[Draft] Demo: The KAK theorem

demonstrations/tutorial_kak_theorem.metadata.json

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+{
+    "title": "The KAK theorem",
+    "authors": [
+        {
+            "username": "dwierichs"
+        }
+    ],
+    "dateOfPublication": "2024-12-05T00:00:00+00:00",
+    "dateOfLastModification": "2024-12-05T00:00:00+00:00",
+    "categories": [
+        "Quantum Computing",
+        "Algorithms"
+    ],
+    "tags": [],
+    "previewImages": [
+        {
+            "type": "thumbnail",
+            "uri": "/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_kak_theorem.png"
+        },
+        {
+            "type": "large_thumbnail",
+            "uri": "/_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_kak_theorem.png"
+        }
+    ],
+    "seoDescription": "Learn about the KAK theorem and how it powers circuit decompositions.",
+    "doi": "",
+    "canonicalURL": "/qml/demos/tutorial_kak_theorem",
+    "references": [
+    ],
+    "basedOnPapers": [],
+    "referencedByPapers": [],
+    "relatedContent": [
+        {
+            "type": "demonstration",
+            "id": "tutorial_liealgebra",
+            "weight": 1.0
+        }
+    ]
+}

demonstrations/tutorial_kak_theorem.py

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+r"""The KAK theorem
+===================
+
+The KAK theorem is a beautiful mathematical result from Lie theory, with
+particular relevance for quantum computing research. It can be seen as a
+generalization of the singular value decomposition, and therefore falls
+under the large umbrella of matrix factorizations. This allows us to
+use it for quantum circuit decompositions. However, it can also
+be understood from a more abstract point of view, as we will see.
+
+In this demo, we will discuss so-called symmetric spaces, which arise from
+subgroups of Lie groups. For this, we will focus on the algebraic level
+and introduce Cartan involutions/decompositions, horizontal
+and vertical subspaces, as well as horizontal Cartan subalgebras.
+With these tools in our hands, we will then learn about the KAK theorem
+itself.
+We conclude with a famous application of the theorem to circuit decomposition
+by Khaneja and Glaser [#khaneja_glaser]_, which provides a circuit
+template for arbitrary unitaries on any number of qubits, and proved for
+the first time that single and two-qubit gates are sufficient to implement them.
+
+While this demo is of more mathematical nature than others, we will include
+hands-on examples throughout.
+
+.. figure:: ../_static/demo_thumbnails/opengraph_demo_thumbnails/OGthumbnail_kak_theorem.png
+    :align: center
+    :width: 60%
+    :target: javascript:void(0)
+
+.. note::
+
+    In the following we will assume a basic understanding of vector spaces,
+    linear maps, and Lie algebras. For the former two, we recommend a look
+    at your favourite linear algebra material, for the latter see our
+    :doc:`introduction to (dynamical) Lie algebras </demos/tutorial_liealgebra/>`.
+
+
+Introduction
+------------
+
+Basic mathematical objects
+--------------------------
+
+Introduce the mathematical objects that will play together to yield
+the KAK theorem.
+
+(Semi-)simple Lie algebras
+~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+- Introduce the notion of a Lie algebra very briefly, refer to existing demo(s).
+- Focus on vector space notion being clear.
+- [optional] Briefly say what a simple/semisimple Lie algebra is.
+- [optional] In particular mention that the adjoint representation is faithful for semisimple algebras.
+
+Group and algebra interaction
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+- Exponential map
+- adjoint action of group on algebra
+- adjoint action of algebra on algebra -> adjoint representation
+- adjoint identity (-> g-sim demo)
