Sampler-based VQE for diagonal operators, plus QAOA

[Cryoris]

Summary

Add a VQE variant based on the sampler primitive, which only allows for optimization of diagonal ("classical") Hamiltonians.

Adds QAOA based on the above.
closes #8472

Details and comments

This differs from the estimator-based, standard VQE in that this version

• takes a BaseSampler and not a BaseEstimator
• allows for a custom aggregation function (such as CVaR)
• keeps track of the best overall measurement (not possible/useful in estimator-based case)
• returns samples of the final state (not possible in estimator-based case we generally have to measure in more than one basis)

Gradient support

In this PR, the gradient support has not yet been added. That's because it's not quite clear what's the best way to integrate the gradients. Here are the options we came up with, and we decided that we will first not support them and then directly implement option 4 if possible:

Option 1 The SamplingVQE takes a sampler and sampler gradient as input
pro: it is consistent as it takes a sampler and a sampler gradient
problem: I cannot support aggregation functions like CVaR, since I cannot easily compute the gradient if I have only access to the samples

Option 2 The SamplingVQE takes a sampler and an empty estimator gradient (i.e. without an estimator set)
pro: I build the diagonal estimator inside the SamplingVQE, and plug it into the estimator gradient
problem: consistency suffers, since I have to make the estimator in the gradients optional

Option 3 Make the SamplingVQE take a DiagonalEstimator and a EstimatorGradient equipped with said diagonal estimator
pro: consistent
problem: users have to build out the diagonal estimator, and the Sampling VQE will take an estimator

Option 4 Change how gradients work and introduce a general ParameterShift() (& co) which is not sampler or estimator specific
pro: makes my life easy
pro: makes users life easy as they don’t have to write out these nasty gradient names
problem: doesn’t exist yet

Todo

• Support batching
• See TODOs in tests

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[woodsp-ibm]

@Cryoris See also my DM to you for some comments

[coveralls]

Pull Request Test Coverage Report for Build 3154881397

• 268 of 296 (90.54%) changed or added relevant lines in 5 files are covered.
• 13 unchanged lines in 2 files lost coverage.
• Overall coverage increased (+0.2%) to 84.593%

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Changes Missing Coverage	Covered Lines	Changed/Added Lines	%
qiskit/algorithms/minimum_eigensolvers/qaoa.py	24	28	85.71%
qiskit/algorithms/minimum_eigensolvers/sampling_mes.py	47	53	88.68%
qiskit/algorithms/minimum_eigensolvers/sampling_vqe.py	116	122	95.08%
qiskit/algorithms/minimum_eigensolvers/diagonal_estimator.py	78	90	86.67%

Files with Coverage Reduction	New Missed Lines	%
qiskit/extensions/quantum_initializer/squ.py	2	79.78%
qiskit/compiler/transpiler.py	11	89.71%

Totals
Change from base Build 3154878985:	0.2%
Covered Lines:	61115
Relevant Lines:	72246

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💛 - Coveralls

[declanmillar]

This PR will also include a QAOA implementation that subclasses SamplingVQE.

[woodsp-ibm]

@declanmillar @Cryoris Thanks for the new function and primitive conversions. LGTM!