= Solution
Write
$$
|\psi_0\rangle=\frac12\sum_{x\in\mathcal B_2}|x\rangle
=\sin\theta|x_0\rangle+\cos\theta|\beta\rangle,
\qquad \theta=\frac\pi6,
$$
where $|\beta\rangle$ is the normalized component orthogonal to $|x_0\rangle$. On this real plane, $I_{x_0}$ is reflection in the line perpendicular to $|x_0\rangle$, while
$$
-J=2|\psi_0\rangle\langle\psi_0|-I
$$
is reflection in the line spanned by $|\psi_0\rangle$. Their product $Q=-JI_{x_0}$ is a rotation through $2\theta=\pi/3$ toward $|x_0\rangle$. This is the geometric core of <Grover search algorithm>[Grover search].
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