Add simulate of analytic case for DefaultMixedNewAPI

[JerryChen97]

Before submitting

Please complete the following checklist when submitting a PR:

• All new features must include a unit test.
  If you've fixed a bug or added code that should be tested, add a test to the
  test directory!

• All new functions and code must be clearly commented and documented.
  If you do make documentation changes, make sure that the docs build and
  render correctly by running make docs.

• Ensure that the test suite passes, by running make test.

• Add a new entry to the doc/releases/changelog-dev.md file, summarizing the
  change, and including a link back to the PR.

• The PennyLane source code conforms to
  PEP8 standards.
  We check all of our code against Pylint.
  To lint modified files, simply pip install pylint, and then
  run pylint pennylane/path/to/file.py.

When all the above are checked, delete everything above the dashed
line and fill in the pull request template.

---

Context:
The implementation of method simulate is the most important one under the architecture of new API standard. The point of simulate, is basically abstract both the analytic version (controlled by shots=None, or infinite shots in a fancier way) and the MC version (finite shots). Meanwhile, the functionality of such method should not truly depend on our details of qml implementationi; they better only rely on either the fundamental computation libs, e.g. math lib and qnp, or the most essential infras, e.g. Tape/Scripts and Operations.

Description of the Change:

Benefits:

Possible Drawbacks:

Related GitHub Issues:
[sc-73320]

[JerryChen97]

The three fails up to now are all expected; will fix when the blockers were removed and this branch got updated

[JerryChen97]

@lillian542 I just rebased to the measure branch. There might be still some confusing out-of-scope changes somehwere e.g. a little fix regarding init_state over the other PR; but this PR's primary purpose is to introduce the added file simulate.py as well as its test suite

[codecov]

Codecov Report

Attention: Patch coverage is 94.87179% with 2 lines in your changes missing coverage. Please review.

Project coverage is 99.64%. Comparing base (ff09567) to head (da9e8af).
Report is 20 commits behind head on add-qubit_mixed/measure.

Files with missing lines	Patch %	Lines
pennylane/devices/default_mixed.py	60.00%	2 Missing ⚠️

Additional details and impacted files

@@                     Coverage Diff                     @@
##           add-qubit_mixed/measure    #6618      +/-   ##
===========================================================
- Coverage                    99.65%   99.64%   -0.01%     
===========================================================
  Files                          456      456              
  Lines                        43019    43084      +65     
===========================================================
+ Hits                         42870    42933      +63     
- Misses                         149      151       +2     

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

[JerryChen97]

Put DO NOT MERGE since currently we are only comparing it with another in-progress branch instead of requesting to merge.
After the measure branch is merged to master, we remove the DO NOT MERGE.