Add warm-start QAOA tutorial

[henryzou50]

Replaces #5039 — moved branch to upstream so CI can access the QiskitRuntimeService token. See #5039 for prior discussion.

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[qiskit-bot]

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

• @nathanearnestnoble

[abbycross]

All set from my side, thanks! @nathanearnestnoble final review?

[nathanearnestnoble]

@t-imamichi plans to review this -- lets come back once that is done.

[t-imamichi]

The tutorial looks good overall. I have some minor comments as follows.

1. Cell [3]: Since opt-mapper includes Maxcut class , Cell [3] and "Large Scale Hardware Example" section can be simplified with the class.
2. Cell [7]: I suggest using qaoa_ansatz link to generate the standard QAOA ansatz.
3. Cell [14]: It would be nice to compare the probabilities of bitstrings with the optimal cut value 4 so that we can see the difference between the standard QAOA and the warm-start QAOA. (not a strong opinion)

[henryzou50]

The tutorial looks good overall. I have some minor comments as follows.

1. Cell [3]: Since opt-mapper includes `Maxcut` [class](https://qiskit.github.io/qiskit-addon-opt-mapper/stubs/qiskit_addon_opt_mapper.applications.Maxcut.html) , Cell [3] and "Large Scale Hardware Example" section can be simplified with the class.

2. Cell [7]: I suggest using `qaoa_ansatz` [link](https://quantum.cloud.ibm.com/docs/en/api/qiskit/qiskit.circuit.library.qaoa_ansatz) to generate the standard QAOA ansatz.

3. Cell [14]: It would be nice to compare the probabilities of bitstrings with the optimal cut value 4 so that we can see the difference between the standard QAOA and the warm-start QAOA. (not a strong opinion)

Thanks @t-imamichi for the review! I've addressed all three comments:

Cell [3]: Switched to the Maxcut application class, both the small-scale Cell [3] and the "Large Scale Hardware Example" section now build the problem with Maxcut(G).to_optimization_problem() instead of manually expanding the QUBO objective. Updated the imports and surrounding markdown accordingly.

Cell [7]: The standard QAOA ansatz is now generated with qaoa_ansatz from qiskit.circuit.library. I kept the custom builder for WS-QAOA only, since its per-qubit mixer $R_Y(\theta),R_Z(-2\beta),R_Y(-\theta)$ isn't expressible as a sum of Paulis — let me know if you'd prefer it routed through qaoa_ansatz with a custom mixer/initial state as well.

Cell [14]: Added a bar chart comparing the probability distribution over cut values for standard QAOA vs. WS-QAOA, with the optimal cut (4) highlighted, plus a printed P(cut = 4) comparison. This makes the contrast clear, as WS-QAOA concentrates almost all of its probability mass on the optimal cut, while standard QAOA spreads it across suboptimal partitions.

The notebook has been re-run with refreshed outputs.

[t-imamichi]

Thank you @henryzou50 for reflecting my comments. LGTM.
As for qaoa_ansatz, you are right. It cannot generate WS-QAOA ansatz easily. I posted a comment.
It makes sense to have a custom ansatz function for WS-QAOA.