[Demo] The KAK theorem (#1227)

**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:**
See end of demo file or the metadata file

**Possible Drawbacks:**
N/A

**Related GitHub Issues:**
#1261 

[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](https://pennylane.ai/qml/demos/tutorial_liealgebra/) demo and
it is related to content about Lie algebraic simulation
([g-sim](https://pennylane.ai/qml/demos/tutorial_liesim/)|
[(g+P)-sim](https://pennylane.ai/qml/demos/tutorial_liesim_extension/)).
- There will be a close follow-up demo on "Fixed-depth Hamiltonian
simulation" (#1261), 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 (to be used for Khaneja-Glaser circuit
templates, for example).
- 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 coming up in #1261.

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

"KAK theorem"
"Lie algebra"
"Symmetric space"
"Cartan decomposition" and/or "Cartan involution"
"Circuit templates"

* 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: Korbinian Kottmann <[email protected]>
Co-authored-by: Ivana Kurečić <[email protected]>

conf.py

@@ -84,7 +84,7 @@
     "backreferences_dir"  : "backreferences",
     "doc_module"          : ("pennylane"),
     "junit": "../test-results/sphinx-gallery/junit.xml",
-    'reset_modules': ("module_resets.reset_jax", "matplotlib", "seaborn"),
+    "reset_modules": ("module_resets.reset_jax", "matplotlib", "seaborn"),
 }
 
 

demonstrations/tutorial_kak_theorem.metadata.json

@@ -0,0 +1,174 @@
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