Let $\Pi_{\mathcal G}$ be the <orthogonal projection> onto the good <linear subspace> and write
$$
|\psi\rangle
=\sin\theta\,|\psi_G\rangle
+\cos\theta\,|\psi_B\rangle,
\qquad
\sin^2\theta=\langle\psi|\Pi_{\mathcal G}|\psi\rangle,
$$
where the two displayed states are normalized projections into $\mathcal G$ and $\mathcal G^\perp$. Define the <reflection in a hyperplane>[reflections]
$$
R_G=I-2\Pi_{\mathcal G},
\qquad
R_\psi=2|\psi\rangle\langle\psi|-I.
$$
The <amplitude amplification> iterate $Q=R_\psi R_G$ preserves the good-bad plane, and its $k$th iterate satisfies
$$
Q^k|\psi\rangle
=\sin((2k+1)\theta)|\psi_G\rangle
+\cos((2k+1)\theta)|\psi_B\rangle.
$$
Thus repeated reflections rotate amplitude toward the good subspace, reaching constant success probability after $O(1/\sin\theta)$ iterations.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 324 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-324/2/a/solution