# Quantum Fourier Transform of Periodic States

1. Oct 17, 2011

### maverick280857

Hi,

This is probably trivial, but I don't see it and would therefore appreciate receving inputs.

Suppose we define a state

$$|\phi_{lr}\rangle = \sum_{n=0}^{N/r - 1}\sqrt{\frac{r}{N}}|l + n r\rangle$$

How is the quantum Fourier transform of this state equal to

$$|\tilde{\phi}_{lr}\rangle = \sum_{m=0}^{r-1}\alpha_{m}\left|\frac{m N}{r}\right\rangle$$

where $|\alpha| = \sqrt{1/r}$ for all $m$?

This is from http://www-bcf.usc.edu/~tbrun/Course/lecture13.pdf.

2. Apr 10, 2012

### i2000s

I have the same question on it... Any help?

3. Apr 10, 2012

### Halcyon-on

you must use the Poisson summation formula.

4. Apr 10, 2012

### i2000s

I have just proved it using similar formulas. Thanks!