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How can we prove these two quantum oracles are equivalent: $$O_x:|x,b\rangle\mapsto|x,b\oplus f(x)\rangle$$ and $$O_z:|x⟩ \mapsto(−1)^{f(x)}|x⟩$$

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  • $\begingroup$ They're not based on the information you provided. I feel like there's some missing context here. $\endgroup$
    – Dani007
    Nov 28, 2022 at 13:06

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The two oracles are not equivalent. But if you have either one of these oracles, you can trivially construct the other. In that sense they are equivalent.

Converting a phase oracle into a standard oracle is discussed here.

To convert a standard oracle into a phase oracle is discussed in the Wikipedia article on Grover's Algorithm. Put $|-\rangle$ into the "result" qubit before running the algorithm.

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  • $\begingroup$ Thank you very much for your reply! $\endgroup$
    – venki
    Nov 28, 2022 at 22:02
  • $\begingroup$ @venki If the answer helped you/solved your problem, please remember to upvote it and mark it as accepted! :) $\endgroup$
    – Tristan Nemoz
    Nov 29, 2022 at 0:46
  • $\begingroup$ Surely conversion from phase into standard only works of you've got a controlled version of the phase oracle? So if you've only got the non-controlled version, that doesn't help with an equivalence claim? $\endgroup$
    – DaftWullie
    Nov 29, 2022 at 16:36

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