Deprecate v1 primitives

[jyu00]

What should we add?

We should have deprecated V1 primitives in Qiskit 1.1, given its replacement (v2) had co-existed for 1 minor release. We should make sure it's deprecated in 1.2.

cc @t-imamichi, @ihincks

[1ucian0]

related to #11490

[1ucian0]

I'm putting this PR on hold as there are still open conversations on if or how are we going to do this.

[1ucian0]

After some off line conversations, we agree too:

• Deprecate the Interface type aliases qiskit.primitives.BaseEstimator (while leaving qiskit.primitives.BaseEstimatorV1). The recommendation is to use qiskit.primitives.BaseEstimatorV2 directly.
• Deprecate the Interface type aliases qiskit.primitives.BaseSampler (while leaving qiskit.primitives.BaseSamplerV1). The recommendation is to use qiskit.primitives.BaseEstimatorV2 directly.
• Deprecate qiskit.primitives.Estimator as reference implementation. The recommendation is to use qiskit.primitives.EstimatorV2 directly.
• Deprecate qiskit.primitives.Sampler as reference implementation. The recommendation is to use qiskit.primitives.SamplerV2 directly.
• Deprecate qiskit.primitives.backend_estimator.BackendEstimator as a tool. The recommendation is to use qiskit.primitives.BackendEstimatorV2 directly.
• Deprecate qiskit.primitives.backend_estimator.BackendSampler as a tool. The recommendation is to use qiskit.primitives.BackendSamplerV2 directly.

[t-imamichi]

How about other items in the PR #12575 (comment), e.g., BasePrimitivesResult , EstimatorResult, and SamplerResult? They are used for only EstimatorV1 and SamplerV1 while V2 uses PubResult and SamplerPubResult.

BasePrimitives is a base class of BaseSamplerV1 and BaseEstimatorV1.

[ElePT]

Given that BasePrimitives is not part of the deprecation (according to @1ucian0's message), I think it would make sense not to deprecate BasePrimitivesResult.

For EstimatorResult and SamplerResult, they are documented as the result types for the run method of BaserSamplerV1/BaseEstimatorV1, so I see a point for keeping them. Especially if we want to avoid breaking external implementations of the V1 interface (which I think is why the Base interface should be kept). However, I see how the names can lead to confusion. Would it make sense to "rename" them to SamplerV1Result or SamplerResultV1/ EstimatorV1Result or EstimatorResultV1 and deprecate the non-V1 name? In practice, we could deprecate the class and introduce a new type alias that doesn't raise a warning. This is however, this is still a breaking change, so I am not 100% sure it's the right way to go. The safest way would be keeping and not renaming them.

[t-imamichi]

Sorry, I misunderstood BasePrimitiveResult status. It's already deprecated #11054 at 0.46 (renamed as a private class _BasePrimitiveResult) and BasePrimitiveResult was removed #11576 at 1.0. We need to decide when we remove _BasePrimitiveResult.

[t-imamichi]

I have a concern to rename SamplerResult and EstimatorResult because it's a breaking change. I updated the docstrings to clarify that they are for V1 0b66c6a. How about this approach?