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What is the thought process for circuit making after seeing input and output of a matrix?

Quantum Computing Asked by quantum_boy on January 26, 2021

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?

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