[WIP] [g-sim 3] (g + P)-sim (#1118)

Merged the approach from #1106
with the branching approach here. This has neater illustrations anyway 👍

If you are writing a demonstration, please answer these questions to
facilitate the marketing process.

* GOALS — Why are we working on this now?
Lie algebras in QC and simulation techniques like g-sim are a hot topic
due to their relevance to barren plateaus. This is an extension of g-sim
to a small number of non-DLA gates. It bears some intriguing theoretical
features but, like g-sim itself, is of rather limited practical
relevance.

* AUDIENCE — Who is this for?
This is an expert demo targeted towards people familiar with Lie
algebras, g-sim in particular.

* KEYWORDS — What words should be included in the marketing post?
  - dynamical Lie algebra / Lie groups
  - g-sim

* 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 

[sc-62859]

---------

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

demonstrations/tutorial_liesim.metadata.json

@@ -6,7 +6,7 @@
         }
     ],
     "dateOfPublication": "2024-06-07T00:00:00+00:00",
-    "dateOfLastModification": "2024-06-07T00:00:00+00:00",
+    "dateOfLastModification": "2024-06-18T00:00:00+00:00",
     "categories": [
         "Quantum Computing",
         "Getting Started"
@@ -154,6 +154,11 @@
             "type": "demonstration",
             "id": "tutorial_How_to_optimize_QML_model_using_JAX_and_Optax",
             "weight": 1.0
+        },
+        {
+            "type": "demonstration",
+            "id": "tutorial_liesim_extension",
+            "weight": 1.0
         }
     ]
 }

demonstrations/tutorial_liesim.py

@@ -68,8 +68,8 @@
     operators and explicitly add imaginary units in the exponents where appropriate.
     For more details, see the note in the "Lie algebras" section of our :doc:`Intro to (Dynamical) Lie Algebras for quantum practitioners </demos/tutorial_liealgebra//#lie-algebras>`.
 
-g-sim theory
-------------
+:math:`\mathfrak{g}`-sim theory
+-------------------------------
 
 In Lie algebra simulation, :math:`\mathfrak{g}`-sim, we are interested in how expectation values of Lie algebra elements are transformed under unitary evolution.
 We start from an initial expectation value vector of the input state :math:`\rho^0` with respect to each DLA element,
@@ -148,8 +148,8 @@
 vector in :math:`\mathfrak{g}`-sim, which is more efficient whenever :math:`\text{dim}(\mathfrak{g}) < 2^n`. In general, it is efficient
 whenever :math:`\text{dim}(\mathfrak{g}) = O\left(\text{poly}(n)\right)`.
 
-g-sim in PennyLane
-------------------
+:math:`\mathfrak{g}`-sim in PennyLane
+-------------------------------------
 
 Let us put this into practice and write a differentiable :math:`\mathfrak{g}`-simulator in PennyLane.
 We start with some boilerplate PennyLane imports.
@@ -403,6 +403,8 @@ def forward(theta):
 # and saw how this connection can be used for classical simulation. In particular, for specific systems like the TFIM we can efficiently simulate circuit 
 # expectation values.
 #
+# In case you are more curious about Lie algebraic simulation techniques, check out the follow-up demo on :doc:`(g+P)-sim </demos/tutorial_liesim_extension>`
+# an extension of :math:`\mathfrak{g}`-sim.
 
 
 

demonstrations/tutorial_liesim_extension.metadata.json

@@ -0,0 +1,99 @@
+{
+    "title": "(g + P)-sim: Extending g-sim by non-DLA observables and gates",
+    "authors": [
+        {
+            "id": "korbinian_kottmann"
+        }
+    ],
+    "dateOfPublication": "2024-06-18T00:00:00+00:00",
+    "dateOfLastModification": "2024-06-18T00:00:00+00:00",
+    "categories": [
+        "Quantum Computing",
+        "Quantum Machine Learning"
+    ],
+    "tags": [],
+    "previewImages": [
+        {
+            "type": "thumbnail",
+            "uri": "/_static/demonstration_assets/liesim_extension/thumbnail_liesim_extension.png"
+        },
+        {
+            "type": "large_thumbnail",
+            "uri": "/_static/large_demo_thumbnails/thumbnail_large_liesim_extension.png"
+        }
+    ],
+    "seoDescription": "(g + P)-sim: Extending g-sim by non-DLA observables and gates",
+    "doi": "",
+    "canonicalURL": "/qml/demos/tutorial_liesim_extension",
+    "references": [
+        {
+            "id": "Wiersema",
+            "type": "preprint",
+            "title": "Classification of dynamical Lie algebras for translation-invariant 2-local spin systems in one dimension",
+            "authors": "Roeland Wiersema, Efekan Kökcü, Alexander F. Kemper, Bojko N. Bakalov",
+            "year": "2023",
+            "publisher": "",
+            "journal": "",
+            "doi": "10.48550/arXiv.2309.05690",
+            "url": "https://arxiv.org/abs/2309.05690"
+        },
+        {
+            "id": "Goh",
+            "type": "preprint",
+            "title": "Lie-algebraic classical simulations for variational quantum computing",
+            "authors": "Matthew L. Goh, Martin Larocca, Lukasz Cincio, M. Cerezo, Frédéric Sauvage",
+            "year": "2023",
+            "publisher": "",
+            "journal": "",
+            "doi": "10.48550/arXiv.2308.01432",
+            "url": "https://arxiv.org/abs/2308.01432"
+        },
+        {
+            "id": "Somma",
+            "type": "preprint",
+            "title": "Quantum Computation, Complexity, and Many-Body Physics",
+            "authors": "Rolando D. Somma",
+            "year": "2005",
+            "publisher": "",
+            "journal": "",
+            "doi": "10.48550/arXiv.quant-ph/0512209",
+            "url": "https://arxiv.org/abs/quant-ph/0512209"
+        },
+        {
+            "id": "Somma2",
+            "type": "preprint",
+            "title": "Efficient solvability of Hamiltonians and limits on the power of some quantum computational models",
+            "authors": "Rolando Somma, Howard Barnum, Gerardo Ortiz, Emanuel Knill",
+            "year": "2006",
+            "publisher": "",
+            "journal": "",
+            "doi": "10.48550/arXiv.quant-ph/0601030",
+            "url": "https://arxiv.org/abs/quant-ph/0601030"
+        },
+        {
+            "id": "Galitski",
+            "type": "preprint",
+            "title": "Quantum-to-Classical Correspondence and Hubbard-Stratonovich Dynamical Systems, a Lie-Algebraic Approach",
+            "authors": "Victor Galitski",
+            "year": "2010",
+            "publisher": "",
+            "journal": "",
+            "doi": "10.48550/arXiv.1012.2873",
+            "url": "https://arxiv.org/abs/1012.2873"
+        }
+    ],
+    "basedOnPapers": ["10.48550/arXiv.2308.01432"],
+    "referencedByPapers": [],
+    "relatedContent": [
+        {
+            "type": "demonstration",
+            "id": "tutorial_liealgebra",
+            "weight": 1.0
+        },
+        {
+            "type": "demonstration",
+            "id": "tutorial_liesim",
+            "weight": 1.0
+        }
+    ]
+}