[Demo] Add: Unitary synthesis with recursive KAK decompositions

[dwierichs]

Title:
Unitary synthesis with recursive KAK decompositions

Summary:
Unitary synthesis, the process of compiling a unitary matrix into a quantum circuit, has been a topic of study for over
25 years. Three interesting techniques for unitary synthesis, namely the first, the (probably) most widely known, and the minimal-CNOT-count ("the best"), all use the same recursive KAK decomposition under the hood, as we recently
elaborated in a paper.

This demo builds on our demo on KAK decompositions (added in #1227) and explains how such decompositions can be applied recursively to construct a many-qubit circuit and how the three mentioned techniques all are powered by the same recursion.
It defers some optimization details to the literature but provides a brief gate count overview and showcases a simplified numerical implementation of all three techniques.

Relevant references:
Quite a few, see metadata file.

Possible Drawbacks:
N/A

Related GitHub Issues:

[sc-87548]

To do

• Redo circuit diagrams
• Add .tex files to repo
• add examples/code

---

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

• GOALS — Why are we working on this now?

• Promote qml.liealg features
• [promote a paper we recently did (see link above)]
• Follow up on started line of demos ([Demo] The KAK theorem #1227, Fixed Depth Hamiltonian Simulation via Cartan Decomposition demo #1261)

• AUDIENCE — Who is this for?

• Researchers in quantum compilation and quantum information
• Practicioners that want to learn where the complexity of their operations comes from

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

• (Recursive) KAK decomposition
• Quantum Shannon decomposition
• Block-ZXZ decomposition
• Unitary synthesis
• Compilation

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

• Tutorial
• Demo
• How-to

[github-actions]

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

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Please hide this comment once the field(s) are updated. Thanks!

[github-actions]

Thank you for opening this pull request.

You can find the built site at this link.

Deployment Info:

• Pull Request ID: 1372
• Deployment SHA: 2eafab5edf9899ed4c60fd18b393b356de0b2045
  (The Deployment SHA refers to the latest commit hash the docs were built from)

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[Qottmann]

First look and this looks great, learned a lot !

I think in the first part some numerical or analytic examples could help

I dont know how much you planned for the gate counts, stating the logic of counting and then the results should be sufficient imo but feel free to go wild 😆 🚀

[Qottmann]

No preview :(

TypeError: In-place matrix multiplication is not (yet) supported. Use 'a = a @ b' instead of 'a @= b'.

[Qottmann]

Pretty happy with this now :) thanks for adding the diagrams, I think they make a huge difference!