Update VQE tutorial to have SPSA optimizer

[nathanearnestnoble]

URL to the relevant tutorial

https://github.com/Qiskit/documentation/blob/main/docs/tutorials/spin-chain-vqe.ipynb

Select all that apply

• new content request
• typo
• code bug
• out-of-date content
• broken link
• other

Describe the fix or the content request.

right now spin chain vqe tutorial uses cobyla optimizer. SPSA is generally best for noise in hardware, and can be easily coded up.

For new content requests - if the request is accepted, do you want to write the content?

I only want to review the material when it's finished

[ritajitmajumdar1]

Good to see this coming through.

[jlapeyre]

@nathanearnestnoble I am happy to do this. I've been looking into it a little. I think the proposed change could be done fairly quickly. But, I might post a followup issue afterward. Just LMK.

EDIT: Just trying to reproduce the existing notebook has been a bit of an odyssey. I wonder when this last ran? The earliest version of the tutorial that I can find is this one. (I think an earlier one can be found, there is a reference to a version that is one year older).

I have found that it is difficult to get the optimization to more or less converge. I suspect it never did. That seems to be what the plot in the version linked above shows.

I wanted to reproduce known solid results. But that might not be possible. On the other hand, the existing SPSA code in qiskit algorithms seems to work in the current version of qiskit. Even if problems occur, it looks like the big module in algorithms is mostly features and polish. I've seen this elsewhere in algorithms. What I mean is grabbing just the core might be a good approach

[abbycross]

This will be closed by #4744