Upgrade qml.lie_closure to handle dense matrices

[Qottmann]

Upgrading from labs functionality

labs.dla.lie_closure_dense is integrated into the logic of qml.lie_closure

Also adding qml.pauli.trace_inner_product as a utility function for Pauli matrices and operators.

Integrate lie closure dense to qml lie closure [sc-81965]

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 99.59%. Comparing base (56044fa) to head (6b6a5c2).
Report is 1 commits behind head on master.

Additional details and impacted files

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##           master    #6811   +/-   ##
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  Coverage   99.59%   99.59%           
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  Files         480      481    +1     
  Lines       45551    45625   +74     
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+ Hits        45366    45440   +74     
  Misses        185      185           

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[DSGuala]

Looks good! Left a few comments about doc readability and arguments/UI

[DSGuala]

I think we usually avoid having a "utils" submodule if we can name it something more specific or group functions by their usage.

Since we only have one function in here for now (trace_inner_product), does it still make sense to add the util.py file?

[Qottmann]

I'm expecting there to be more "utility functions", see e.g. here

[Qottmann]

Surely not all of them, but some might make sense to keep public because they are used in different places and might be handy for users

[JerryChen97]

I totally agree that trace_inner_product is very important and handy for any interested physicists. But why did we put it under pauli/dla? I feel logically they should be directly under pauli, although I feel right now the public documentation is also good with trace_inner_product having a separate subsection; probably @DSGuala can help double check from the UI perspective here.

[Qottmann]

I also like the idea of just putting it in pennylane.pauli 👍

[Qottmann]

I moved trace_inner_product to a level higher to just the pauli module.
We might end up needing a util section for DLA stuff but that is independent of this discussion. I'm happy with having trace_inner_product in pauli

[Qottmann]

Also worth mentioning that the plan is to move DLA functioanlity to a new liealg module. I just opted to not do this here since this PR is already quite large

[dwierichs]

Nice and thorough job, @Qottmann 🎉
This basically looks good to go for me, just with the change dense -> matrix pending on the earlier files (by order of git diff :D)

I am wondering whether the trace inner product should be a qml.math feature?

[dwierichs]

Thanks for the quick updates!
Looks good to go from my side now :) 🎉
Nice work @Qottmann 💯

[JerryChen97]

Happy to see the merge of new dla features into PL! Especially the very helpful trace_inner_product method. Just some minor questions left on my side

[JerryChen97]

I totally agree that trace_inner_product is very important and handy for any interested physicists. But why did we put it under pauli/dla? I feel logically they should be directly under pauli, although I feel right now the public documentation is also good with trace_inner_product having a separate subsection; probably @DSGuala can help double check from the UI perspective here.

[JerryChen97]

LGTM! Thanks!

[Qottmann]

@DSGuala do you want to give this another look before I merge? :)