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.

Closes #8491
Closes #8492

Details and comments

Blocked by:

• Gradients with the primitives #8528
• Fidelity interface using primitives #8303
• Adds observable evaluator with primitives. #8683
• VQE implementation with estimator primitive #8702

Closely related to (but not blocked by):

• Rename run_options to options in fidelity and gradients and fidelity bug fix #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:

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

TO-AGREE-ON:

• Keyword vs. positional args
• Require optimizer??
• Including metadata in EigensolverResult, adding variance/shots from aux. ops, changing aux ops return type (only eigenvalue). Blocked by VQE implementation with estimator primitive #8702.

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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%

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

---

💛 - Coveralls

[woodsp-ibm]

I imagine we could build up these circuits just once as we go step by step. Here it will be rebuilding (binding) them each time which will be seen as different circuits on down through the primitives. Maybe this can be some future improvement in the interests of time if its not easily changed.

[ElePT]

Will leave as a future improvement. Not resolving the conversation for that purpose.

[woodsp-ibm]

It seems with lists we are going beyond the typehints for VariationalAlgo. And even on the initial point getter/setter though its Sequence float I guess ideally it would be a list of initial points not a single one, though I think here we only support an single value that is used on every step.

[ElePT]

Removed extension of VariationalResult

[woodsp-ibm]

Maybe we need to think about improving initial_point as input gong forwards since I think it ought to be a list of points one for each step, maybe we support a single value, like now for all. But ideally I should be able to feed in the optimal points outcome which is a list of points so each step has its optimizer set accordingly.

[ElePT]

I also agree, I will note it as a future improvement (again, not resolving the comment to be able to find it later).

[woodsp-ibm]

Suggested change

	knowledge to do this in that application domain.These algorithms are pending depreciation.
	knowledge to do this in that application domain. These algorithms are pending depreciation.

[ElePT]

I also realized it says depreciation instead of deprecation 🤣. Changing both egeinsolvers and minimum eigensolvers

[woodsp-ibm]

depreciation

Well they are losing their value as such! 🤣

[woodsp-ibm]

@ElePT Thanks for the great work!