Ensure aux_operator eigenvalues are normalized

[mrossinek]

Summary

As reported on multiple occasions by @MariaSapova (#1460 and #1467), the eigenvalues of
the auxiliary operators are not correctly normalized when the QASM
backend is used.

This commit fixes this short-coming by applying the same normalization
to the VQE's eigenstate when obtained from a QASM backend (in which case
this is a dictionary) as done by the CircuitSampler in its
sample_circuits function.
The case of the statevector backend is unaffected by this change, as it
will return the eigenstate as a list.

[ikkoham]

Hello, this fix sounds nice, but why didn't you add this processsing here?

[mrossinek]

Hi @ikkoham! I am not sure to be honest.. but that does indeed look like a better place to fit this. I will update the PR now.

[Cryoris]

We actually thought of putting it there in the first place -- but the reason we didn't is to not change the current behavior that get_optimal_vector gives you the circuit output. Personally, I'd be good with changing it there but maybe @pbark @woodsp-ibm or @manoelmarques have an opinion here

[mrossinek]

We actually thought of putting it there in the first place -- but the reason we didn't is to not change the current behavior that get_optimal_vector gives you the circuit output.

Ah I didn't remember correctly then.. well I agree though that this makes more sense even though it changes the current behavior. The reason being that I think it improves the current behavior..

[Cryoris]

I'm good with changing this at the higher level. We'll have to wait for the merge of the algos to Terra though and then do the change there.

[woodsp-ibm]

I feel there ought to be a unit test around this - maybe augment one of those that exist. Just wondering was this specific to VQE-Adapt. I do not recall any issue with VQE in the past.

[mrossinek]

Yes a unittest should be added. I will look into that once we re-open this PR when everything is merged into Terra.

Just wondering was this specific to VQE-Adapt
@woodsp-ibm This was reported both, for AdaptVQE (#1467) and for QEOM (#1460)

I also never noticed this before but I am not sure when this broke. I tried bisecting it but I didn't have enough time to go through with this because I would have to adapt the test script a lot because of the many refactorings we did lately (i.e. the test script to check whether it breaks needs to be changed).
I might look into this in more detail once revisiting the PR for terra.

[mrossinek]

I migrated this PR over to Terra and also added a unittest while I was at it.
Closing this PR.