Integration into the current PL device ecosys

[JerryChen97]

Context:
We finished all the necessary components of the new API interface of default_mixed. However, since this is an internal change, which involves many different files and paths, we would like to have one more round of check, success and merge of which should concludes the whole epic.

Description of the Change:

• DefaultMixedLegacy removed! keep DefaultMixed, but rename it to DefaultMixedLegacy (note: still the legacy one will be purged before this PR merged into master; it's kept right now due to great help in debug)
• rename DefaultMixedNewAPI to DefaultMixed
• a previous TODO: renaming operations_mixed back to operations, and copy it to DefaultMixedLegacy
• add a intermediate step to apply readout_error just as BitFlip with arbitrary input error channels
• remove redundant measure process functions
• add an IntegrationTest for our new DefaultMixed
• modify all the device tests in tests/devices accordingly. Basically the tests/devices/test_default_mixed*.py Deleted legacy specified device tests. These should be covered in the integration test suite mentioned above
• if no errors come up, then separate the legacy code into another file Lots of errors, see drawback section for a record along with updates.
• some meaningless tests were deleted:
• • all the legacy-specific tests
• • some tests that appear to be all-device, but actually skip all the other dev but legacy default mixed (e.g. tests in test_return_types_qnode
• with completed PL package, test against all qml demos. REQUIREMENT: literally nothing should be modified for any downstreaming applications, except for few necessary breaking
• • After search we found the following 9 demos explicitly using default.mixed
• • tutorial_noisy_circuits.py the usage of dev._state needs to be removed and may be replaced by equivalent or not. Sol: had a PR to improve this line
• • tutorial_error_mitigation.py other irrelavant breaks involved (removal of top-level access to QubitStateVec) (!Note: to reproduce this one has to install the latest version of pennylane qiskit via pip install git+https://github.com/PennyLaneAI/pennylane-qiskit.git@master)
• • tutorial_haar_measure.py no issue atl
• • tutorial_unitary_designs.py no issue atl
• • tutorial_how_to_import_qiskit_noise_models.py why user warning here? Didn't read like sensible. Should double check. when running on notebook everything is fine
• • tutorial_how_to_use_noise_models.py no issue atl
• • tutorial_diffable-mitigation.py
• • !The following two demos are super heavy and resource consuming.
• • tutorial_mitigation_advantage.py
• • tutorial_quantum_circuit_cutting.py

some by-the-way improvement:

• applied the new get_canonical_interface_name and removed the hardcoded dictionary mimic of enum.
• aligned the implementation of qubit_mixed much more with qubit module

Benefits:

• Legacy code was preserved for reference purpose as well as catering for potential users
• Better separation between new interface and legacy
• Maintained the functionalities from demos
• Device default.mixed is ported directly with our new API.

Possible Drawbacks:

• Some advanced users that involved the src code deep within legacy default_mixed might find it annoying for a while. However, we would include a direct indication in changelog as well as the new class.
• Fixed unexpected bug 1: finite-diff support for PurityMP on torch, tf, and jax down
• Fixed unexpected bug 2: gradient calculation at PhaseFlip(0) breaks with nan returned at initial state |0> (all good for other other channels or other state with PhaseFlip`
• Fixed. expected bug 3: classical shadow is down for most of tests. This is due to the fact that the implementation is hidden inside QubitDevice which was missed out before. Need to be fixed asap

Related GitHub Issues:
[sc-73324]

[JerryChen97]

Let's merge this after the v0.40 release is live

[mudit2812]

Sorry for missing this earlier, but this seems like a check that should be performed in the initialization of QubitDensityMatrix rather than here.