[WIP] [g-sim 3] (g + P)-sim

[Qottmann]

Merged the approach from #1106 with the branching approach here. This has neater illustrations anyway 👍

If you are writing a demonstration, please answer these questions to facilitate the marketing process.

• GOALS — Why are we working on this now?
  Lie algebras in QC and simulation techniques like g-sim are a hot topic due to their relevance to barren plateaus. This is an extension of g-sim to a small number of non-DLA gates. It bears some intriguing theoretical features but, like g-sim itself, is of rather limited practical relevance.

• AUDIENCE — Who is this for?
  This is an expert demo targeted towards people familiar with Lie algebras, g-sim in particular.

• KEYWORDS — What words should be included in the marketing post?

  • dynamical Lie algebra / Lie groups
  • g-sim

• Which of the following types of documentation is most similar to your file?
  (more details here)

• Tutorial
• Demo
• How-to

[sc-62859]

[github-actions]

👋 Hey, looks like you've updated some demos!

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[github-actions]

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Deployment Info:

• Pull Request ID: 1118
• Deployment SHA: 1de9b1b1439d37e2cf7801ecf55f6431a4a71fe0
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[Qottmann]

In the end I managed to combine both approaches into one, as they are effectlvely the same. The branching one is nice to understand the underlying transformations and the required maximum moment order. The g-sim in moment vector spaces approach is more practical and what one would do in the end, though I finally argue based on some scaling numbers that this approach is overall of limited practical utility and should be understood as a mere theoretical insight.

[dwierichs]

Nice that you made this work, and in two ways, even! Congrats for this nice work @Qottmann 💯
Very nice and concise read, really liked it.

I left a number of comments, as usual a good bit of them are subjective suggestions.

Regarding the direction of the drawings: I think reverting it might actually be a good idea. However, I think it should be consistent with the other demo, and as the depicted matrices/tensors are not a circuit diagram directly, it might be useful to have this visual differentiation. Also, the current ordering works with matrix product notation, which is nice :) So I think both directions have their perks.

[dwierichs]

Not sure this demo is for beginners? 😅

[Qottmann]

haha, definitely not 😆 The metadata is actually just a copy-paste from the liesim demo, I need to still update that but figured this is lowest priority atm

[dwierichs]

Thanks for all the updates! 🎉 Sorry to keep you waiting but I'd like to do one more math-only pass before approving.

[dwierichs]

Took another look and I'm comfortable with the second approach now as well :)
Really well done pulling this one off 💯 It's definitely a funky mathematical foundation to walk on! :o

Just had some tiny additional comments, including one typo for math mode.

Remember to update metadata before merging 😇

[Qottmann]

I will update the metadata together with the thumbnail

A note: I found some inconsistency in gate nomenclature — it would be worth it to check that the P gate is consistently capitalized and in mathcal or not.

The mathcal $\mathcal{P}$ is referring to the unitary gate $\mathcal{P} = e^{-i \theta h_{\mu}}$ whereas the $P$ is the adjoint action "gate" $P = e^{-i \theta ad_{h_{\mu}}}$. There was one case where a regular $p$ was not capitalized that you found, I corrected that 👍

Also, why is it called tutorial_liesim3? I assume this is for historical reasons; it might be better to pick something different (_extension)?

Good point! Will update the file names 👍

[Qottmann]

Can the website handle that? I removed all inline math from headings for this demo due to the rendering issues

[ikurecic]

Thanks, @Qottmann .
Two items remain:

• I'll look into fixing the rendering issue in the first note.
• If you could check if the demo title would display correctly if math-mode is used within it, that would be great. This issue in section titles was fixed recently.

[ikurecic]

Something went wrong here.
I'll look into it.

[Qottmann]

Yeah I think I've seen this before, just gonna remove the mathmode from the link 🙃