[Feature Request] Built in converter for quantum circuit to Clifford circuits

Required prerequisites

• Search the issue tracker to check if your feature has already been mentioned or rejected in other issues.

Describe the feature

Feature Request: Clifford Converter

Summary

Add a utility to convert a cudaq Circuit into Clifford form:

Project – replace non-Clifford gates with nearest Clifford gates.

This would directly enable Clifford Data Regression (CDR), canary Clifford circuits for fidelity checks, and workflows used in recent error mitigation literature.

Motivation

• CDR: requires generating nearby Clifford circuits from user-defined ansätze.
• Canary circuits: fast simulation baselines for fidelity estimation.
• Error mitigation papers: recent works rely on Clifford projection/approximation.

References

1. [arXiv:2406.15721] Clapton: Clifford-Assisted Problem Transformation for Error Mitigation in Variational Quantum Algorithms
2. [arXiv:2408.06482] Demonstration of a CAFQA-bootstrapped Variational Quantum Eigensolver on a Trapped-Ion Quantum Computer
3. Clifford Data Regression [arxiv:2005.10189].

The issue describes a utility for converting a cudaq Circuit into Clifford form but names no files, tests, or entry points. Start by locating the Circuit representation and existing gate-conversion utilities, then determine how “nearest Clifford” should be defined for each non-Clifford gate. Done requires an agreed conversion policy and coverage for the stated CDR and canary-circuit use cases.