[Demo] Add: Unitary synthesis with recursive KAK decompositions (#1372)

**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](https://arxiv.org/abs/2503.19014).

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

- [x] Redo circuit diagrams
- [x] Add .tex files to repo
- [x] 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 (#1227, #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](https://www.notion.so/xanaduai/Different-kinds-of-documentation-69200645fe59442991c71f9e7d8a77f8))
    
- [ ] Tutorial
- [x] Demo
- [ ] How-to

---------

Co-authored-by: cognigami <53974528+cognigami@users.noreply.github.com>

Author: dwierichs

_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_unitary_synthesis_kak.png

@@ -0,0 +1,18 @@
+\documentclass[]{standalone}
+\usepackage{amsmath,quantikz,graphicx,xcolor}
+
+\begin{document}
+\begin{quantikz}
+    & & \gate{R_Y}\wire[d][3]{q} & \\
+    & \qwbundle{n-3} & |[operator]| & \\
+    & & |[operator]| & \\
+    & & |[operator]| &
+\end{quantikz}
+=
+\begin{quantikz}
+    & & \gate{R_Y(\theta_0)} & \gate{R_Y(\theta_1)} & \gate{R_Y(\theta_2)} & \ \ldots \ & \gate{R_Y(\theta_{2^{n-1}-1})} & \\
+    & \qwbundle{n-3} & \octrl{-1} & \octrl{-1} & \octrl{-1} & \ \ldots \ & \ctrl{-1} & \\
+    & & \octrl{-1} & \octrl{-1} & \ctrl{-1} & \ \ldots \ & \ctrl{-1} & \\
+    & & \octrl{-1} & \ctrl{-1} & \octrl{-1} & \ \ldots \ & \ctrl{-1} &
+\end{quantikz}
+\end{document}

_static/demo_thumbnails/opengraph_demo_thumbnails/OGthumbnail_unitary_synthesis_kak.png

@@ -0,0 +1,15 @@
+\documentclass[]{standalone}
+\usepackage{amsmath,quantikz,graphicx,xcolor}
+
+\begin{document}
+\begin{quantikz}
+    & & \gate[2]{U(2^n)} &\\
+    & \qwbundle{n-1} & &
+\end{quantikz}
+=
+\begin{quantikz}
+    & & |[operator]| & \gate{R_Y}\wire[d]{q} & |[operator]| &\\
+    & \qwbundle{n-1} & \gate{U(2^{n-1})}\wire[u]{q} & |[operator]| & \gate{U(2^{n-1})}\wire[u]{q} & 
+\end{quantikz}
+
+\end{document}