[WIP] Add demo on groups, FFTs and QFTs #1469
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👋 Hey, looks like you've updated some demos! 🐘 Don't forget to update the Please hide this comment once the field(s) are updated. Thanks! |
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| "dateOfPublication": "2025-07-30T10:00:00+00:00", |
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These new metadata fields will let you control the executability of the demo. Not needed now, but in case there are dependency issues in the future you can set these to false to make the demo non-executable temporarily, without having to change the name of the demo:
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| "dateOfPublication": "2025-07-30T10:00:00+00:00", | |
| ], | |
| "executable_stable": true, | |
| "executable_latest": true, | |
| "dateOfPublication": "2025-07-30T10:00:00+00:00", |
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I applied these to trigger a rebuild 😄
Co-authored-by: Paul Finlay <[email protected]>
Your preview is ready 🎉!You can view your changes here |
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@alvaro-at-xanadu the demo is now in a final draft stage, and I tagged Vasilis and Guillermo for a technical review. What else would I need to do to proceed? Shall I ask Tarik for images? :) |
Hi @mariaschuld ! Yes, feel free to touch base with Tarik for images, and let us know when technical review is ready so we can do a quick product/content review. |
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| # .. admonition:: Subgroup | ||
| # :class: note | ||
| # | ||
| # A subgroup is a subset of the original set that, under the group operation, fulfills all group axioms. |
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If you are defining the bases, you could also say what dual G was before, I actually forgot what that means 🙈
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Ah, it's complicated...that's a can of worms I'd like to keep closed! :)
| # The entire trick of the Cooley-Tukey algorithm is a change of variables, | ||
| # | ||
| # .. math:: | ||
| # \begin{align} | ||
| # x &\rightarrow 3 x_1 + x_2, \;\; x_1 = 0,1, \; x_2 = 0,1,2 \\ | ||
| # k &\rightarrow 2k_2 + k_1 , \;\; k_1 = 0,1,2 \; k_2 = 0,1 | ||
| # \end{align} | ||
| # |
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Where this come from? What should I do if I have a different m Z_m?
| # | ||
| # Essentially, the change in variables turns the Fourier basis function over period 6 into a Fourier basis function with period 2, | ||
| # which makes the dependency on :math:`k_2` disappear. This effectively turns the Fourier transform into a sum of "smaller" Fourier transforms | ||
| # over :math:`3x_1`, namely :math:`\{0, 3\}`, :math:`\{1, 4\}`, and :math:`\{2, 5\}`. These are combined with |
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not sure what (0,3),(1,4), (2,5) is referring to
| # \begin{align} | ||
| # x &\rightarrow 3 x_1 + x_2, \;\; x_1 = 0,1, \; x_2 = 0,1,2 \\ | ||
| # k &\rightarrow 2k_2 + k_1 , \;\; k_1 = 0,1,2 \; k_2 = 0,1 | ||
| # \end{align} |
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render is not ideal, maybe \quad instead of ; solve the issue
| # | ||
| # The splitting of the variable :math:`k` is related to the concept of "restricting a character" to the subgroup. | ||
| # Essentially, it allows us turn the characters, or Fourier basis functions :math:`e^{\frac{2 \pi i}{6} x k }` | ||
| # related to the original group, into characters of the subgroup, :math:`e^{\frac{2 \pi i}{2} x_1 k_1 }` by changing the period. |
| # | ||
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| ####################################################################### | ||
| # From FFTs to QFTs |
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In this section I have missed the connection with the previous section. After developing the mathematics I thought it would be clearer the connection. I saw the previous version where the formulas were and it was not clear to me neither, so I understand that it has been tried and it has not been easy 😅 (not a blocker but those were my thoughts after reading)
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Thanks @KetpuntoG I implemented all of your corrections, and it was reviewed by Joseph and Vasilis for technical content. @tarik-elkhateeb has a ticket open for the figures. Handing this over to the product team now, let me know if more work is needed :) |
This demo was internally discussed, and is supposed to enlarge our portfolio of Fourier-theoretic methods in quantum algorithms.