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I'm reading about quantum circuits.

I've just read about CNOT gate and after that, I read about the controlled U gate.

Can someone please clarify what is the difference between the two circuits below? The first one is the controlled-Hadamard gate (c-H). But what is the second one?! What is the difference between the matrix of the second and the first?

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What I'm guessing is that the first one is going to be controlled by the first compound qubit, and the second by the second compound qubit, is that right?!

Thanks!

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The first gate embodies the unitary transform:$$\begin{align}|00\rangle &\mapsto |00\rangle,\\ |01\rangle &\mapsto |01\rangle,\\ |10\rangle &\mapsto \sqrt{\frac12}|10\rangle + \sqrt{\frac12}|11\rangle,\\ |11\rangle &\mapsto \sqrt{\frac12}|10\rangle - \sqrt{\frac12}|11\rangle.\end{align}$$That is, it is a Hadamard gate on the second qubit conditioned on the first qubit being $|1\rangle.$ The second gate embodies the unitary transform, $$\begin{align}|00\rangle &\mapsto |00\rangle,\\ |01\rangle &\mapsto \sqrt{\frac12}|01\rangle + \sqrt{\frac12}|11\rangle,\\ |10\rangle &\mapsto |10\rangle,\\ |11\rangle &\mapsto \sqrt{\frac12}|01\rangle - \sqrt{\frac12}|11\rangle.\end{align}$$That is, it is a Hadamard gate on the first qubit conditioned on the second qubit being $|1\rangle.$

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  • $\begingroup$ Perfect, I was guessing that, but wasn't sure! Thanks, mate. $\endgroup$ – Bruno Reis Sep 11 '17 at 22:11

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