# Oblivious Amplitude Amplification & Eigenstate decomposition

Looking at the oblivious AA (OAA) : https://docs.microsoft.com/en-us/azure/quantum/user-guide/libraries/standard/algorithms

I am trying to figure out the eigendecomposition of Q, which leads to below. Any help or suggestion on how to derive these would be highly appreciated!

$$|\psi\rangle=\frac{-i}{\sqrt{2}}\left(e^{i \theta}\left|\psi_{+}\right\rangle+e^{-i \theta}\left|\psi_{-}\right\rangle\right)$$ where $$\left|\psi_{\pm}\right\rangle$$are eigenvectors of $$Q$$ with eigenvalues $$e^{\pm 2 i \theta}$$ and only have support on the $$+1$$ eigenvectors of $$P_{0}$$ and $$P_{1}$$.

Here are the key sentences from the website

Each iteration of Amplitude amplification requires that two reflection operators be specified. Specifically, if $$Q$$ is the amplitude amplification iterate and $$P_{0}$$ is a projector operator onto the initial subspace and $$P_{1}$$ is the projector onto the marked subspace then $$Q=-\left(\mathbf{1}-2 P_{0}\right)\left(\mathbf{1}-2 P_{1}\right)$$. Recall that a projector is a Hermitian operator that has eigenvalues $$+1$$ and 0 and as a result $$\left(\mathbf{1}-2 P_{0}\right)$$ is unitary because it has eigenvalues that are roots of unity (in this case $$\pm 1$$ ). As an example, consider the case of Grover's search with initial state $$H^{\otimes n}|0\rangle$$ and marked state $$|m\rangle, P_{0}=H^{\otimes n}|0\rangle\langle 0| H^{\otimes n}$$ and $$P_{1}=|m\rangle\langle m| .$$ In most applications of amplitude amplification $$P_{0}$$ will be a projector onto an initial state meaning that $$P_{0}=\mathbf{1}-2|\psi\rangle\langle\psi|$$ for some vector $$|\psi\rangle ;$$ however, for oblivious amplitude amplification $$P_{0}$$ will typically project onto many quantum states (for example, the multiplicity of the $$+1$$ eigenvalue of $$P_{0}$$ is greater than 1 ).

The logic behind amplitude amplification follows directly from the eigen-decomposition of $$Q .$$ Specifically, the eigenvectors of $$Q$$ that the initial state has non-zero support over can be shown to be linear combinations of the $$+1$$ eigenvectors of $$P_{0}$$ and $$P_{1}$$. Specifically, the initial state for amplitude amplification (assuming it is a $$+1$$ eigenvector of $$P_{0}$$ ) can be written as $$|\psi\rangle=\frac{-i}{\sqrt{2}}\left(e^{i \theta}\left|\psi_{+}\right\rangle+e^{-i \theta}\left|\psi_{-}\right\rangle\right)$$ where $$\left|\psi_{\pm}\right\rangle$$are eigenvectors of $$Q$$ with eigenvalues $$e^{\pm 2 i \theta}$$ and only have support on the $$+1$$ eigenvectors of $$P_{0}$$ and $$P_{1}$$. The fact that the eigenvalues are $$e^{\pm i \theta}$$ implies that the operator $$Q$$ performs a rotation in a two-dimensional subspace specified by the two projectors and the initial state where the rotation angle is $$2 \theta$$. This is why after $$m$$ iterations of $$Q$$ the success probability is $$\sin ^{2}([2 m+1] \theta)$$