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My question relates to the theoretical application of the diffusion operator in Grover's algorithm.

I understand that a single Grover iteration on an equal superposition will change the amplitude of one specific basis and give it a higher probability of being measured.

The diffusion operation is defined as

$H^{\otimes n}(2|0\rangle\langle0|-I)H^{\otimes n} = 2|s\rangle\langle s|-I$

where it basically assumes that the input states are in equal superposition. But this assumption is wrong as soon as more than one operation is applied, right? So how does this shape for the diffusion operator still hold when there is more than a single iteration of the Grover cycle?

Or is this definition simply like this for any initial state in the register and will always result in the inversion about the mean? I just haven't really found any good explanation on how did Grover/anybody else come up with this formula and what are the assumptions of it...

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