how can we prove the quantum oracle gates are equivalent. Ox with input |x,b⟩ returns |x,b⊕f(x)⟩ and Oz with input |x⟩ and returns(−1)^f(x)|x⟩

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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 $$O_z:|x⟩ \mapsto(−1)^{f(x)}|x⟩$$

score 1 · Answer 1 · answered Nov 28 '22 at 20:03

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.

how can we prove the quantum oracle gates are equivalent. Ox with input |x,b⟩ returns |x,b⊕f(x)⟩ and Oz with input |x⟩ and returns(−1)^f(x)|x⟩

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