Implement eigensolvers (VQD) with primitives

[ElePT]

Summary

This PR addresses #8491 and #8492 and provides an implementation of VQD using the new primitive interfaces. The new implementation comes with a change of namespace for the whole eigensolvers module (from eigen_solvers to eigensolvers), which is more consistent with the naming conventions used throughout the stack.

Details and comments

Blocked by:

• [x] #8528
• [x] #8303
• [x] #8683
• [ ] #8702

Closely related to (but not blocked by):

• [ ] #8755

To ensure consistency with the changes made in VQE (#8702), the following elements have been removed:

• The final eigenstate that used to be part of the EigensolverResult. To reproduce this one may use a Sampler and the optimal point from VQD outside the algorithm.
• The setting property and print_settings method.
• All trivial getters and setters, just use public attributes instead (expect for initial_point, which requires it for the VariationalAlgorithm interface).
• expectation and include_custom. These are not required anymore when using an Estimator.

Additionally, the following changes have been applied:

• Removed the option to send a custom Gradient through the VQD intialization. This option will be on hold until FidelityGradients using primitives are enabled.
• Created a new NumPyEigensolverResult class that allows the NumPyEigensolver to still return the eigenstate.
• Enforced "non-essential" arguments to be keyword-only.

This PR also fixes a bug found in qiskit/algorithms/observables_evaluator.py where the variance/shots are not correctly extracted from the estimator metadata, and adds a corresponding unit test to verify the fix.

TO-DO:

• [x] Add attributes to doctstring
• [x] Add support and unit tests for BaseOperator inputs in numpy eigensolver
• [x] Create property defaults in compute instead of init
• [ ] Import _validate_bounds, _validate_initial_point from utils (blocked by #8702)
• [ ] Modify VQE import (blocked by #8702)

TO-AGREE-ON:

• [x] Keyword vs. positional args
• [x] Require optimizer??
• [ ] Including metadata in EigensolverResult, adding variance/shots from aux. ops, changing aux ops return type (only eigenvalue).

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[coveralls]

Pull Request Test Coverage Report for Build 3150026480

• 372 of 398 (93.47%) changed or added relevant lines in 8 files are covered.
• No unchanged relevant lines lost coverage.
• Overall coverage increased (+0.05%) to 84.551%

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Changes Missing Coverage	Covered Lines	Changed/Added Lines	%
qiskit/algorithms/eigensolvers/eigensolver.py	32	33	96.97%
qiskit/algorithms/eigensolvers/vqd.py	184	194	94.85%
qiskit/algorithms/eigensolvers/numpy_eigensolver.py	145	160	90.63%
<!--	Total:	372	398

Totals
Change from base Build 3150022884:	0.05%
Covered Lines:	60756
Relevant Lines:	71857

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💛 - Coveralls

[woodsp-ibm]

In #8702, since the algorithms are now in a subfolder and need to be imported from there to avoid name collisions, the unit tests for the new algorithms were added in a like-named subfolder. That maybe a better bet here too - leave the old NumPy/VQD tests be where they were testing the pending deprecated existing versions and add tests for new algos in a new folder