Hadamard-phase-Hadamard qubit rotation ($H{P}_{\alpha }H=e^{i\alpha /2}{e}^{-i\alpha {\sigma }_x/2}$)

For $P_{\alpha }=\mathrm{diag}(1,e^{i\alpha })$, conjugation by the Hadamard gate turns a relative quantum phase into an $x$-axis Bloch vector rotation. In the convention $\rho =(I+\mathbf{P}\cdot \mathbf{\sigma })/2$, the map is $(P_x,P_y,P_z)\mapsto (P_x,P_y\cos \alpha -P_z\sin \alpha ,P_y\sin \alpha +P_z\cos \alpha )$. An $x$-polarized input is unchanged, whereas a $z$-polarized input gives a sinusoidal population difference. A maximally mixed input remains maximally mixed under every unitary time evolution.