Let $P_G$ project onto the good subspace and write
$$
|\psi\rangle=\sin\theta|\psi_G\rangle
+\cos\theta|\psi_B\rangle.
$$
The <amplitude amplification theorem> says that alternating the reflection $I-2P_G$ with the reflection $2|\psi\rangle\langle\psi|-I$ rotates this two-dimensional plane by $2\theta$. After $j$ amplification iterations the good probability is
$$
\boxed{\sin^2((2j+1)\theta)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 324 / 3 / a / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-324/3/a/i/solution