}{2}}$ and $P_{\frac{\pi}{4}}$ are all (up to global phase) rational rotations of the Bloch sphere. Give the axis of rotation and the angle of rotation for each of these gates, and also the gate $S = HP_{\frac{\pi}{4}}H$.

b) Show that the transformation $V = P_{\frac{\pi}{4}}S$ is an irrational rotation of the Bloch sphere.