[Draft] Demo: The KAK theorem #1227

dwierichs · 2024-10-03T07:16:35Z

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

github-actions · 2024-10-03T07:16:46Z

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

🐘 Don't forget to update the dateOfLastModification in the associated metadata files so your changes are reflected in Glass Onion (search and recommendations).

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

dwierichs · 2024-10-14T12:52:55Z

demonstrations/tutorial_kak_theorem.py

+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
+----------


Hi @Qottmann,
can I ask you for some super-preliminary feedback on this, regarding

overall structure/logic of the outline

amount of content (also happy about suggestions of things I missed/I should throw out!)

Thanks! 🙏

It's a lot of ground to cover with quite involved topics. Do you reckon it is realistic to put this all in a demo? This seems like review paper level thoroughness (which makes me think, should we write a review paper on this stuff for physicists?) I actually wouldnt want to scratch anything because I'd be curious to read all of this myself, I am just wondering if you think it's realistic for you to cover all these things in reasonable time (your own time effort) and in a presentation that is not too long (time effort of a reader)?

It is a very fair question. Taking some of this offline.
For the length I see the following options:

I try to do the shortest version of the demo that I deem understandable and we see whether that already is a good length/complexity. This will have to skip over quite a few things from above.

We split content up into multiple bits.

We cover all the content but go very superficial. Not sure that's so nice to have 🤔

Qottmann · 2024-10-14T13:14:37Z

demonstrations/tutorial_kak_theorem.py

+- 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.


oh good to know! I wasnt aware!

The precise criterion is that the center is trivial, which is a bit more general. But a center would be an Abelian ideal, which semisimple Lie algebras don't have, so semisimple implies faithful ad.rep.

Qottmann · 2024-10-14T13:20:05Z

demonstrations/tutorial_kak_theorem.py

+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
+----------


It's a lot of ground to cover with quite involved topics. Do you reckon it is realistic to put this all in a demo? This seems like review paper level thoroughness (which makes me think, should we write a review paper on this stuff for physicists?) I actually wouldnt want to scratch anything because I'd be curious to read all of this myself, I am just wondering if you think it's realistic for you to cover all these things in reasonable time (your own time effort) and in a presentation that is not too long (time effort of a reader)?

Qottmann · 2024-10-14T13:20:42Z

demonstrations/tutorial_kak_theorem.py

+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


If things had to be scratched, Perhaps this could be made a separate demo (though I am very interested in the contents here so maybe we can already talk about it)

Yeah, maybe we could actually have this demo focus on KAK theorem, including only the 2-qubit application.
Then we can have a Khaneja-Glaser decomposition demo as a follow-up?

Yeah something like that, or even more generally a demo on recursive decomposition, but I think that would depend on our findings this quarter and how much we want to invest (seems like a lot of work to do all of this 😅 )

Qottmann · 2024-10-14T13:21:23Z

demonstrations/tutorial_kak_theorem.py

+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)


Not sure how much you plan on talking about these points but could be kept brief

github-actions · 2024-10-23T14:00:45Z

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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 · 2024-10-25T14:31:11Z

demonstrations/tutorial_kak_theorem.py

+.. math::
+
+    \mathfrak{su}(2)
+    &= \left\{x \in \mathbb{C}^{(2\times 2)} {\large|} x^\dagger = -x , \text{tr}[x]=0\right\}\\
+    &= \left\{\left(\begin{array} i a & b + ic \\ -b + ic & -i a \end{array}\right)
+    {\large |} a, b, c \in \mathbb{R}\right\}.


why not start with pauli matrices from the get-go? 😅

Qottmann · 2024-10-25T14:33:00Z

demonstrations/tutorial_kak_theorem.py

+    There is a simple way to see this; Multiplying a skew-Hermitian matrix
+    :math:`x\in\mathfrak{su}(n)` by a complex number :math:`c\in\mathbb{C}` will yield
+    :math:`(cx)^\dagger=\overline{c} x^\dagger=-\overline{c} x,` so that
+    the result might no longer be in the algebra! If we keep it to real scalars
+    :math:`c\in\mathbb{R}` only, we have :math:`\overline{c}=c,` so that
+    :math:`(cx)^\dagger=-cx` and we're fine.


Suggested change

There is a simple way to see this; Multiplying a skew-Hermitian matrix

:math:`x\in\mathfrak{su}(n)` by a complex number :math:`c\in\mathbb{C}` will yield

:math:`(cx)^\dagger=\overline{c} x^\dagger=-\overline{c} x,` so that

the result might no longer be in the algebra! If we keep it to real scalars

:math:`c\in\mathbb{R}` only, we have :math:`\overline{c}=c,` so that

:math:`(cx)^\dagger=-cx` and we're fine.

