Add Hamiltonian simulation with GQSP demo

[Mostafa-Atallah2020]

Title:

Hamiltonian simulation with Generalized Quantum Signal Processing

Summary:

The demo implements the time-evolution operator exp(-iHt) with qml.GQSP in three steps. First it block-encodes H as a qubitization walk W (qml.Qubitization) whose eigenphases are +/- arccos(E/lambda). Then it chooses the polynomial via the Jacobi-Anger expansion, so that a single complex polynomial of W equals exp(-iHt) on the walk's eigenphases. Finally it runs qml.GQSP with angles from qml.poly_to_angles(..., "GQSP"), undoes the Laurent z^K shift, and reads exp(-iHt) off the top-left block, verified against scipy.linalg.expm.

Relevant references:

• D. Motlagh and N. Wiebe, Generalized Quantum Signal Processing, PRX Quantum 5, 020368 (2024).
• G. H. Low and I. L. Chuang, Hamiltonian Simulation by Qubitization, Quantum 3, 163 (2019).

Possible Drawbacks:

The worked example is a small 2-qubit Heisenberg-type Hamiltonian, chosen so the full exp(-iHt) matrix can be formed and compared directly. The method itself is general; the small size is purely for an exact, fast check. For the same reason the demo reads the evolution from the dense circuit matrix on default.qubit, rather than running it as a sampled circuit, which is appropriate for a correctness-focused demo at this scale.

Related GitHub Issues:

Closes #1807.

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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?

  PennyLane ships qml.GQSP and a GQSP resource-estimation demo, but there is no executable, end-to-end recipe taking qml.GQSP to its flagship application, Hamiltonian simulation. This demo fills that gap, showing the full runnable pipeline and verifying it against the exact evolution.

• AUDIENCE: Who is this for?

  Graduate students and researchers comfortable with qubitization and QSP basics who want to run GQSP-based Hamiltonian simulation, not just estimate its cost.

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

  GQSP, Generalized Quantum Signal Processing, Hamiltonian simulation, qubitization, quantum signal processing, Jacobi-Anger, block encoding.

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

• Tutorial
• Demo
• How-to

[github-actions]

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[github-actions]

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Deployed at: 2026-07-24 22:58:24 UTC

[drdren]

Hi @Mostafa-Atallah2020 , thanks for this great demo contribution! I completed my review in addition to @nobese 's. Overall, it reads very well and truly shows the reader how to do Hamiltonian simulation with GQSP well. I added a conclusion section, whose skeleton code may still be tweaked, but at least the conclusion mirrored the introduction. Thanks again!

[drdren]

Hi @Mostafa-Atallah2020 , it's great to read your updated demo! I left some comments to shape this demo further.

[Mostafa-Atallah2020]

Hi @nobese I am sharing the thumbnails pencil scratchs here as I don't know how to share them with the pennylane artist. they are not as pretty as the ones in the demos I wish they will be representative enough let me know if there is anything else

[nobese]

Thanks @Mostafa-Atallah2020! This is very helpful : ) I will share them with our designer today, there is typically a minimum 5 work day turnaround for thumbnails so I will send you the draft when it is ready. I will ping you if any questions come up!