I have been studying on Quantum Fourier Transform (QFT) by myself, and I am little bit confused about how could QFT be used. For example, if the QFT of three quantum bits is
$a_1|000\rangle + a_2|001\rangle + a_3|010\rangle + a_4|011\rangle + a_5|100\rangle + a_6|101\rangle + a_7|110\rangle+ a_8|111\rangle$
three questions arising in my mind:
- Whether or not the useful information is represented by the probability amplitude coefficients ($a_1,..., a_8$) of the superposition state? Are these coefficients the same as their counterparts in DFT?
- To my best knowledge, once a quantum system is measured, the superposition state will be destroyed, and the information represented by the probability coefficients ($a_1, ..., a_8$) will be lost. Then how could the coefficients be extracted from by measurements?
- How is QFT related with Shor’s Algorithm?
Thank you by advance for answers.