[Demo] The KAK theorem

[dwierichs]

Title:
The KAK theorem

Summary:
The KAK theorem is a group theoretical tool to decompose operators into a sequence of smaller operators.
It brings an abstract mathematical structure to direct use in compilation and simulation tasks.

In this demo we will explain the mathematical objects "Lie subalgebra", "Cartan involution", and "symmetric space", which are prerequisites to the KAK theorem.
Then we state the theorem and explain how it powers a standard circuit decomposition/template construction technique, which also proves the universality of single- and two-qubit operations for quantum computing.
All steps are illustrated with mathematical and code examples.

Relevant references:

TBD

Possible Drawbacks:
N/A

Related GitHub Issues:
TBD: FDHS demo PR

[sc-74884]

---

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

• GOALS — Why are we working on this now?

• The theorem uses Lie algebra theory, for which functionalities were recently added to PennyLane.
• It can be seen as a follow-up/extension/application of the more basic Introduction to Lie algebras demo and it is related to content about Lie algebraic simulation (g-sim| (g+P)-sim).
• There will be a close follow-up demo on "Fixed-depth Hamiltonian simulation", which makes use of the KAK theorem.

• AUDIENCE — Who is this for?

• Researchers/Learners that would like to understand the mathematical structure behind the KAK theorem and Khaneja-Glaser circuit templates.
• Researchers/Learners that would like to familiarize themselves with Lie algebraic tools in PennyLane beyond a basic introduction and the simulation-related tools in the existing content pieces.
• Researchers/Learners that would like to read/understand the "Fixed-depth Hamiltonian simulation" demo.

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

"KAK theorem"
"Lie algebra"
"Symmetric space"
"Cartan decomposition" and/or "Cartan involution"
"Khaneja-Glaser decomposition"
"Circuit templates"
"Universality"

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

• Tutorial
• Demo
• How-to

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[github-actions]

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[Qottmann]

I absolutely love this and can see this become a super valuable resource for anyone with a physics brackground wanting to dive deeper into the topic (think, in particular, future residents)

In terms of length, it is indeed a chonker but honestly that is fine for this kind of demo. I'd even go as far as saying that it could do with more content, in particular a more non-trivial example beyond su(2). At least for my taste, feel free to go all in.

We should perhaps have someone that is unfamiliar with these concepts also read the demo

[Qottmann]

Suggested change

	# :math:`g'' K` if :math:`g^{-1} K g\subset K.` Subgroups for which this condition holds
	# :math:`g'' K` if :math:`g^{-1} K g\subset G.` Subgroups for which this condition holds

?

[dwierichs]

Mhm. I think K is correct 🤔 The condition is that some arbitrary $g$ that might not be in $K$ also leaves $K$ untouched, which is a strong condition ($K$ being a normal subgroup) and usually is not true.
But maybe I confused myself 😅