[WIP] Implement a conversion from Clifford+T ops to PPR ops

[sengthai]

Context:
This PR implements a new pass Clifford_ppr that converts Clifford+T quantum operations to their equivalent Pauli Product Rotation (PPR) representations. This conversion is a crucial step in enabling Quantum Error Correction (QEC) compilation capabilities.

Implementation Details

• The new dialect for PPR and PPM was introduced in Add Pauli Product dialect (QEC) #1486.
• Introduces a new lowering pass that converts:
  • Basic Clifford+T gates (H, S, T, CNOT) to their PPR decompositions
  • Measurement operations to Pauli Product Measurements (PPM)
• Gate decompositions follow the conventions from "A Game of Surface Codes"
• Supported gate conversions:
  • Hadamard (H) → PPR with Pauli sequence [Z, X, Z] and θ = π/4
  • S gate → PPR with Pauli sequence [Z] and θ = π/4
  • T gate → PPR with Pauli sequence [Z] and θ = π/8
  • CNOT → PPR with Pauli sequence [Z, X] and θ = π/4

Benefits:

• Enables compilation pathways for QEC
• Provides a standardized representation of quantum operation in term of Pauli rotations

TODO:

• Convert Clifford gates to pi/4 rotations and T to pi/8 rotation
• Convert CNOT gates to C(Z, X)
• Convert measurement op to PPR measurement
• Implement to support running in Pennylane

[sc-82894]

[dime10]

@sengthai Are you able to rebase this PR onto main now that #1486 is merged? :)

[sengthai]

@sengthai Are you able to rebase this PR onto main now that #1486 is merged? :)

Yes, @dime10. I've just rebased the conv_quantum2ppr branch onto main. Thanks for the reminder!

[dime10]

@sengthai Are you able to rebase this PR onto main now that #1486 is merged? :)

Yes, @dime10. I've just rebased the conv_quantum2ppr branch onto main. Thanks for the reminder!

Oh right, my commits from that PR still show up in the history, so it's not actually rebased, but a merge is also fine 😅

[dime10]

I guess this is still in progress, so feel free to ignore some of the style comments until later 😄 But it's looking great so far 👍

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