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I saw an example which takes a 2 level matrix. Which is a $8\times8$ matrix that acts non trivially only on 2 levels of only states $|000\rangle$ and $|111\rangle$. The way they do it is by using a gray code from $|000\rangle$ to $|111\rangle$ and then shifting $|000\rangle$ to $|011\rangle$ and performing the $U$ operation only on the most left qubit conditional on the two $|11\rangle$ qubits to the right.

The thing I am trying to do, is to show that in the end this is equivalent to the original $U$ operation of $8\times 8$ matrix. How can I show their equivalency?

What I was trying to do is, to take the $U$ acting on the left hand side qubit and tensor product it with two identity matrices acting on the two other qubits. But this doesn't yeild the original $U$ ($8\times 8$ matrix). What did I get wrong here?

And how do you actually prove that the original $8\times 8$ matrix operation $U$ acting only on states $|000\rangle$ and $|011\rangle$ can be translated by a small $U$ matrix acting only on the most left qubit plus CNOT gates. How can I show it? It would be even preferred to show it by using matrix manipulation of sorts to get the original $8\times 8$ matrix. Or even some intuition would be good as well. Thanks!

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