[BUGFIX] Fix qml.lie_closure by using SVD based linear independence check

[Qottmann]

fixes #6065

Additionally introduces normalization of vectors to avoid exploding coefficients
Could alternatively also introduce orthonormalization with orthonormal_basis = U[:, :rank] in the SVD step, but I dont think this is necessary.

[sc-70526]

[Qottmann]

@dwierichs @vincentmr tagged you for review, but no rush on this. I did this change on my local research branch and found it to work quite well.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 99.70%. Comparing base (2188799) to head (a47e549).
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Additional details and impacted files

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##           master    #6232      +/-   ##
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- Coverage   99.71%   99.70%   -0.01%     
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  Files         444      444              
  Lines       42234    42235       +1     
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- Hits        42114    42112       -2     
- Misses        120      123       +3     

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[vincentmr]

We now have

eps = 1e-3
g = qml.lie_closure([X(0), X(0) + eps * Y(0)])
>> [X(0), X(0) + 1e-03 * Y(0), 1e-03 * Z(0)]

but

eps = 1e-9
g = qml.lie_closure([X(0), X(0) + eps * Y(0)])
>> [X(0), X(0) + 1e-09 * Y(0)]

This is because the commutator is nullified by com.simplify() in lie_closure. Should we pass tol along to obtain a more consistent or predictable behaviour?
Similarly,

eps = 1e-5
g = qml.lie_closure([X(0), X(0) + eps * Y(0)], tol=1e-2)
>> [X(0)]

but

eps = 1e-5
g = qml.lie_closure([X(0), X(0) + eps * Y(0)], tol=1e-3)
>> [X(0), X(0) + 1e-05 * Y(0), 1e-05 * Z(0)]

which looks unintuitive to me.

[dwierichs]

Should we pass tol along to obtain a more consistent or predictable behaviour?

I agree that we should do this: If a user wants to take care of tolerances themselves, they should be able to do so consistently

which looks unintuitive to me.

For default tolerances, I think we can give our best to create a consistent default, but if there are hickups in edge cases, I'd personally say that's fine :)
Regarding your last code snippet, @vincentmr , I can't seem to reproduce it (the result is just [X(0)] for both tol values passed to lie_closure.

[vincentmr]

Regarding your last code snippet, @vincentmr , I can't seem to reproduce it (the result is just [X(0)] for both tol values passed to lie_closure.

You're right, I reinstalled PennyLane in a fresh venv and the effect went away. Never mind that.

[dwierichs]

Nice upgrade @Qottmann!
I had a few small comments to be addressed :)

[Qottmann]

Thanks for the great feedback @vincentmr and @dwierichs !

[dwierichs]

LGTM! Nice upgrade @Qottmann 💯

[dwierichs]

This fix was unblocking some research I was doing today, might be nice to get in soon :)

[PietropaoloFrisoni]

LGTM! Just a small suggestion for the changelog : )