Quantum Fourier Transform of Periodic States

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Discussion Overview

The discussion centers on the quantum Fourier transform of a specific periodic quantum state, denoted as |\phi_{lr}\rangle. Participants are exploring the mathematical transformation of this state and its representation in terms of another state |\tilde{\phi}_{lr}\rangle, as referenced in a lecture document.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant presents a state |\phi_{lr}\rangle and inquires how its quantum Fourier transform results in |\tilde{\phi}_{lr}\rangle, suggesting a specific form for the coefficients |\alpha|.
  • Another participant expresses a similar question, indicating that they are seeking clarification on the transformation process.
  • A third participant suggests that the Poisson summation formula must be utilized in the transformation process.
  • A fourth participant claims to have proved the transformation using similar formulas, indicating some progress in understanding the topic.

Areas of Agreement / Disagreement

There is no consensus on the method to achieve the quantum Fourier transform, as participants have differing approaches and levels of understanding. Some propose using specific mathematical tools, while others are still seeking clarification.

Contextual Notes

The discussion may be limited by assumptions regarding the definitions of the states involved and the mathematical steps required for the transformation, which remain unresolved.

maverick280857
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Hi,

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

Suppose we define a state

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

How is the quantum Fourier transform of this state equal to

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

where [itex]|\alpha| = \sqrt{1/r}[/itex] for all [itex]m[/itex]?

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

Thanks in advance!
 
Physics news on Phys.org
I have the same question on it... Any help?
 
you must use the Poisson summation formula.
 
I have just proved it using similar formulas. Thanks!
 

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