What should we add?

Pauli-based gates, such as PauliEvolutionGate, PauliGate and the in-progress PauliProductMeasurement, allow for efficient commutative analysis via their Paulis. Current Qiskit however, doesn't take this into account and just treats these as generic gates investigating their properties via the exponentially expensive matrix representation.

This is of particular interest for circuits in PBC format but is something we should generally do. It would also fix problems where the commutation checker could be efficient but fails (like #15021).

We should add

• Support for Pauli-based gates in the CommutationChecker
• Build a commutative optimization pass able to merge & cancel Pauli-based gates (Unified commutative optimization #15047)