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The CNOT gate is usually written as

$|0\rangle\langle0|\otimes I + |1\rangle\langle1|\otimes X$

(with $X,Y,Z$ beign the Pauli Basis and $I$ the Identity).

I have yet to stumble across the representation Wikipedia gives when looking at books on the topic:

$e^{i\frac{\pi}{4}\left(I-Z\right)\otimes\left(I-X\right)}$

How can I show these forms are equivalent? Where can I find a derivation that shows this?


Cross-posted on qc.SE

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  • $\begingroup$ Maybe ask in quantumcomputing.stackexchange.com $\endgroup$
    – Mauricio
    Commented May 18, 2022 at 9:36
  • $\begingroup$ Thank you for the hint! $\endgroup$
    – manuel459
    Commented May 18, 2022 at 9:44
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    $\begingroup$ There might be a $\otimes$ missing in the argument of the exponential $\endgroup$
    – Mauricio
    Commented May 18, 2022 at 10:01
  • $\begingroup$ Thank you. That is absolutely right. $\endgroup$
    – manuel459
    Commented May 18, 2022 at 10:04

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