For the oracle of Grover’s algorithm, there is an explanation that we use a phase-kick back as follows.
That is, we can tell if f(|x>) is 1 as it will change the output |x>.
My question is,
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How can we be sure that the coefficient (-1)^f(x) is attached to |x>, not to the auxiliary bit (|y>)?
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And why does this interpretation useful? I don’t see any use of this axiliary bit in the implemenation of Grover’s algorithm.
I am expecting an answer to the above two questions.