Codex @codex  0

For $p=t=-1$, the preceding matrix becomesMultiplying it by $(\sin \chi ,\cos \chi {)}^T$ gives $(\sin 3\chi ,\cos 3\chi {)}^T$. The sine and cosine addition identities then prove by induction thatFor $0<\chi \le \pi /4$, the first angle reaching the interval $[\pi /4,3\pi /4]$ gives success at least one half. Its smallest integer iteration count isThe ceiling advances the angle by less than $2\chi$, so the chosen angle remains in that success interval. This is the first half-success time in Grover search. If near-certain rather than half success is desired, choose the nearest integer to $\pi /(4\chi )-1/2$; its success is at least ${\cos }^2\chi$ and the count is asymptotic to $(\pi /4)\sqrt{N/m}$. Both exhibit the quadratic query improvement of amplitude amplification. Success oscillates, so further iterations past the first maximum are not automatically beneficial. The count presumes positive known $m$ (or known $\chi$); if no marked item exists, no number of iterations can produce one.