GQE demo using Pennylane (#1119)

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**Title:**
Generative quantum eigensolver demo using Pennylane

**Summary:**
We use Pennylane to generate a static molecular dataset and calculate
the corresponding energies to train a small GPT model as described by
https://arxiv.org/abs/2401.09253. We show that as training progresses,
the GPT model generates operator sequences whose predicted energies more
accurately resembles the true energies calculated by Pennylane. In
addition, the sampling process is shown to better generate the ground
state for better performing models.

- Story ticket:
https://app.shortcut.com/xanaduai/story/64095/contribute-gqe-demo-as-a-pennylane-demo

**Relevant references:**

**Possible Drawbacks:**

**Related GitHub Issues:**

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

* GOALS — Why are we working on this now?

*Eg. Promote a new PL feature or show a PL implementation of a recent
paper.*


* AUDIENCE — Who is this for?

*Eg. Chemistry researchers, PL educators, beginners in quantum
computing.*


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


* 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
- [ ] Demo
- [ ] How-to

---------

Co-authored-by: JosephRRB <joseph.bunao@xanadu.ai>
Co-authored-by: Josh Izaac <josh146@gmail.com>
Co-authored-by: Ivana Kurečić <ivana@xanadu.ai>

Author: 4 people

_static/authors/joseph_bunao.jpg

@@ -0,0 +1,4 @@
+.. bio:: Joseph Bunao
+   :photo: ../_static/authors/joseph_bunao.jpg
+
+   Joseph is a machine learning specialist at Xanadu. His main focus is to accelerate quantum hardware research and processes using machine learning.

_static/authors/joseph_bunao.txt

@@ -0,0 +1,74 @@
+{
+    "title": "Generative quantum eigensolver training using PennyLane data",
+    "authors": [
+        {
+            "username": "Joseph"
+        }, 
+        {
+            "username": "zy_n"
+        }
+    ],
+    "dateOfPublication": "2024-09-20T00:00:00+00:00",
+    "dateOfLastModification": "2024-09-20T00:00:00+00:00",
+    "categories": ["Quantum Machine Learning", "Quantum Chemistry", "Algorithms"],
+    "tags": [],
+    "previewImages": [
+        {
+            "type": "thumbnail",
+            "uri": "_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_generative_quantum_eigensolver.png"
+        },
+        {
+            "type": "large_thumbnail",
+            "uri": "/_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_generative_quantum_eigensolver.png"
+        }
+    ],
+    "seoDescription": "Learn how you can train a small GPT model using the generative quantum eigensolver (GQE) technique and PennyLane data.",
+    "doi": "",
+    "canonicalURL": "/qml/demos/gqe_training",
+    "references": [
+        {
+            "id": "nakaji2024",
+            "type": "article",
+            "title": "The generative quantum eigensolver (GQE) and its application for ground state search",
+            "authors": "K. Nakaji, L. B. Kristensen, J. A. Campos-Gonzalez-Angulo, M. G. Vakili, H. Huang, M. Bagherimehrab, C. Gorgulla, F. Wong, A. McCaskey, J. S. Kim, T. Nguyen, P. Rao, A. Aspuru-Guzik",
+            "year": "2024",
+            "journal": "",
+            "url": "https://arxiv.org/abs/2401.09253"
+        },
+        {
+            "id": "radford2019",
+            "type": "article",
+            "title": "Language Models are Unsupervised Multitask Learners",
+            "authors": "A. Radford, J. Wu, R. Child, D. Luan, D. Amodei, I. Sutskever",
+            "year": "2019",
+            "journal": "OpenAI blog",
+            "url": "https://openai.com/index/better-language-models/"
+        },
+        {
+            "id": "vaswani2017",
+            "type": "article",
+            "title": "Attention is All you Need",
+            "authors": "A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, I. Polosukhin",
+            "year": "2017",
+            "journal": "Advances in Neural Information Processing Systems",
+            "url": "https://papers.nips.cc/paper_files/paper/2017/hash/3f5ee243547dee91fbd053c1c4a845aa-Abstract.html"
+        }
+    ],
+    "basedOnPapers": [
+        "10.48550/arXiv.2401.09253"
+    ],
+    "referencedByPapers": [],
+    "relatedContent": [
+        {
+            "type": "demonstration",
+            "id": "tutorial_vqe",
+            "weight": 1.0
+        },
+        {
+            "type": "demonstration",
+            "id": "tutorial_vqe_qng",
+            "weight": 1.0
+        }
+    ],
+    "hardware": []
+}