Matrix square root for a not-perfectly PSD matrix

[abhinavd]

This method zeros out any negative eigenvalues before taking the matrix square root. This ensures that the matrix square root of a known-to-be PSD matrix continues to have real eigenvalues. This is useful sometimes in cases where one wants something like $\sqrt{A^\dagger A}$ but $A^\dagger A$ is imperfectly computed and has numerically small negative eigenvalues.

[refraction-ray]

what about directly including relu line in sqrtmh function implementation as non-PSD matrix has no well defined sqrtm anyway?

[abhinavd]

That should also work, although hopefully no one relies on the current behaviour of sqrtmh. That's why I thought it might be safest to have a different function itself. Or maybe add an optional keyword in sqrtmh to enforce PSD-ness.

[refraction-ray]

@abhinavd , ah, yes, sqrt can return complex values if the input is of negative value and in complex dtype. Then I think the better way is to add an optional keyword in sqrtmh, default as psd=False.

[codecov]

Codecov Report

Attention: Patch coverage is 20.00000% with 4 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
tensorcircuit/backends/abstract_backend.py	20.00%	4 Missing ⚠️

Files with missing lines	Coverage Δ
tensorcircuit/backends/abstract_backend.py	69.10% <20.00%> (-0.83%)	⬇️

[refraction-ray]

Thanks for the contribution, fixed now by e105763