Make changes to the functions in mitiq.shadows

[purva-thakre]

This issue will be used to keep track of all the things that could be changed, removed etc. from mitiq.shadows.

• A function to calculate fidelity is in the utilities. But a similar function also exists in cirq.fidelity. The function is mitiq also has some lines flagged by pytest as uncovered.
  

  mitiq/mitiq/shadows/shadows_utils.py

  Line 140 in 05be477

  def fidelity(

• Re-word all the function docstrings to clarify what the function is supposed to do. Right now, some of the functionalities in this module are not clear for this reason.
• Remove random strings in the middle of a function. Are these supposed to be comments?
  

  mitiq/mitiq/shadows/shadows.py

  Lines 138 to 144 in 05be477

  	r"""
  	Additional information:
  	Shadow stage 1: Sample random unitary form
  	:math:`\mathcal{g}\subset \mathrm{U}(2^n)` and perform computational
  	basis measurement. In the current state, we have implemented
  	local Pauli measurement, i.e. :math:`\mathcal{g} = \mathrm{Cl}_2^n`.
  	"""

  
  

  mitiq/mitiq/shadows/shadows.py

  Lines 191 to 195 in 05be477

  	"""
  	Additional information:
  	Shadow stage 2: Estimate the expectation value of the observables OR
  	reconstruct the state
  	"""

  
  

  mitiq/mitiq/shadows/classical_postprocessing.py

  Lines 45 to 48 in 05be477

  	"""
  	The following functions are used in the classical post-processing
  	of calibration
  	"""

  
  

  mitiq/mitiq/shadows/classical_postprocessing.py

  Lines 176 to 179 in 05be477

  	"""
  	The following functions are used in the classical post-processing of
  	classical shadows.
  	"""

• If some error is raised, make sure it is mentioned exactly in the docstring as Raises: SomeError instead of as a Warning.
• Is there some kind of check to verify classical shadow size chosen by user is appropriate for the functionality in mitiq.shadows to work as expected?
• n_measurements_tomography_bound is only used in a tutorial in the docs. it might be better to define this function there instead of having it in mitiq/shadows/shadows_utils.py

More to be added later.

[Misty-W]

Partially addressed in
Refactor classical postprocessing in shadows module #2152 , but more work to be done to close this issue. Remove from 0.34.0.

[purva-thakre]

Compared to other techniques, mitiq.shadows does not follow the pattern of having a function called execute_with_shadows, shadows.mitigate_executor, and shadows.decorator.

Whenever we get to the cleanup of this module, we should consider a refactor of the above.