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code review; undo accidental minuscilization
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dwierichs committed Nov 11, 2024
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35 changes: 22 additions & 13 deletions demonstrations/tutorial_kak_theorem.py
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see our :doc:`intro to (dynamical) Lie algebras </demos/tutorial_liealgebra/>`).
A *Lie algebra* :math:`\mathfrak{g}` is a vector space with an additional operation
that takes two vectors to a new vector, the *Lie bracket*.
that takes two vectors to a new vector, the *Lie bracket*. :math:`\mathfrak{g}` must be
closed under the Lie bracket to form an algebra.
For our purposes, the vectors will always be matrices and the Lie bracket will be the matrix
commutator.
Expand All @@ -63,9 +64,9 @@
.. math::
\mathfrak{su}(2)
&= \left\{\left(\begin{array} i a & b + ic \\ -b + ic & -i a \end{array}\right)
&= \left\{i\left(\begin{array}{cc} a & b-ic \\ b+ic & -a \end{array}\right)
{\large |} a, b, c \in \mathbb{R}\right\}\\
&= \left\{i(a Z + b Y + c X)| a, b, c \in \mathbb{R}\right\}.
&= \left\{i(a Z + b X + c Y)| a, b, c \in \mathbb{R}\right\}.
We will also look at a more involved example at the end of the demo.
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# .. admonition:: Math detail: (semi-)simple Lie algebras
# :class: note
#
# Our main result for this demo will be the KAK theorem, which applies to
# so-called *semisimple* Lie algebras, which are in turn composed of *simple* Lie algebras
# as building blocks. Without going into detail, it often is sufficient to think of these
# Our main result for this demo will be the KAK theorem, which applies to so-called
# *semisimple* Lie algebras, which are in turn composed of *simple* Lie algebras as building
# blocks. Without going into detail, it often is sufficient to think of these building
# blocks as (1) special orthogonal algebras :math:`\mathfrak{so}(n),` (2) unitary symplectic
# algebras :math:`\mathfrak{sp}(n),` and (3) special unitary algebras :math:`\mathfrak{su}(n).`
# In particular, our example here is of the latter type, so it is not only semisimple,
Expand All @@ -140,7 +141,7 @@
#
# .. math::
#
# \exp : \mathfrak{g} \to \exp(\mathfrak{g})=\mathcal{G}, \ x\mapsto \exp(x).
# \mathcal{G}=\exp(\mathfrak{g}).
#
# We will only consider Lie groups :math:`\exp(\mathfrak{g})` arising from a Lie algebra
# :math:`\mathfrak{g}` here.
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#
# where we applied the exponential map to :math:`\text{ad}_x`, which maps from :math:`\mathfrak{g}`
# to itself, via its series representation.
# We will refer to this relationship as *adjoint identity*.
# We will refer to this relationship as the *adjoint identity*.
# We talk about Ad and ad in more detail in the box below, and refer to our demo
# :doc:`g-sim: Lie algebraic classical simulations </demos/tutorial_liesim/>` for
# further discussion.
Expand Down Expand Up @@ -309,9 +310,11 @@ def is_orthogonal(op, basis):
# .. admonition:: Nomenclature: Cartan decomposition/pair
# :class: warning
#
# Depending on context and field, there sometimes are additional requirements
# for :math:`\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}` to be called a Cartan decomposition
# and for :math:`(\mathfrak{k}, \mathfrak{p})` to be a Cartan pair.
# Depending on context and field, there sometimes is an additional requirement
# for :math:`\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}` to be called a Cartan decomposition.
# Without going into detail, this requirement is that the so-called *Killing form* must be
# negative definite on :math:`\mathfrak{k}` and positive definite on :math:`\mathfrak{p}`
# [#helgason]_.
#
# .. admonition:: Math detail: quotient space
# :class: note
Expand All @@ -328,7 +331,8 @@ def is_orthogonal(op, basis):
# condition holds for any :math:`g\in \mathcal{G}` are called *normal subgroups*.
# We are interested in cases where the symmetric property
# :math:`[\mathfrak{p}, \mathfrak{p}] \subset \mathfrak{k}` holds, which excludes (non-Abelian)
# normal subgroups, and thus :math:`\mathcal{G`/\mathcal{K}` will not be a group.
# normal subgroups, and thus our quotient space :math:`\mathcal{G`/\mathcal{K}` will not be
# a group.
#
# **Example**
#
Expand Down Expand Up @@ -649,6 +653,8 @@ def theta_Y(x):
