Look at the diagram in the middle of page 6-3 here, http://stellar.mit.edu/S/course/6/fa14/6.845/courseMaterial/topics/topic3/lectureNotes/qctlec6/qctlec6.pdf
I am confused as to how should one think of the Hadamard gate.
As per the mathematics shown below it seems that the Hadamard gate, $H$ acts on tensor products of states as follows,
$$H( \otimes ^n | 0 \rangle) = \otimes ^n ( H | 0 \rangle ) = \otimes ^n ( \frac { |0\rangle + |1\rangle }{\sqrt{2} } ) = \frac{1}{\sqrt{2^n} } \sum_{x \in \{ 0,1\}^n} |x\rangle $$
But then the above interpretation and the diagram and the text have a few apparent discrepancies between them,
In the diagram there is no step which looks like $\otimes ^n ( H | 0 \rangle ) $. It seem that all the wires carrying $H|0\rangle$ are straight away sent to the $f$ oracle without this tensoring between them.
In terms of circuit complexity is this to be thought of as a "single" Hadamard gate or as $n$ Hadamard gates?
Lastly how does the gate $H$ know about the state $|1\rangle$ ? It only sees the $|0\rangle$ states.
And the last step is totally unclear: on which state is the $i$th Hadamard gate acting to get $ |s_i\rangle$? It doesn't seem to be there in the equations below.
Can someone help clarify this?