Question regarding the notation of QFT

Question

I have a question about the notation of QFT. I would like to present briefly what my problem is. So given is the QFT as a mapping with:

$$|j_1,...,j_n\rangle \rightarrow \frac{(|0\rangle + e^{2\pi i 0.j_n}|1\rangle)(|0\rangle + e^{2\pi i0.j_{n-1}j_n}|1\rangle)...(|0\rangle + e^{2\pi i 0.j_1j_2 ... j_n}|1\rangle)}{\sqrt{2^n}} \quad\quad\text{eq.1}$$

Based on this notation, I would number the individual qubits as follows:

$$|j_1,...,j_n\rangle = \overbrace{|j_1\rangle}^{\text{Qubit 1}}\overbrace{|j_2\rangle}^{\text{Qubit 2}}...\overbrace{|j_n\rangle}^{\text{Qubit $n$}} \rightarrow \frac{\overbrace{(|0\rangle + e^{2\pi i0.j_n}|1\rangle)}^{\text{Qubit 1}}\overbrace{(|0\rangle + e^{2\pi i0.j_{n-1}j_n}|1\rangle)}^{\text{Qubit 2}}...\overbrace{(|0\rangle + e^{2\pi i0.j_1j_2 ... j_n}|1\rangle)}^{\text{Qubit $n$}}}{\sqrt{2^n}}$$

If what I said before is correct, I am a bit confused about the following statement I read in "Physics 191 Lecture, Fall 14", there it says:

Lets take the $l$th qubit, $|j_l\rangle$. According to eq. (2) above. this needs to be transformed from $|j_l\rangle$ into the state $$\frac{1}{\sqrt{2}}(|0\rangle + e^{2\pi i0.j_l ... j_n}|1\rangle)$$

A few words regarding the quote:

• What is described in the quote as equation 2 corresponds to my first equation here eq.1 (the mapping).

Let's assume a very simple setup, say we have two qubits $|1_{1}0_{2}\rangle$ (Indices explicitly added) that we want to transform with QFT. According to the quote, however, I would get the following result:

Let's just look at the first qubit $l = 1$ then (QFT on first Qubit):

$$|1\rangle \rightarrow \frac{1}{\sqrt{2}}(|0\rangle + e^{2\pi i0.j_1 ... j_2}|1\rangle) = \frac{1}{\sqrt{2}}(|0\rangle + e^{2\pi i0.10}|1\rangle)$$

Second qubit $l = 2$ then (QFT on second Qubit):

$$|0\rangle \rightarrow \frac{1}{\sqrt{2}}(|0\rangle + e^{2\pi i0.j_2}|1\rangle) = \frac{1}{\sqrt{2}}(|0\rangle + e^{2\pi i0.0}|1\rangle)$$

But then this does not correspond to what I marked/interpreted above as qubit 1 and 2.

In my opinion, however, the first and second qubit should be displayed like this:

$$|10\rangle = \overbrace{|1\rangle}^{\text{Qubit 1}}\overbrace{|0\rangle}^{\text{Qubit 2}} \rightarrow \frac{\overbrace{(|0\rangle + e^{2\pi i0.0}|1\rangle)}^{\text{Qubit 1}}\overbrace{(|0\rangle + e^{2\pi i0.10}|1\rangle)}^{\text{Qubit 2}}}{\sqrt{2^2}}$$

My question now is, what is correct here? It may also be that my interpretation and numbering of the qubits is wrong?

I know that the question may go into great detail, but hope to have expressed myself understandably here :)

Answer 1

You are missing the fact that after applying all the Hadamard gates and Rotation Gates, Swap Gates are used. As shown below:

Now, QFT$|j_1j_2...j_n\rangle=\frac{1}{\sqrt(2^n)}((|0\rangle+e^{2\pi i0.j_n})(|0\rangle+e^{2\pi i0.j_{n-1}j_n})...(|0\rangle+e^{2\pi i0.j_1j_2...j_n}))$, if you apply the swap gates also.

Now, if you don't apply swap gates then you end up getting for the $l^{th}$ state as,

$|j_l\rangle = |0\rangle+e^{2\pi i0.j_l...j_n}|1\rangle$, of course normalized state.