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Improve the Clifford+T compiler pipeline #15457

@Cryoris

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@Cryoris

The Clifford+T compilation pipeline currently struggles with prioritizing discrete gate sets in a few places. For example

  • it might replaces discrete sequences with continuous gates that need re-synthesis, or
  • the basis translation doesn't always prefer Clifford-based decompositions over continuous gate ones (e.g. use SX over RX(pi/2))

We should improve the pipeline such that we only always decrease T count and minimize the number of times we need to call the expensive T gate synthesis algorithms.

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