Demo quantum chebyshev transform

[colinjdale]

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Title:

Quantum Chebyshev Transform

Summary:

Inspired by recent work in quantum generative modeling in the Chebyshev basis. Introduces Chebyshev polynomials and their properties, briefly describing the classical discrete Chebyshev transform. Defines quantum Chebyshev basis states, and implements the Quantum Chebyshev transform in PennyLane. Finishes by showing how the transformed states have the expected amplitudes and orthogonality relation.

Relevant references:

Paper that introduces the Quantum Chebyshev transform, and uses it for generative modeling of probability distributions.

Chelsea A. Williams, Annie E. Paine, Hsin-Yu Wu, Vincent E. Elfving, Oleksandr Kyriienk, "Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions", https://arxiv.org/abs/2306.17026, (2023).

Further work that applies the toolkit introduced above for learning and sampling distributions that appear in high-energy physics.

Jorge J. Martínez de Lejarza, Hsin-Yu Wu, Oleksandr Kyriienko, Germán Rodrigo, Michele Grossi, "Quantum Chebyshev Probabilistic Models for Fragmentation Functions", https://arxiv.org/abs/2503.16073, (2025).

Possible Drawbacks:

Demo does not show what to do with the transform. Perhaps could include a section on QPE of a qubitized Hamiltonian. Still thinking about it.

Related GitHub Issues:

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

• GOALS — Why are we working on this now?

To show a PL implementation of a paper from late 2023 by William's et al. The implementation here is part of the toolbox introduced in that paper which seems useful for generative modeling and high-energy physics applications.*

• AUDIENCE — Who is this for?

Quantum machine learning researchers and quantum computing for high-energy physics researchers.

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

Chebyshev, generative modeling, probability distributions, PennyLane

• Which of the following types of documentation is most similar to your file?
  (more details here)

• Tutorial
• Demo
• How-to

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[alvaro-at-xanadu]

The results look great 🚀 I'll send it to Guillermo too so he can take a quick peek. Overall, my main feedback is to try and make it more self-contained, especially in the introductory part. I also wonder if we can say anything about the QChT being advantageous over the classical one in any way? (If it is, otherwise we just stay quiet!)