I have a question about the Quantum Fourier Transform. I would like to understand it because I have a re-take for an exam. I have studied the provided / recommended literature extensively. Unfortunately, studying the literature did not provided me a good understanding of this subject. I hope that some one could provide me some help. Let me first provide the Quantum Fourier Transform: \begin{align} F_{M} | k \rangle &= \frac{1}{\sqrt{M}} \sum_{j = 0}^{2^{m}-1} e^{2 \pi i \frac{1}{2^m} jk } | j \rangle= \\ &= \frac{1}{\sqrt{2^m}} \bigotimes(|0\rangle + e^{2\pi i k / 2^{l}} |1\rangle) \\ &= \frac{1}{\sqrt{2^m}} (|0\rangle + e^{2 \pi i 0.k_{m}} | 1 \rangle) \otimes(|0\rangle + e^{2 \pi i 0.k_{m-1}k_{m}} | 1 \rangle) \dots (|0\rangle + e^{2 \pi i 0.k_{1}k_{2} \dots k_{m}} | 1 \rangle) \end{align} I am able to understand the following part: \begin{align} F_{M}| k \rangle &= \frac{1}{\sqrt{M}} \sum_{j = 0}^{2^{m}-1} e^{2 \pi i \frac{1}{2^m} jk }| j \rangle \end{align} I am also able to understand the second part: \begin{align} &= \frac{1}{\sqrt{2^m}} (|0\rangle + e^{2 \pi i 0.k_{m}} | 1 \rangle) \cdot (|0\rangle + e^{2 \pi i 0.k_{m-1}k_{m}} | 1 \rangle) \dots (|0\rangle + e^{2 \pi i 0.k_{1}k_{2} \dots k_{m}} | 1 \rangle) \end{align} I get stuck at the third part. I do understand that applying a Walsh-Hadamard transform on a set of bits produces the outcome: \begin{align} H^{\bigotimes n}|y\rangle_{n} &= \sum_{x = 0}^{2^n-1} (-1)^{x\cdot y }|y\rangle \\ &= \frac{1}{\sqrt{2^{m}}} (|0\rangle + |1\rangle)(|0\rangle + |1\rangle) \dots (|0\rangle + |1\rangle) \end{align} In order to get a better understanding of the Quantum Fourier Transform, I would like to know why do I get the following? \begin{align} &= \frac{1}{\sqrt{2^m}} (|0\rangle + e^{2 \pi i 0.k_{m}} | 1 \rangle) \cdot (|0\rangle + e^{2 \pi i 0.k_{m-1}k_{m}} | 1 \rangle) \dots (|0\rangle + e^{2 \pi i 0.k_{1}k_{2} \dots k_{m}} | 1 \rangle) \end{align} I am aware of that $0.k_{1}k_{2} \dots k_{m}$ is a fraction in binary representation. What confuses me is why is this so important? What does that binary representation plays such a big role? I would like to note that an example on Wikipedia also has a fractional representation: http://en.wikipedia.org/wiki/Quantum_Fourier_transform#Example
Finally, to get an understanding I wrote on a sheet of a paper a specific case where $M = 2^3 =8$ then $m = 3$. Again I get the binary fractional part. From that I am unable to work further.