Add grouping support for arithmetic ops

[mudit2812]

Context:
Adding support for Prod and SProd to be used with grouping.

Description of the Change:

• Update Tensor.__matmul__ to stop mutating in-place and return a new op instead.
• Added Pauli rep to Tensor. Consequently, Tensor.copy() and Tensor.map_wires() also had to be updated.
• Moved around code inside Tensor__init__ to fix incorrect validation after the previous changes. The warning about overlapping wires is now raised in Tensor.queue() instead of at the beginning of construction, so that raising the warning pre-maturely can be prevented.
• qml.pauli.group_observables now returns Prods if the observable list contains new arithmetic ops.
• Update pauli_to_binary to rely on the Pauli rep. All operations that can be grouped should now always have a valid Pauli rep, so we don't need to worry about other cases, as pauli_to_binary is only a utility function for grouping.
• Update are_pauli_words_qwc to correctly handle ops with Pauli reps. This change however means that any new arithmetic ops that have pauli_rep=None will still not work with this function. If we eventually start doing more simplification eagerly, this can be addressed.
• split_non_commuting now creates Prods instead of Tensors for creating observables for measurements that use wires. Also made a change so that if a measurement without any wires is made, the observable that is created uses tape.wires.

Benefits:

Possible Drawbacks:

Related GitHub Issues:

[mudit2812]

[sc-53504]

[Qottmann]

Thanks @mudit2812
Great to see progress on grouping with arithmetic ops!

I gave it a first pass and there are honestly a lot of things that I dont quite understand yet. Could you help me by providing some context to the changes where I ask questions and generally add a description to the PR what the goal is as well as developer comments. Since there are no added tests it is hard to understand the context of the PR.

Re caching and Sum not having a coeffs attribute, I think this is something we wanted to add for feature parity anyway. If this attribute existed, would this make this PR any easier?

[codecov]

Codecov Report

Attention: 1 lines in your changes are missing coverage. Please review.

Comparison is base (4e2e344) 99.68% compared to head (3fce75c) 99.67%.
Report is 1 commits behind head on master.

Files	Patch %	Lines
pennylane/operation.py	94.73%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #5070      +/-   ##
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- Coverage   99.68%   99.67%   -0.02%     
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  Files         392      392              
  Lines       35844    35565     -279     
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- Hits        35732    35448     -284     
- Misses        112      117       +5     

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[timmysilv]

looks awesome ⚛️ just a few little questions

[Qottmann]

Great job @mudit2812 🥳

Left some comments on the pauli checks. Looks overall good though 👍

[codecov-commenter]

Codecov Report

All modified and coverable lines are covered by tests ✅

Comparison is base (c1a813d) 99.68% compared to head (6a9c799) 99.67%.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #5070      +/-   ##
==========================================
- Coverage   99.68%   99.67%   -0.02%     
==========================================
  Files         394      394              
  Lines       36160    35875     -285     
==========================================
- Hits        36047    35759     -288     
- Misses        113      116       +3     

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

[timmysilv]

looks awesome. just left one suggestion to make sure we clean up in case of errors

[Qottmann]

looks good to me now 👍