Add helper func apply_operation for default mixed

[JerryChen97]

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---

Context:
Before getting into the implementation of new default_mixed we need to separate most of the useful logics from the older version one. One of the most important things is the apply_operation. Basically, what we want to do is to condensate everything into a new sub-module under qml/devices/qubit_mixed such that previous empirical heuristics regarding how to select between einsum and tensordot can be carried over and wrapped independently.

Description of the Change:
New script containing the helper funcs as described above.

Benefits:
Better modularization. Getting ready for the implementaion of a new default_mixed in the future.

Possible Drawbacks:

Related GitHub Issues:

Related Shortcut Stories
[sc-73319]

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[JerryChen97]

RN I've done the implementation of both einsum and tensordot.

• The einsum method is directly adopted from current qutrit mixed module
• The tensordot method is from the default_mixed but several necessary modifications are applied.

Due to the fact that there are quite some other shortcuting regeisters for many special cases to implement, currently I would like to put this as draft without the full CI. After all the other registers complete I will load the CI along with more tests.

[JerryChen97]

In previous devices.default_mixed the apply_operation even includes some state preparation. Apparently this is a wrong scope and i will create another story for this

[JerryChen97]

The following methods involve additional stories and will be neglected for now:

• snapshot: this involves measurement submodule, missing for now
• evolve: this involves extra refinement over how the odeint can be applied onto a density matrix

[JerryChen97]

I noticed that seemingly we can directly use those from pure states given the fact that the conjugated operator's wire numbers are simply all +num_wires where num_wires can be calculated from the input dm state. Let's say such transformation for operator is Shift, then for real, symmetric op e.g. X and CNOT and so on. As long as the apply method of qubit device is always only involving some local wire order, any real operator with symmetricity (e.g. Grover) should be able to be applied in same way.

[JerryChen97]

More tests to add:

• identity-like ops
• batch
• channels
• multiCX and Grover of larger sizes (at least 9)

[codecov]

Codecov Report

Attention: Patch coverage is 93.50649% with 15 lines in your changes missing coverage. Please review.

Project coverage is 99.35%. Comparing base (88caa8e) to head (cff4b60).

Files with missing lines	Patch %	Lines
pennylane/devices/qubit_mixed/apply_operation.py	95.45%	9 Missing ⚠️
pennylane/devices/qubit_mixed/utils.py	80.00%	6 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #6379      +/-   ##
==========================================
- Coverage   99.38%   99.35%   -0.04%     
==========================================
  Files         447      451       +4     
  Lines       42418    42648     +230     
==========================================
+ Hits        42158    42373     +215     
- Misses        260      275      +15     

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