There is a simple way to see this; Multiplying a skew-Hermitian matrix

:math:`x\in\mathfrak{su}(n)` by a complex number :math:`c\in\mathbb{C}` will yield

:math:`(cx)^\dagger=\overline{c} x^\dagger=-\overline{c} x,` so that

the result might no longer be skew-Hermitian, i.e. no longer in the algebra! If we keep it to real scalars

:math:`c\in\mathbb{R}` only, we have :math:`\overline{c}=c,` so that

:math:`(cx)^\dagger=-cx` and we're fine.

(personal preference)

Qottmann · 2024-10-25T14:33:56Z

demonstrations/tutorial_kak_theorem.py

+all_hermitian = all(qml.equal(qml.adjoint(op).simplify(), op) for op in su2)
+print(f"The operators are all Hermitian: {all_hermitian}")
+
+su2_lie_closed = qml.pauli.dla.lie_closure(su2)


Suggested change

su2_lie_closed = qml.pauli.dla.lie_closure(su2)

su2_lie_closed = qml.lie_closure(su2)

Qottmann · 2024-10-25T15:01:45Z

demonstrations/tutorial_kak_theorem.py

+# :doc:`g-sim: Lie algebraic classical simulations </demos/tutorial_liesim/>` for
+# further discussion.
+#
+# .. admonition:: Derivation: Adjoint representations


Qottmann · 2024-10-25T15:02:16Z

demonstrations/tutorial_kak_theorem.py

+#
+#     .. math::
+#
+#         \text{Ad}_{\exp(x)} : \mathfrak{g} \to \mathfrak{g},


in the previous definition its mapping $\mathcal{G} \times \mathfrak{g} \rightarrow \mathfrak{g}$

Qottmann · 2024-10-25T15:57:59Z

demonstrations/tutorial_kak_theorem.py

+# factorizations in general, and for quantum circuit decompositions in particular.
+#


would be nice to also discuss kak decompositions for specific ops (i.e. writing some_op in g = K_1 a K_2) as well as the version where the op is in m and we can get away with KAK^dagger

Qottmann · 2024-10-25T15:59:42Z

demonstrations/tutorial_kak_theorem.py

+p = check_cartan_decomposition(su4, su2_su2, space_name)
+
+######################################################################
+# .. admonition:: Mathematical detail


super nice!

Qottmann · 2024-10-25T16:00:33Z

demonstrations/tutorial_kak_theorem.py

+#
+#     \mathfrak{a} = \text{span}_{\mathbb{R}}\{iX_0X_1, iY_0Y_1, iZ_0Z_1\}
+#
+# Clearly, these three operators commute, making :math:`\mathfrak{a}` Abelian.


Suggested change

# Clearly, these three operators commute, making :math:`\mathfrak{a}` Abelian.

# These three operators commute, making :math:`\mathfrak{a}` Abelian.

Qottmann · 2024-10-25T16:02:07Z

demonstrations/tutorial_kak_theorem.py

+fig, ax = qml.draw_mpl(su4_gate, wire_order=[0, 1])(params)
+
+######################################################################
+# And that's a wrap on our KAK theorem application for two-qubit gates!


can we easily get the optimal 2qubit decomposition from this somehow?

Qottmann · 2024-10-25T16:02:59Z

demonstrations/tutorial_kak_theorem.py

+# :math:`y_{1,2}` and :math:`a` for a given gate :math:`U.` If you are
+# curious about this question, watch out for follow-up demos!


Suggested change

# :math:`y_{1,2}` and :math:`a` for a given gate :math:`U.` If you are

# curious about this question, watch out for follow-up demos!

# :math:`y_{1,2}` and :math:`a` for a given gate :math:`U.`

wouldnt make promises, I'd directly refer to the Kökcü paper or other resources

add dummy files

15c4690

dwierichs added 3 commits October 3, 2024 14:35

some text and structure ideas

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Merge branch 'master' into kak-theorem

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dwierichs changed the title ~~[Empty Draft] Demo: The KAK theorem~~ [Draft] Demo: The KAK theorem Oct 23, 2024

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[Draft] Demo: The KAK theorem #1227

[Draft] Demo: The KAK theorem #1227

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	su2_lie_closed = qml.pauli.dla.lie_closure(su2)
	su2_lie_closed = qml.lie_closure(su2)

		# factorizations in general, and for quantum circuit decompositions in particular.
		#

	# Clearly, these three operators commute, making :math:`\mathfrak{a}` Abelian.
	# These three operators commute, making :math:`\mathfrak{a}` Abelian.

		# :math:`y_{1,2}` and :math:`a` for a given gate :math:`U.` If you are
		# curious about this question, watch out for follow-up demos!

[Draft] Demo: The KAK theorem #1227

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[Draft] Demo: The KAK theorem #1227

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