add pvqnn (#1154)

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**Title:**
Post-Variational Quantum Neural Networks

**Summary:**
In this demo, we discuss “post-variational strategies”, where we take
the classical combination of multiple fixed quantum circuits and find
the optimal combination through feeding our combinations through a
classical multilayer perceptron. We shift tunable parameters from the
quantum computer to the classical computer, opting for ensemble
strategies when optimizing quantum models.

**Relevant references:**
P.-W. Huang, P. Rebentrost (2023). Post-variational quantum neural
networks. arXiv:2307.10560 [quant-ph]

**Possible Drawbacks:**
Scalability. 

**Related GitHub Issues:**
NA
----
If you are writing a demonstration, please answer these questions to
facilitate the marketing process.

* GOALS — Why are we working on this now?

Introduce a new architecture for quantum machine learning.

* AUDIENCE — Who is this for?

Academic Researchers and Students, Quantum Technology enthusiasts 

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

Quantum Machine Learning, Neural Networks, Post-Variational

* 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
- [ v] Demo
- [ ] How-to

---------

Co-authored-by: Elaina Gray <[email protected]>
Co-authored-by: Huang Po-Wei <[email protected]>
Co-authored-by: Ivana Kurečić <[email protected]>
Co-authored-by: Po-Wei (George) Huang <[email protected]>
Co-authored-by: Ivana Kurečić <[email protected]>

_static/authors/elaina_zhu.txt

@@ -0,0 +1,4 @@
+.. bio:: Elaina Zhu
+   :photo: ../_static/authors/elaina_zhu.jpeg
+
+   Elaina is a undergraduate student at Oxford University, interning at CQT. She is interested in research in the intersection of quantum technologies and computer science.

_static/authors/po_wei_huang.txt

@@ -0,0 +1,4 @@
+.. bio:: Po-Wei Huang
+   :photo: ../_static/authors/po_wei_huang.jpeg
+
+   Po-Wei is a research assistant at CQT and a DPhil student at the University of Oxford. He is interested in research on algorithms and machine learning for quantum computation. You can find more about him at georgepwhuang.github.io

demonstrations/tutorial_post-variational_quantum_neural_networks.metadata.json

@@ -0,0 +1,135 @@
+{
+    "title": "Post Variational Quantum Neural Networks",
+    "authors": [
+        {
+            "username": "elainazhu"
+        },
+        {
+            "username": "georgepwhuang"
+        }
+    ],
+    "dateOfPublication": "2024-10-07T00:00:00+00:00",
+    "dateOfLastModification": "2024-10-07T00:00:00+00:00",
+    "categories": [
+        "Quantum Machine Learning",
+        "Algorithms",
+        "Quantum Computing"
+    ],
+    "tags": [],
+    "previewImages": [
+        {
+            "type": "thumbnail",
+            "uri": "_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_post-variational_quantum_neural_networks.png"
+        },
+        {
+            "type": "large_thumbnail",
+            "uri": "_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_post-variational_quantum_neural_networks.png"
+        }
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+    "doi": "",
+    "canonicalURL": "/qml/demos/tutorial_post-variational_quantum_neural_networks",
+    "references": [
+        {
+            "id": "cerezo2021variational",
+            "type": "article",
+            "title": "Variational quantum algorithms",
+            "authors": "Marco Cerezo and Andrew Arrasmith and Ryan Babbush and Simon C. Benjamin and Suguru Endo and Keisuke Fujii and Jarrod R. McClean and Kosuke Mitarai and Xiao Yuan and Lukasz Cincio and Patrick J. Coles",
+            "year": "2021",
+            "publisher": "Nature Portfolio",
+            "journal": "Nat. Rev. Phys.",
+            "doi": "10.1038/s42254-021-00348-9",
+            "url": "https://doi.org/10.1038/s42254-021-00348-9"
+        },
+        {
+            "id": "mcclean2018barren",
+            "type": "article",
+            "title": "Barren plateaus in quantum neural network training landscapes",
+            "authors": "Jarrod R. McClean and Sergio Boixo and Vadim N. Smelyanskiy and Ryan Babbush and Hartmut Neven",
+            "year": "2018",
+            "publisher": "Nature Portfolio",
+            "journal": "Nat. Commun.",
+            "doi": "10.1038/s41467-018-07090-4",
+            "url": "https://doi.org/10.1038/s41467-018-07090-4"
+        },
+        {
+            "id": "du2020expressive",
+            "type": "article",
+            "title": "Expressive power of parametrized quantum circuits",
+            "authors": "Du, Yuxuan and Hsieh, Min-Hsiu and Liu, Tongliang and Tao, Dacheng",
+            "year": "2020",
+            "publisher": "American Physical Society",
+            "journal": "Phys. Rev. Res.",
+            "doi": "10.1103/PhysRevResearch.2.033125",
+            "url": "https://doi.org/10.1103/PhysRevResearch.2.033125"
+        },
+        {
+            "id": "schuld2019evaluating",
+            "type": "article",
+            "title": "Evaluating analytic gradients on quantum hardware",
+            "authors": "Schuld, Maria and Bergholm, Ville and Gogolin, Christian and Izaac, Josh and Killoran, Nathan",
+            "year": "2019",
+            "publisher": "American Physical Society",
+            "journal": "Phys. Rev. A",
+            "doi": "10.1103/PhysRevA.99.032331",
+            "url": "https://doi.org/10.1103/PhysRevA.99.032331"
+        },
+        {
+            "id": "huang2024postvariational",
+            "type": "article",
+            "title": "Post-variational quantum neural networks",
+            "authors": "Po-Wei Huang and Patrick Rebentrost",
+            "year": "2024",
+            "doi": "10.48550/arXiv.2307.10560",
+            "url": "https://arxiv.org/abs/2307.10560"
+        },
+        {
+            "id": "huang2020predicting",
+            "type": "article",
+            "title": "Predicting many properties of a quantum system from very few measurements",
+            "authors": "Hsin-Yuan Huang and Richard Kueng and John Preskill",
+            "year": "2024",
+            "doi": "10.1038/s41567-020-0932-7",
+            "url": "https://doi.org/10.1038/s41567-020-0932-7"
+        }
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+            "id": "tutorial_local_cost_functions",
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+        {
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+            "id": "tutorial_How_to_optimize_QML_model_using_JAX_and_Optax",
+            "weight": 1.0
+        },
+        {
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+            "id": "tutorial_jax_transformations",
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+        {
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+            "id": "tutorial_classical_shadows",
+            "weight": 1.0
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+        {
+            "type": "demonstration",
+            "id": "tutorial_diffable_shadows",
+            "weight": 1.0
+        },
+        {
+            "type": "demonstration",
+            "id": "ensemble_multi_qpu",
+            "weight": 1.0
+        }
+    ]
+}