Full Qudit Compilation Automation

[KevinMTO]

✨ Automated Mixed-Dimensional Circuit Compilation

Description

This PR introduces automated compilation support for circuits mixing different qudit dimensions (2-7 dimensions per qudit, with new fake backends) and establishes the interface layer for quantum hardware integration with quantum computers.

Type of Changes

✨ New Features

• Integrated compiler passes in wrapper methods that can abstract the user's setting
• Added stochastic search for single qudit operation compilation
• New simple transpilation passes

🐛 Bug Fixes

• Bug fix of mapping aware entangling gates compilation
• Corrected QASM representation of dagger gates

⚡️ Performance Improvements

• Optimized routing in energy level graphs
• Simplified VirtZ-gate propagation
• Improved gate decomposition transpilation by removing permutation gates

♻️ Refactoring

• Restructured quantum gate daggering system
• Simplified QASM representation handling

Checklist:

• The pull request only contains commits that are related to it.
• [] I have added appropriate tests and documentation.
• I have made sure that all CI jobs on GitHub pass.
• The pull request introduces no new warnings and follows the project's style guidelines.