Fix QASM Pauli exponential Rz angle

[marko1olo]

Fixes #721.

Summary

• doubles the uncontrolled Rz angle emitted by pauli_exp_to_qasm so it implements exp(-i * theta * Z) rather than exp(-i * theta * Z / 2)
• updates existing QASM expectations for Pauli-Z strings
• adds a direct regression test with non-default evolution_time

Validation

• C:\hades\oss\.venvs\openfermion-1317\Scripts\python.exe -m pytest src\openfermion\circuits\trotter_exp_to_qgates_test.py -q
• C:\hades\oss\.venvs\openfermion-1317\Scripts\python.exe -m black --check src\openfermion\circuits\trotter_exp_to_qgates.py src\openfermion\circuits\trotter_exp_to_qgates_test.py
• C:\hades\oss\.venvs\openfermion-1317\Scripts\python.exe -m compileall -q src\openfermion\circuits\trotter_exp_to_qgates.py src\openfermion\circuits\trotter_exp_to_qgates_test.py
• git diff --check

AI-assisted contribution: implemented with OpenAI Codex and reviewed locally before submission.

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[gemini-code-assist]

Code Review

This pull request doubles the Rz rotation angle in pauli_exp_to_qasm for uncontrolled exponentiation to correct the rotation angle, updating the corresponding test cases and adding a new test. However, the reviewer identified that the controlled version of the exponentiation (when ancilla is not None) still contains a sign inversion bug, implementing exp(i * theta * P) instead of exp(-i * theta * P). To resolve this, the signs of the angles in the controlled block should be inverted, and the corresponding tests should be updated.

[gemini-code-assist]

While doubling the uncontrolled Rz angle correctly fixes the rotation for the uncontrolled case, the controlled version of the exponentiation (when ancilla is not None on lines 242-255) still contains a sign inversion bug.

Specifically, to implement $\exp(-i \theta P)$ controlled on the ancilla, the relative phase on the ancilla should be $e^{-i \theta \lambda}$ (where $\lambda \in {1, -1}$ is the eigenvalue of the target state):

• For $\lambda = 1$, we need a relative phase of $e^{-i \theta}$.
• For $\lambda = -1$, we need a relative phase of $e^{i \theta}$.

With the current implementation:

• Rz on the ancilla has angle $\theta$, giving a base relative phase of $e^{i \theta}$.
• C-Phase has angle $-2\theta$, giving an additional phase of $e^{-2i \theta}$ when the target is $|1\rangle$ ($\lambda = -1$).
  This results in:
• For $\lambda = 1$: relative phase is $e^{i \theta}$ (should be $e^{-i \theta}$).
• For $\lambda = -1$: relative phase is $e^{-i \theta}$ (should be $e^{i \theta}$).

This means the controlled version implements $\exp(i \theta P)$ instead of $\exp(-i \theta P)$. To fix this, the signs of the angles in the ancilla is not None block should be inverted:

1. Change the C-Phase angle to 2 * term_coeff * evolution_time.
2. Change the Rz angle on the ancilla to -1 * term_coeff * evolution_time (both in the if len(qids) > 0 and else branches).

Note that you will also need to update the corresponding tests in trotter_exp_to_qgates_test.py (such as test_qasm_string_Controlled_XYZ and test_qasm_string_Controlled_XYZ_ancilla_list) to reflect the corrected signs.