Grover search with different diffusion operators

Question

I was reading about the Grover Search algorithm on https://qiskit.org/textbook/ch-algorithms/grover.html#example. I understood the method but I have a few questions. My question regards the two-qubit case.

Does the diffusion operator $D=2|s\rangle\langle s|-1$, depend upon the initial state i.e $|+\rangle|+\rangle$ and the marked state?

Actually I was reading an article https://journals.aps.org/pra/pdf/10.1103/PhysRevA.68.022306, which had an equation \begin{equation} -U_{S_j}|S_j\rangle_{w}=|w\rangle \end{equation} with $U_x=1-2|x\rangle\langle x|$, $S_1=\left(\dfrac{0+1}{\sqrt{2}}\right)^{\otimes 2}$, and $w$ is the marked state. The other $S_{j's}$ can be the states for instance $|+\rangle|-\rangle$, $|-\rangle|-\rangle$, $|-\rangle|+\rangle$ etc. with total such $S_j$ being $16$. My question is how does one make the diffusion operator for a state $|+\rangle|-\rangle$. As an example from the table in the article it states for instance if $j=2$, $S_2=|+\rangle|-\rangle$ $$-U_{S_2}|S_1\rangle_{10}=-|00\rangle,$$ where $10=w$ is the marked state. Can somebody explain how this equation came? can somebody atleast hint at some references?

Answer 1

1. it should be clear that the core of the Grover algorithm includes 3 steps

   a) prepare initial state $|s\rangle$

   b) apply $U=1-2|\omega\rangle\langle \omega|$

   c) apply $D=2|s\rangle\langle s|-1$

   then repeat step b and c

2. In the original Grover algorithm, the diffusion operator is fixed as $D=2|s\rangle\langle s|-1$, which you can say it depends upon the initial state. Actually in the image of qiskit textbook, you can see the initial state is $|s\rangle=H^{\otimes n}|0\rangle^{n}$

3. In the paper your reference, it extends the Grover algorithm, especially extends the diffusion operator from $|s\rangle$ (named as $\left|S_{1}\right\rangle$) to another 15 states $\left|S_{j}\right\rangle$

4. For the equation you mentioned in the case of $j = 2$, the detail derivation is as follows: the tensor product formula can learn from here