For $\phi \in[0,2 \pi)$ and $|\psi\rangle \in \mathbb{C}^{4}$ consider the operator

$$R_{\psi}^{\phi}=\mathbb{I}-\left(1-e^{i \phi}\right)|\psi\rangle\langle\psi|$$

Let $U$ be a unitary operator on $\mathbb{C}^{4}=\mathbb{C}^{2} \otimes \mathbb{C}^{2}$ with action on $|00\rangle$ given as follows

$$\tag{†} U|00\rangle=\sqrt{p}|g\rangle+\sqrt{1-p}|b\rangle=:\left|\psi_{\mathrm{in}}\right\rangle$$

where $p$ is a constant in $[0,1]$ and $|g\rangle,|b\rangle \in \mathbb{C}^{4}$ are orthonormal states.

(i) Give an explicit expression of the state $R_{g}^{\phi} U|00\rangle$.

(ii) Find a $|\psi\rangle \in \mathbb{C}^{4}$ for which $R_{\psi}^{\pi}=U R_{00}^{\pi} U^{\dagger}$.

(iii) Choosing $p=1 / 4$ in equation ($\dagger$), calculate the state $U R_{00}^{\pi} U^{\dagger} R_{g}^{\phi} U|00\rangle$. For what choice of $\phi \in[0,2 \pi)$ is this state proportional to $|g\rangle$ ?

(iv) Describe how the above considerations can be used to find a marked element $g$ in a list of four items $\left\{g, b_{1}, b_{2}, b_{3}\right\}$. Assume that you have the state $|00\rangle$ and can act on it with a unitary operator that prepares the uniform superposition of four orthonormal basis states $|g\rangle,\left|b_{1}\right\rangle,\left|b_{2}\right\rangle,\left|b_{3}\right\rangle$ of $\mathbb{C}^{4}$. [You may use the operators $U$ (defined in (†)), $U^{\dagger}$ and $R_{\psi}^{\phi}$ for any choice of $\phi \in[0,2 \pi)$ and any $|\psi\rangle \in \mathbb{C}^{4}$.]