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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