Add warm-start QAOA tutorial

[henryzou50]

Summary

Adds a new tutorial, Warm-starting QAOA with the Optimization Addon, demonstrating how to improve QAOA convergence on Max-Cut problems by initializing the circuit from a continuous (QP) relaxation of the binary problem, following Egger, Mareček, and Woerner (arXiv:2009.10095). Problem modeling uses the qiskit-addon-opt-mapper package.

The tutorial walks through:

• Mapping a Max-Cut problem to a QUBO with qiskit-addon-opt-mapper
• Running standard QAOA on a simulator as a baseline
• Building the WS-QAOA initial state and custom mixer from the QP-relaxed solution
• Comparing energy convergence and solution quality between the two

Files changed

• docs/tutorials/warm-start-qaoa.ipynb — the new tutorial
• public/docs/images/tutorials/warm-start-qaoa/ — output images
• docs/tutorials/_toc.json, docs/tutorials/index.mdx — registered under Verifiable sampling algorithms
• qiskit_bot.yaml — reviewer notifications
• scripts/config/notebook-testing.toml — added to the tutorials exclude list (tutorials are not run in CI)

[qiskit-bot]

One or more of the following people are relevant to this code:

• @nathanearnestnoble

[review-notebook-app]

Check out this pull request on 

See visual diffs & provide feedback on Jupyter Notebooks.

---

Powered by ReviewNB

[henryzou50]

Re: failing Pull request / Execute notebooks check

Just flagging that this CI failure isn't caused by changes in this PR, but instead it's a regression from a recently merged PR on main.

What's failing:
Several notebooks in [groups.normal] are erroring out, including docs/guides/estimator-noise-management.ipynb and others, all for the same underlying reason: they call QiskitRuntimeService() directly, but there are no IBM credentials available in fork CI and no mock patch is applied to that group's ci strategy. (See the full list of failures in the job log.)

Why this PR is hitting it:
The CI script scripts/ci/pr-execute-notebooks.py only runs the specific notebooks a PR changed unless a file under scripts/ is touched, in which case it runs every notebook in [groups.normal]. This PR updates scripts/config/notebook-testing.toml (to add the new tutorial to the exclude list) and scripts/nb-tester/requirements.txt, which trips that condition and causes the full suite to run, surfacing the existing breakage on main.

Where the regression seems to came from:
PR #4640 ("Separate out primitives") added the new estimator-noise-management.ipynb and sampler-noise-management.ipynb notebooks to [groups.normal] in notebook-testing.toml. Since [groups.normal]'s ci strategy doesn't apply the qiskit-ibm-runtime-open patch, any notebook in that group that instantiates QiskitRuntimeService() will fail in fork CI.

[henryzou50]

Quick update, I tested locally with tox -- path/to/notebook and it runs successfully, so this PR is good to go from the notebook side. Flagging for a maintainer: the suggested path is to merge this to a temporary branch in Qiskit/documentation first, then PR that branch into main so CI can run with the token.

[henryzou50]

Closing this PR to open a new one