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I'm pretty new to the field. I was reading Preskill Ph219 course notes and came across this. I am a bit confused about the derivation and wondered if someone can write down some skipped steps here.

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glS
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thongn98
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    please see https://quantumcomputing.meta.stackexchange.com/questions/49/tutorial-how-to-use-tex-mathjax-to-render-math-notation and https://quantumcomputing.stackexchange.com/help/how-to-ask. Please edit the post to state explicitly what you don't find clear about the stated calculation – glS Jul 11 '23 at 21:38

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$f(x)$ is a function that takes in $x$ and outputs a single bit 0 or 1.

To start, if we assume that $f(x)$ returns 0, then the state is

$|x\rangle \frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$.

If instead, we assume the $f(x)$ returns 1, then the state is

$|x\rangle \frac{1}{\sqrt{2}}(|1\rangle-|0\rangle)$.

But the only difference between these is a factor of -1!, so we can factor out $(-1)^{f(x)}$ which will give us the desired phase shift as a function of $f(x)$.

cpoole
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