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The matrix contains information about the vectors(states). To see this, the matrix form can be written as $U_{ij}=\langle i\mid U\mid j\rangle$ or the total form of $U$ mentioned by @Quantum Mechanic, i.e., $U=\sum_{ij}U_{ij}\mid i\rangle\langle j\mid$. To show it more vividly, the stabilizer code will be a good example. Another easier example is that: when ...

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Where did the article say M/N is the probability of error? M/N in the article is only for the use of normalization. For example(2 qubits), after $H^{\otimes 2}$ acted on initial $\mid 0\rangle^{\otimes 2}$, the state becomes $\mid\psi\rangle\equiv1/2(\mid 00\rangle+\mid01\rangle+\mid10\rangle+\mid11\rangle)$ . If the answer is $| 01\rangle$, then the $\mid\... 2 The original Amplitude Estimation algorithm (Brassard et al., 2002) uses Phase Estimation. So, it is OK to notice the similarity between them. Other approaches, however, have emerged in recent years that do not use PE for Amplitude Estimation like this and this. 4 To find the differences between them, you only need to know the aim of the problems. The aim of AA is to find the answers from unstructured data(or more directly, amplify the probability of the right answer). The aim of PE is to find the phase, more specifically the$\phi$in the book of Nielsen: Suppose a unitary operator$U$has an eigenvector$\mid u\...

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Given that $\mathcal{A}$ is a unitary matrix, $\mathcal{A}^{-1} = \mathcal{A}^*$  \begin{aligned} \mathcal{A}S_{0}^{\phi}\mathcal{A}^{-1}|Y\rangle = & \mathcal{A}(I - (1-\phi)|0\rangle\langle0|)\mathcal{A}^{-1}|Y\rangle \\ =& |Y\rangle - (1-\phi)\mathcal{A}|0\rangle\langle0|\mathcal{A}^{-1}|Y\rangle\\ =& |Y\rangle - (1-\phi)\mathcal{A}|0\...

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