Use SVD-based fidelity for density matrices and add numerical stability test

[RevanthGundala]

Summary

Fixes #4819

This PR replaces the existing eigen‑decomposition formula for two density matrices

state1_sqrt = _sqrt_positive_semidefinite_matrix(state1)
eigs = eigvalsh(state1_sqrt @ state2 @ state1_sqrt)
trace = sum(sqrt(abs(eigs)))
fidelity = trace**2

with an equivalent but more accurate form

rho1_sqrt = scipy.linalg.sqrtm(rho1)
rho2_sqrt = scipy.linalg.sqrtm(rho2)
fidelity = np.sum(scipy.linalg.svdvals(rho1_sqrt @ rho2_sqrt)) ** 2

This ensures fidelity never exceeds 1 due to round‑off error in higher dimensions.

What’s Changed

In cirq/qis/measures.py, the density‑matrix branch of _fidelity_state_vectors_or_density_matrices now uses sqrtm + svdvals. A new unit test test_fidelity_numerical_stability_high_dim in measures_test.py constructs a 10‑qubit pure state, traces out one qubit, and verifies that the old eigen‑value‑based fidelity exceeds 1, and the new SVD‑based fidelity ≈ 1.

Testing

Ran pytest cirq-core/cirq/qis/measures_test.py::test_fidelity_numerical_stability_high_dim — old formula fails (>1), new passes (≈1).

[codecov[bot]]

Codecov Report

:white_check_mark: All modified and coverable lines are covered by tests. :white_check_mark: Project coverage is 98.68%. Comparing base (e9a592d) to head (9a46b19). :warning: Report is 249 commits behind head on main.

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #7292   +/-   ##
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  Coverage   98.68%   98.68%           
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  Files        1112     1112           
  Lines       97599    97616   +17     
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+ Hits        96315    96334   +19     
+ Misses       1284     1282    -2     

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[mhucka]

@RevanthGundala would you be able to take a look at the merge conflicts?

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[github-actions[bot]]

This pull request has been automatically labeled as stale because 90 days have passed without comments or other activity. If no further activity occurs and the status/stale label is not removed by a maintainer within 60 days, this pull request will be closed. If you would like to restore its active status, please leave a comment here; doing so will cause the staleness handler to remove the label.

If you have questions or feedback about this process, we welcome your input. You can open a new issue to let us know (please also reference this issue there, for continuity), or reach out to the project maintainers at [email protected].