Ask here about the “Hadamard Transform” Codebook topic from the “Basic Quantum Algorithms” module.

Starting from the sentence

This two-qubit example makes it clear that we can generalize …

Tensor products \vert\pm^i\rangle\otimes\vert\pm^j\rangle of states are written more succinctly with juxtaposition as \vert\pm^i\rangle\vert\pm^j\rangle. This initially gave me some confusion, since the change in notation was not mentioned. I personally would prefer if it were written as \vert\pm^i\pm^j\rangle, which is as succinct and also clear due to precedent, and makes for more intuitive notation in later expressions, e.g. \langle\mathbf{y}\vert\pm^{x_1}\cdots\pm^{x_n}\rangle rather than \langle\mathbf{y}\vert\pm^{x_1}\rangle\cdots\vert\pm^{x_n}\rangle.

The connection of the LHS in Eq. (3) to the prior discussion was not immediately clear to me, but those are states after applying HUfH to 0. I figured out the connection by inserting two Hadamard transforms in the middle of the binomial expansion, then distributing them into either clause; this recovers the post-oracle, pre-Hadamard transform states in either clause.

Thanks for your feedback @jhanschoo !

Just to clarify, are these two equations the ones causing confusion? Or is it another equation?

Do you mean here that the solution of Exercise A.5.2 isn’t clear?

Can you maybe be a bit more specific about what exactly is causing confusion?

Hi there, regarding the first comment, yes the confusing notation appears in those two equations, but also in the solution to * Exercise A.5.1*. Those are the only places where I’ve found them appear, so just changing these parts should be good.

Regarding the second comment,

the appearance of Eq. (3) (prior to the solution) seemed to be unmotivated and pops out of thin air, in context of the exposition. In particular, it was not immediately clear to me that \vert\mathbf{0}\rangle-\frac{2}{\sqrt{2}}H^{\otimes n}\vert\mathbf{s}_1\rangle=H^{\otimes n}U_{f_{\mathbf{s}_1}}\vert\psi\rangle, and the RHS here is clearly the post-oracle state. (Similarly, that \langle\mathbf{0}\vert-\frac{2}{\sqrt{2}}\langle\mathbf{s}_2\vert H^{\otimes n}=(H^{\otimes n}U_{f_{\mathbf{s}_2}}\vert\psi\rangle)^{\dagger})

You should probably ignore the confusing discussion I previously gave in the second comment; it explained how I made the connection between the expressions, but is not the shortest or clearest way of relating the expressions in Eq. (3) back to the discussion.

Finally, I’d like to express my appreciation to you and the team for providing this resource and actively responding and working on it.

Thanks for your feedback and your kind words @jhanschoo

It’s definitely a team effort, including people like you who use the Codebook and provide feedback!