Create a finite state machine analysis pass to propagate Pauli basis states, and use it to disentangle CNOTs

[paul0403]

Context: In circuits that mostly involve the six Pauli eigenstates, the effect of gates can be described by a simple six-state finite state machine. This allows us to deduce their midcircuit state at compile time.

The knowledge of the snapshot midcircuit states can be useful when:

• Deducing CNOT/Toffoli/SWAP/... effect and decouple them. We decouple CNOT in this PR.
• Replace complicated circuits whose results are |01+-LR> with standard preparation routines
• Potentially more

Description of the Change: Added an analysis pass --propagate-simple-states (and tests).

Benefits: More compile time circuit information for potential compile time transformation/optimization.

Possible Drawbacks: This algorithm is from an external paper. We need to properly cite it. https://arxiv.org/abs/2012.07711 (fig.5)

Related GitHub Issues:

sc epic: https://app.shortcut.com/xanaduai/epic/74389

[sc-74649]

[paul0403]

Note that this is a hackweek project and is not on any roadmaps, so it's not urgent

Edit (2024/12/20): this is a P2 item now

[paul0403]

Adding do-not-merge label to freeze this branch for assessment

[codecov[bot]]

Codecov Report

All modified and coverable lines are covered by tests :white_check_mark:

Project coverage is 96.68%. Comparing base (48edd02) to head (e5401f4). Report is 166 commits behind head on main.

Additional details and impacted files

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  Coverage   96.68%   96.68%           
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  Files          75       75           
  Lines        8266     8266           
  Branches      867      867           
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  Hits         7992     7992           
  Misses        221      221           
  Partials       53       53           

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