[Classical Shadows 5] Shadows/RShadows tutorial

[Min-Li]

Fixes #1930

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Description

Notebook on classical shadow and calibration, state reconstruction, and expectation value estimation
Users guide:
shadows-1-intro.md
shadows-5-theory.md

Examples:
shadows_tutorial.md
rshadows_tutorial.md

I also write a very brief theory intro part for Clifford data regression:
CDR-5-theory.md

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License

• I license this contribution under the terms of the GNU GPL, version 3 and grant Unitary Fund the right to provide additional permissions as described in section 7 of the GNU GPL, version 3.

Before opening the PR, please ensure you have completed the following where appropriate.

• I added unit tests for new code.
• I used type hints in function signatures.
• I used Google-style docstrings for functions.
• I updated the documentation where relevant.
• Added myself / the copyright holder to the AUTHORS file

[nathanshammah]

Docs build fail due to shadows-1-intro.md with this message:

/home/runner/work/mitiq/mitiq/docs/source/guide/shadows-1-intro.md: WARNING: Executing notebook failed: CellExecutionError [mystnb.exec]
/home/runner/work/mitiq/mitiq/docs/source/guide/shadows-1-intro.md: WARNING: Notebook exception traceback saved in: /home/runner/work/mitiq/mitiq/docs/build/html/reports/guide/shadows-1-intro.err.log [mystnb.exec]
looking for now-outdated files... none found

[natestemen]

This PR will need to be updated again as #1989 moved some of the functions required in the notebooks to different files. The actual error that's occurring is

---------------------------------------------------------------------------
ImportError                               Traceback (most recent call last)
Cell In[15], line 1
----> 1 from mitiq.shadows.shadows_utils import operator_ptm_vector_rep
      3 ghz_state = circuit.final_state_vector().reshape(-1, 1)
      4 ghz_true = ghz_state @ ghz_state.conj().T

ImportError: cannot import name 'operator_ptm_vector_rep' from 'mitiq.shadows.shadows_utils' (/home/docs/checkouts/readthedocs.org/user_builds/mitiq/envs/1945/lib/python3.10/site-packages/mitiq/shadows/shadows_utils.py)

You can see this on RTD.

[andreamari]

There is a link-check error unrelated to this PR. It should go away once #2022 is merged.

[Misty-W]

Thanks for finishing this @Min-Li, great work!

[nathanshammah]

Thanks @Min-Li! Massive work, glad to add it to the Mitiq codebase! We are going to open some issues to improve the documentation and reduce the documentation build time.

[Min-Li]

@nathanshammah Thank you for the positive feedback! I'm glad my work could be a valuable addition to the Mitiq codebase. I look forward to the upcoming issues on improving the documentation and optimizing the build time.