# :math:`\mathfrak{so}(n)` (AI), the unitary symplectic algebra :math:`\mathfrak{sp}(n)` (AII),
# and a sum of (special) unitary algebras
# :math:`\mathfrak{su}(p)\oplus\mathfrak{su}(q)\oplus\mathfrak{u}(1)` (AIII, :math:`p+q=n`).
# For a quick overview, see for example the `Wikipedia entry on symmetric spaces
# <https://en.wikipedia.org/wiki/Symmetric_space#Classification_of_Riemannian_symmetric_spaces>`__.
# Their involutions are usually represented by complex conjugation (AI), by the adjoint
# action with a Pauli operator (AIII, for qubits, :math:`p=q=2^{N-1}`), or by both
# (AII). It is instructive to try and see why those three are *not* equivalent
Expand Down Expand Up @@ -707,6 +713,7 @@ def theta_Y(x):
# .. math::
#
# \mathcal{P}
# =\exp(\mathfrak{p})
# = \{\exp(\exp(-y) \mathfrak{a} \exp(y)) | y\in\mathfrak{k}\}
# = \{\exp(K^{-1} \mathfrak{a} K) | K\in\mathcal{K}\}
# = \{K^{-1} \mathcal{A} K | K\in\mathcal{K}\},
Expand All @@ -719,7 +726,9 @@ def theta_Y(x):
# .. math::
#
# \mathcal{G}
# = \{\exp(y_1) \exp(a) \exp(y_2) | a\in\mathfrak{a}, \ y_{1, 2}\in\mathfrak{k}\}
# =\mathcal{K}\mathcal{P}
# = \mathcal{K}\{K^{-1} \mathcal{A} K | K\in\mathcal{K}\},
# = \{K_1 \mathcal{A} K_2 | K_{1,2}\in\mathcal{K}\},
# = \mathcal{K} \mathcal{A} \mathcal{K} \qquad\textbf{(KAK Theorem).}
#
# It teaches us that any group element can be decomposed into two factors from the Lie subgroup and
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80 changes: 40 additions & 40 deletions demonstrations/tutorial_liesim.metadata.json
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Expand Up @@ -37,108 +37,108 @@
"url": "https://arxiv.org/abs/2309.05690"
},
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"id": "fontana",
"id": "Fontana",
"type": "article",
"title": "the adjoint is all you need: characterizing barren plateaus in quantum ansätze",
"authors": "enrico fontana, dylan herman, shouvanik chakrabarti, niraj kumar, romina yalovetzky, jamie heredge, shree hari sureshbabu, marco pistoia",
"title": "The Adjoint Is All You Need: Characterizing Barren Plateaus in Quantum Ansätze",
"authors": "Enrico Fontana, Dylan Herman, Shouvanik Chakrabarti, Niraj Kumar, Romina Yalovetzky, Jamie Heredge, Shree Hari Sureshbabu, Marco Pistoia",
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"authors": "michael ragone, bojko n. bakalov, frédéric sauvage, alexander f. kemper, carlos ortiz marrero, martin larocca, m. cerezo",
"title": "A Unified Theory of Barren Plateaus for Deep Parametrized Quantum Circuits",
"authors": "Michael Ragone, Bojko N. Bakalov, Frédéric Sauvage, Alexander F. Kemper, Carlos Ortiz Marrero, Martin Larocca, M. Cerezo",
"year": "2023",
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{
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"title": "lie-algebraic classical simulations for variational quantum computing",
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"title": "Lie-algebraic classical simulations for variational quantum computing",
"authors": "Matthew L. Goh, Martin Larocca, Lukasz Cincio, M. Cerezo, Frédéric Sauvage",
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"title": "quantum computation, complexity, and many-body physics",
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"title": "Quantum Computation, Complexity, and Many-Body Physics",
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"year": "2005",
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"title": "efficient solvability of hamiltonians and limits on the power of some quantum computational models",
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"title": "Efficient solvability of Hamiltonians and limits on the power of some quantum computational models",
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"title": "quantum-to-classical correspondence and hubbard-stratonovich dynamical systems, a lie-algebraic approach",
"authors": "victor galitski",
"title": "Quantum-to-Classical Correspondence and Hubbard-Stratonovich Dynamical Systems, a Lie-Algebraic Approach",
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"authors": "m. cerezo, martin larocca, diego garcía-martín, n. l. diaz, paolo braccia, enrico fontana, manuel s. rudolph, pablo bermejo, aroosa ijaz, supanut thanasilp, eric r. anschuetz, zoë holmes",
"title": "Does provable absence of barren plateaus imply classical simulability? Or, why we need to rethink variational quantum computing",
"authors": "M. Cerezo, Martin Larocca, Diego García-Martín, N. L. Diaz, Paolo Braccia, Enrico Fontana, Manuel S. Rudolph, Pablo Bermejo, Aroosa Ijaz, Supanut Thanasilp, Eric R. Anschuetz, Zoë Holmes",
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"title": "quantum computing for chemistry and physics applications from a monte carlo perspective",
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"title": "Quantum computing for chemistry and physics applications from a Monte Carlo perspective",
"authors": "Guglielmo Mazzola",
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{
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"id": "Park",
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"title": "hardware-efficient ansatz without barren plateaus in any depth",
"authors": "chae-yeun park, minhyeok kang, joonsuk huh",
"title": "Hardware-efficient ansatz without barren plateaus in any depth",
"authors": "Chae-Yeun Park, Minhyeok Kang, Joonsuk Huh",
"year": "2024",
"publisher": "",
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"doi": "10.48550/arxiv.2403.04844",
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}
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"basedonpapers": ["10.48550/arxiv.2308.01432"],
"referencedbypapers": [],
"relatedcontent": [
"basedOnPapers": ["10.48550/arXiv.2308.01432"],
"referencedByPapers": [],
"relatedContent": [
{
"type": "demonstration",
"id": "tutorial_liealgebra",
Expand All @@ -151,7 +151,7 @@
},
{
"type": "demonstration",
"id": "tutorial_how_to_optimize_qml_model_using_jax_and_optax",
"id": "tutorial_How_to_optimize_QML_model_using_JAX_and_Optax",
"weight": 1.0
},
{
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