find output after each step in two qubit circuit for grovvers algorithm $|$

question is from neural networks

Box $6\mathrm{.}1$: Quantum search: a two$-$bit example

Here is an explicit example illustrating how the quantum search algorithm works on a search space of size $N=4.$ The oracle, for which $f(x)=0$ for all $x$ except $x=x_0,$ in which case $f(x_0)=1,$ can be taken to be one of the four circuits

corresponding to $x_0=0,1,2,$ or $3$ from left to right, where the top two qubits carry the query $x,$ and the bottom qubit carries the oracle's response. The quantum circuit which performs the initial Hadamard transforms and a single Grover iteration $G$ is

Initially, the top two qubits are prepared in the state $|0$:$|,$ and the bottom one as $|1$:$|.$ The gates in the dotted box perform the conditional phase shift operation $2|00$:$|.$ How many times must we repeat $G$ to obtain $x_0?$ From Equation $(6\mathrm{.}15),$ using $M=1,$ we find that less than one iteration is required. It turns out that because $=\frac{}{3}$ in $(6\mathrm{.}14),$ only exactly one iteration is required, to perfectly obtain $x_0,$ in this special case. In the geometric picture of Figure $6\mathrm{.}3,$ our initial state $|$:$|$ is $30$ from $|$:$|,$ and a single rotation by $=60$ moves $|$:$|$ to $|$:$|.$ You can confirm for yourself directly, using the quantum circuits, that measurement of the top two qubits gives $x_0,$ after using the oracle only once. In contrast, a classical computer $-$ or classical circuit $-$ trying to differentiate between the four oracles would require on average $2\mathrm{.}25$ oracle queries!

solves the search problem. More precisely, we write down a Hamiltonian $H$ which depends on the solution $x$ and an initial state $|$:$|$ such that a quantum system evolving according to $H$ will change from $|$:$|$ to $|x$:$|$ after some prescribed time. Once we've found such a Hamiltonian and initial state, we can move on to the second step, which is to attempt to simulate the action of the Hamiltonian using a quantum circuit. Amazingly, following this procedure leads very quickly to the quantum search algorithm! We have$|$