Matrix/Tensor product between two PauliWords now returns PauliSentence

[Qottmann]

Following discussion here](#5017 (comment)) changing the default behavior of products between two PauliWord instances.

old (see comment in discussion)
One thing I am still unsure about at the moment is for pure tensor products (where no coefficient is raised). E.g.

pw1 = PauliWord({0:"X"})
pw2 = PauliWord({1:"X"})

I would find it counterintuitive to get a PauliSentence from pw1 @ pw2 but would expect PauliWord({0:"X", 1:"X"})

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Comparison is base (3ab3a32) 99.68% compared to head (d04267c) 99.67%.

Additional details and impacted files

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##           master    #5054      +/-   ##
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- Coverage   99.68%   99.67%   -0.02%     
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  Files         394      394              
  Lines       35730    35465     -265     
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- Hits        35619    35351     -268     
- Misses        111      114       +3     

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[Qottmann]

See no significant performance regression in this (admittably) simple test benchmark

def foo(n):
    sentences = [
        PauliSentence({PauliWord({i:"X", i+1:"X"}): 1.23, PauliWord({i:"Y", i+1:"Z"}): 5.})
        for i in range(n)
    ]
    res = sentences[0]
    for ps in sentences:
        res @= ps

ns = range(2, 10, 2)
times = [_timeit(foo, n, reps=10) for n in ns]

[lillian542]

Existing changes look good to me so far, but it will be missing some docs updates once the Pauli4 PR is merged in.

Looks to me like the examples in the docstring for PauliWord will need an update for this change, as well as the qml_pauli.rst file. I guess the exact form of the change might depend on the decision about a deprecation cycle discussed above (and certainly for the Pauli4 PR to merge).

[lillian542]

I would find it counterintuitive to get a PauliSentence from pw1 @ pw2 but would expect PauliWord({0:"X", 1:"X"})

I think it's fine - you also get a PauliSentence from PauliSentence({pw1: 2}) - pw1 that's just the PauliWord with a coefficient of 1.

I can see it either way. I think it's probably better to have the result be a consistent type instead of simplifying in certain cases and ending up with different possible outcomes (in terms of the type of object). But I wouldn't be opposed to just returning a PauliWord either.

[Qottmann]

I can see it either way. I think it's probably better to have the result be a consistent type instead of simplifying in certain cases and ending up with different possible outcomes (in terms of the type of object)

I agree with that, the (marginal) performance increases dont justify that special case. Overall better to have something that just works for a user than something that is complicated to handle and slightly faster here.

[lillian542]

Thanks @Qottmann, looks good to me! 🎉

[albi3ro]

Thanks for suggesting this.

I think the only way to really go about this is with a hard breaking change, but it's one that shouldn't particularily influence anyone, and I can see everyone being happy with.

[trbromley]

Thanks @Qottmann!