Fix qml.center with linear combinations

[dwierichs]

Context:
Linear combinations were not taken into account in qml.center, making it miss center elements for some inputs.

Description of the Change:
Change the algorithm that computes the center.
Kudos to @therooler for the prototype.

Benefits:
Correct code.

Possible Drawbacks:
Slower performance for inputs that would have been fine before anyways.
This has been fixed by reverting to the previous implementation if PauliWord instances are passed only.

Related GitHub Issues:
Fixes #6048
[sc-70014]

[Qottmann]

Looks good 👍

Some minor things/questions. Could you add a bit more description of the involved math, since this is not super trivial and not explained in the doc string, would be good to at least have some explanation in the code as comment. That being said, is it really necessary to compute three kernels here?

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 99.66%. Comparing base (eafd446) to head (1cf8da1).
Report is 1 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #6049      +/-   ##
==========================================
- Coverage   99.66%   99.66%   -0.01%     
==========================================
  Files         430      430              
  Lines       41762    41489     -273     
==========================================
- Hits        41624    41349     -275     
- Misses        138      140       +2     

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

[dwierichs]

@Qottmann

Could you add a bit more description of the involved math, since this is not super trivial and not explained in the doc string, would be good to at least have some explanation in the code as comment.

Yep, good point! Will do!

That being said, is it really necessary to compute three kernels here?

I'm not sure, maybe we can do Gram-Schmidt instead 🤔 I'll give it another attempt!

[dwierichs]

@Qottmann Found a (probably) easier way for the kernel intersections. Added derivation for that new technique to docstring and put the intersection code into a helper function with dev docstring.

[Qottmann]

Very nice now! Just some minor things but overall looks good 👍

[Qottmann]

Thanks for this great upgrade @dwierichs 💪

[Qottmann]

Is this a bug or am I once again not fully getting the definition of the center 🫠

gg = [
    Z(0) @ Z(1),
    Y(0) @ Y(1),
    Z(0) @ Y(1),
    Y(0) @ Z(1)
]
c = qml.center(gg) 
print(c)
# [Z(0) @ Z(1), Y(0) @ Y(1), Z(0) @ Y(1), Y(0) @ Z(1)]


gg = [g_i.pauli_rep for g_i in gg]
for g1 in gg:
    for g2 in gg:
        com = g1.commutator(g2)
        com.simplify()
        assert len(com) == 0 # fails with e.g. third output being -2j * X(1)

[mudit2812]

Thanks @dwierichs ! Your documentation was really helpful in understanding what the function does. Nevertheless, I was still a bit confused so I've mostly just left questions for my own understanding in this review. Other than that, it looks good, and I'm ready to approve once my questions are resolved 🚀 😄

[mudit2812]

Thanks @dwierichs !