How do I approach on solving the below two problems?
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$\begingroup$ Try this article arxiv.org/abs/quant-ph/9503016. It contains a lot of identities among quantum circuits. $\endgroup$– Martin VeselySep 7 at 5:40
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1 Answer
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For (a), and in accordance with the hint:
With controls = 00, the target qubit is not operated on (because all the gates have their controls set to 0 and so they do not affect the target).
With controls = 01, the left $V$ and the middle $V'$ are active but cancel each other, resulting in no change to the target.
With controls = 10, the middle $V'$ and right $V$ are active but cancel each other, resulting in no change to the target.
With controls = 11, the left $V$ and the right $V$ are active resulting in operation: $V^2$.
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$\begingroup$ For controls 01 or 10, I think it should be necessary to specify the qubit ordering $\endgroup$ Sep 7 at 14:09