While reading about Grover's mean estimation algorithm, I came across this post on StackExchange. As an undergrad, there are several points I don't quite understand.
What is the role of $f(x)$. Consider I have a list of numbers, such as $0.1, 0.2, 0.3, 0.4$ and I want to calculate the mean. What exactly is $f(x)$ mapping? What exactly is $x$ here? My wildest guess is that the above numbers are labelled as $00, 01, 10, 11$ and these binary strings are referred to as $x$. In this sense $f(x)$ behaves more like a table to associate $00 \rightarrow 0.1$, $00 \rightarrow 0.2$, etc. Is this the correct understanding?
What computation on the ancilla? The aforementioned post also suggest that perform a computation of $f(x)$ on an ancilla register. What does the computation mean?
How does this controlled-rotation look like? It appears to me that three registers are needed to implement this algorithm. To achieve this controlled-rotation, which bit is used as control?
