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Here is an exercise (4.27) from Nielsen and Chuang and I found the answer (given in the figure below) online without any explanation. The question was to construct a circuit by seeing a matrix (given in the figure).

So I want to know what should be the thought process after seeing the matrix?

enter image description here

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    $\begingroup$ This thread that is about the same exercise might be interesting. $\endgroup$ Sep 16 '20 at 9:48

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