Add a new transformer that performs random pauli insertion

[NoureldinYosri]

implements the error mitigation technique described in appendix D of https://arxiv.org/abs/2503.20870

[eliottrosenberg]

Thanks, Nour! This is great! A few comments:

• Can we have a clear, deterministic way of saying which qubit is which in the probabilities array?
• For applications, we might need finer-grained pair-by-pair control over the probabilities.
• Also see the error_mitigation/noise_adding.py file in my user folder.

[zlatko-minev]

Eliott will chat with you at our chat

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 97.50%. Comparing base (0e87d09) to head (4eb0bdd).
⚠️ Report is 116 commits behind head on main.

Additional details and impacted files

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  Coverage   97.50%   97.50%            
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  Files        1101     1103     +2     
  Lines       99411    99541   +130     
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+ Hits        96928    97058   +130     
  Misses       2483     2483            

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[eliottrosenberg]

A uniform distribution would completely depolarize the state, which is probably not what we want. I think probabilities should be required.

[eliottrosenberg]

Maybe we should require that probabilities be a mapping from qubit pairs to numpy arrays and use the order in the qubit pair (not in the op, which is less easily accessible to the user) to determine which qubit is which in probabilities.

[eliottrosenberg]

Can you change these to ValueErrors and document them in a Raises section of the docstring? Also check that none of them are negative.

[eliottrosenberg]

You should say more clearly that probabilities contains 4x4 arrays and the [i,j] element is the probability of applying _PAULIS[i] to qubit 0 and _PAULIS[j] to qubit 1, where the two qubits now (if you make the other change I suggest) are in the order specified in the key of the dictionary.

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