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In "Quantum Computation and Quantum Information", they talk about Grover search. One part of the Grover search is Amplitude amplification: enter image description here

But i think the red line is not true. Because i do a matrix multiplication, and they did not give the same matrix. enter image description here enter image description here

So what is wrong here?

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Nothing is wrong. You're just forgetting that there's an equivalence up to global phases. Multiply one answer by $-1$ and you have the other. Hence, they are the same up to a global phase.

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  • $\begingroup$ i think you are right. But i dont totally understand it. If the matrix of the first Oracle is multiple by -1, then the answer would be wrong? $\endgroup$
    – Huy By
    Commented Aug 17, 2023 at 3:08
  • $\begingroup$ No, it just means that your overall state vector would be multiplied by $-1$, which has no observable difference. $\endgroup$
    – DaftWullie
    Commented Aug 17, 2023 at 7:14
  • $\begingroup$ In Grover's algorithm, the first Oracle is Phase Inversion right? If the Oracle is multiple by -1, it's mean we are considering a completely different function? $\endgroup$
    – Huy By
    Commented Aug 17, 2023 at 7:29

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