Update AI transpiler introduction tutorial to new template

[henryzou50]

Summary

Revised ai-transpiler-introduction.ipynb to follow the Tutorial_Template structure, simplifying the tutorial to focus on comparing the default transpiler (SABRE) vs the AI transpiler using random circuits.

Key changes from the old notebook:

• Simplified circuit selection: Uses only random circuits with 2-qubit gates (instead of efficient_su2, QFT, BV, Clifford, and permutation circuits)
• Added execution and fidelity testing: The old notebook skipped Steps 3-4 entirely ("no experiments will be executed"). The revised version uses mirror circuits to measure fidelity on both an Aer simulator (with a depolarizing noise model) and real hardware
• Removed Part II benchmarking and Part III permutation synthesis: Consolidated into a focused Default vs AI comparison
• Added quantitative analysis: Summary tables with mean/stdev for 2Q depth, gate count, and transpilation time, plus percentage improvement plots
• Added commentary: Markdown cells discussing the depth vs gate count trade-off (AI optimizes depth, SABRE minimizes gate count), transpilation time scaling, and mirror circuit fidelity limitations at large scale
• Template compliance: Follows the 4-step Qiskit patterns structure (Map, Optimize, Execute, Post-process) for both small-scale simulator and large-scale hardware examples

Tutorial structure:

• Small-scale simulator example (6-25 qubits, depth 4): Full 4-step walkthrough with Aer simulator fidelity test on the 10-qubit case
• Large-scale hardware example (26-50 qubits, depth 8): Compressed workflow with real hardware submission on the 26-qubit case

[qiskit-bot]

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

• @henryzou50
• @nathanearnestnoble

[review-notebook-app]

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[nathanearnestnoble]

The large scale section states "step 1-4 compress into single code block" but is in multiple blocks. Would suggest removing "compress into single code block"

[henryzou50]

The large scale section states "step 1-4 compress into single code block" but is in multiple blocks. Would suggest removing "compress into single code block"

Thanks @nathanearnestnoble , I just pushed a change for that and added the reasoning for why we don't compress the large scale example into a single code block for this tutorial.

[kaelynj]

Looking pretty good so far! Left some comments to address

Also a side note: @henryzou50 you should have write access to the repo now since you're a regular contributor so you shouldn't need to open PRs through a fork.

[henryzou50]

Thanks so much for the review, @kaelynj I've applied your suggestions:

• Swapped the pandas DataFrames for plain printed tables (they weren't rendering correctly on the web page)
• Removed the per-circuit loop output so only the mean/std summary table shows
• The mirror-circuit fidelity plots now show 1 - P(|0...0>) instead of the raw probability, so the results read much more clearly
• Restored the AI-synthesis coupling-map limitation note in the analysis
• Dropped "service" from the heading/prose to match the updated title

While I was in there, I also made a few related changes:

• Removed the pandas dependency entirely (the summary/plot helpers now work directly off the results list)
• Re-ran the notebook and reconciled the commentary with the new numbers (circuit depth, the depth/gate-count percentages, the 10-qubit fidelity discussion, and the large-scale noise-floor wording)
• Fixed a stale "reduced coupling map" description in Step 2 that no longer matched the code
• Converted the absolute doc links to relative paths for consistency with the other recently updated tutorials, and fixed a broken generate_preset_pass_manager link in the Background section

Let me know if you'd like any further changes!