+
+Subalgebras and Cartan pairs
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+- Introduce the notion of a subalgebra.
+- Explain that there can be vector subspaces that are not subalgebras.
+- Define Cartan pairs via commutation relations
+
+Cartan subalgebras
+~~~~~~~~~~~~~~~~~~
+
+- Define Cartan subalgebras of :math:`m`.
+- Dimension of Cartan subalgebras
+- Transition between Cartan subalgebras via :math:`K`
+
+Involutions
+~~~~~~~~~~~
+
+- Explain linear maps on (matrix) algebras (-> homomorphism)
+- Define involutions.
+- Involutions define Cartan pairs (:math:`k = +1 | m = -1` eigenspaces)
+- Cartan pairs define involutions :math:`\theta = \Pi_{\mathfrak{k}} - \Pi_{\mathfrak{m}}`
+
+KAK theorem
+~~~~~~~~~~~
+
+- KP decomposition
+- KAK decomposition
+- [optional] implication: KaK on algebra level
+
+
+Two-qubit KAK decomposition
+---------------------------
+
+- Algebra/subalgebra :math:`\mathfrak{g} =\mathfrak{su}(4) | \mathfrak{k} =\mathfrak{su}(2) \oplus \mathfrak{su}(2)`
+- Involution: EvenOdd
+- CSA: :math:`\mathfrak{a} = \langle\{XX, YY, ZZ\}\rangle_{i\mathbb{R}}`
+- KAK decomposition :math:`U= (A\otimes B) \exp(i(\eta_x XX+\eta_y YY +\eta_z ZZ)) (C\otimes D)`.
+- [optional] Mention Cartan coordinates
+
+Khaneja-Glaser decomposition
+----------------------------
+
+- Important first recursive decomposition showing universality of single- and two-qubit operations
+- Used for practical decompositions, replaced by other, similar decompositions by now
+
+A recursive decomposition
+~~~~~~~~~~~~~~~~~~~~~~~~~
+
+- Show recursion on qubit count
+- display resulting decomposition structure
+- Mention that a two-qubit interaction is enough to get the CSA elements
+- Universality
+
+The recursion step in detail
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+- Two substeps in each recursion step:
+    - Algebra/subalgebra :math:`\mathfrak{g}=\mathfrak{su}(2^n) | \mathfrak{k} = \mathfrak{su}(2^{n-1}) \oplus \mathfrak{su}(2^{n-1})`
+    - Involution TBD
+    - CSA TBD
+    - Algebra/subalgebra :math:`\mathfrak{g}=\mathfrak{su}(2^{n-1}) \oplus \mathfrak{su}(2^{n-1}) | \mathfrak{k} = \mathfrak{su}(2^{n-1})`
+    - Involution TBD
+    - CSA TBD
+
+Overview of resulting decomposition
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+- Count blocks
+- [optional] CNOT count
+
+
+Conclusion
+----------
+
+In this demo we learned about the KAK theorem and how it uses a Cartan
+decomposition of a Lie algebra to decompose its Lie group.
+A famous immediate application of this result is the circuit decomposition, or
+parametrization, for arbitrary qubit numbers by Khaneja and Glaser. It also allowed
+us to prove universality of single and two-qubit unitaries for quantum computation.
+
+If you are interested in other applications of Lie theory in the field of
+quantum computing, you are in luck! It has been a handy tool throughout the last
+decades, e.g., for the simulation and compression of quantum circuits, # TODO: REFS
+in quantum optimal control, and for trainability analyses. For Lie algebraic
+classical simulation of quantum circuits, check the
+:doc:`g-sim </demos/tutorial_liesim/>` and
+:doc:`(g+P)-sim </demos/tutorial_liesim_extension/>` demos, and stay posted for
+a brand new demo on compiling Hamiltonian simulation circuits with the KAK theorem!
+
+
+The props
+---------
+
+Adjoint representation
+~~~~~~~~~~~~~~~~~~~~~~
+
+"""
+
+import pennylane as qml
+
+######################################################################
+#
+# References
+# ----------
+#
+# .. [#khaneja_glaser]
+#
+#     Navin Khaneja, Steffen Glaser
+#     "Cartan decomposition of SU(2^n), constructive controllability of spin systems and universal quantum computing"
+#     `arXiv:quant-ph/0010100 <https://arxiv.org/abs/quant-ph/0010100>`__, 2000
+#
+# About the author
+# ----------------