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Suppose I have an oracle $U_f$ that maps $|x\rangle \mapsto (-1)^{f(x)}|x\rangle$ where $f : \{0,\dots,N\} \to \{0,1\}$ is such that $f(0)=0$. I want to show that I can implement the oracle $|x\rangle|b\rangle \mapsto |x\rangle|b \oplus f(x)\rangle$ using just one application of $U_f$.

I figured a way to do this but by using a controlled version of $U_f$. The following circuit does the trick:

                                         circ

However what if I don't have access to the controlled version of $U_f$? Can I still do it? Does the fact that $f(0)=0$ help here?

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Make a temporary ancillary copy conditional on the control, and apply the operation to the copy. This only applies the varying -1 factors if the control is ON, not OFF, which gives the necessary phase kickback.

Example:

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The $f(0) = 0$ requirement is avoided by classically computing $f(0)$ and applying a $Z$ to the control if it's 1 instead of 0.

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  • $\begingroup$ Thanks! It's all clear now! $\endgroup$ Sep 3, 2022 at 23:21

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