Past exam of the computer science course of the University of Oxford / 2022 / Quantum information / 3 / b

The Pauli Z gate gives a rotation through $\pi$ about the $z$ axis of the Bloch sphere. In Bloch vector coordinates,For example, this follows from $ZXZ=-X$, $ZYZ=-Y$ and $ZZZ=Z$ in the Pauli matrix expansion $\rho =(I+r_{x}X+r_{y}Y+r_{z}Z)/2$.

Since $H=(X+Z)/\sqrt{2}$, the Hadamard gate is, up to a global phase, the rotation gateThus its Bloch sphere action is a rotation through $\pi$ about the axis $(1,0,1)/\sqrt{2}$, halfway between the positive $x$ and $z$ axes:The identities $HXH=Z$, $HYH=-Y$ and $HZH=X$ verify this Bloch vector transformation directly.