Improve the Clifford+T compiler pipeline

[Cryoris]

The Clifford+T compilation pipeline introduced in Qiskit 2.3 needs to be significantly improved.

In particular, it should:

• First translate the circuit into Clifford+RZ basis, and then approximate each RZ-gate in the circuit using the gridsynth algorithm.
• Minimize the number of times we need to call the expensive T gate synthesis algorithms.
• Make sure that circuits already expressed as Clifford+T (in particular, various arithmetic circuits) do not get re-synthesized as circuits with arbitrary RZ-angles. In other words, we should make sure that the T-count is only decreased.
• Make sure that Clifford+T-based decompositions are always preferred to continuous gate ones (e.g., we should use SX over RX(pi/2)).

[debasmita2102]

I want to take this if no one else is working on it

[Cryoris]

@alexanderivrii has been investigating this, but there's likely multiple issues to address here and we can discuss some more concrete tasks to take over 🙂

[debasmita2102]

@alexanderivrii has been investigating this, but there's likely multiple issues to address here and we can discuss some more concrete tasks to take over 🙂

Sure!