Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 324 / 2 / b / Solution

Expand $|b⟩={\sum }_i{b}_i|{\lambda }_i⟩$ in the nondegenerate eigenbasis of $A$. Apply quantum phase estimation to $U_{+}=e^{2\pi iA}$, using $U_{-}$ when inverse powers are needed, to coherently attach the eigenvalue:On an ancillary qubit perform the eigenvalue-controlled rotationUncompute the value register and measure the ancilla. Conditional on outcome one, the system is proportional toand normalization gives the desired $|\psi ⟩$.

The success probability isSince every ${\lambda }_i>0$,a positive lower bound independent of $|b⟩$. The promised precision assumptions permit the eigenvalue-controlled arithmetic and rotation.