Compute variance and standard error

[jjgoings]

Adds functionality to compute the variance and standard error of an expectation value.

• Adds get_variance function to Simulator to compute the variance of expectation value
• Adds get_standard_error function to Simulator to compute the standard error from the mean (shot-dependent)
• Adds simple analytical test to check variance is computed correctly

[alexfleury-sb]

I just left one comment in the thread. Beside this, I am all good with the code (I will let you make changes if necessary and mark it as approved soon after that)!

[jjgoings]

@AlexandreF-1qbit should be good now! Tests for both n_shots = None and n_shots = 10**5 should pass.

[ValentinS4t1qbit]

Thank you for your PR Joshua, really appreciate it ! You did a great job with adopting our coding guidelines and seemed to have found the right place to add your feature.

I found some minor things, that should be fairly easy to resolve / clean up.

There's code that may be redundant (reused from expectation value functions) but I'm not sure yet whether we should refactor this. For now I think it's better to address what has been brought up and resolve ongoing conversations: then it may become more obvious.

[ValentinS4t1qbit]

@AlexandreF-1qbit should be good now! Tests for both n_shots = None and n_shots = 10**5 should pass.

Dammit, I just gave you a review but it may mean it was given on code that's now outdated. Hopefully the stuff is still relevant. I allowed automated tests to run, see "checks".

[jjgoings]

@ValentinS4t1qbit I believe I addressed all your points -- thanks! Let me know if there is anything more I can do.
I wanted to bring up two points for your consideration:

1. Regarding refactoring, I totally agree this code is very similar to the get_expectation, as that's what I used as a template. I wonder if in the future we could have a function like get_moment where you can pass which moment you want to compute for mean, variance, skewness, etc.
2. The functions here for variance assumes that covariance between Pauli products is zero. This is true if each term of a Hamiltonian is measured with an independent circuit. But when you group terms to reduce the number of circuits, this is not necessarily true. Depending on how you group the terms, this can make the variance increase (if Cov(Pa,Pb) > 0) or decrease (Cov(Pa,Pb) < 0). I'm not sure how to handle this, because it seems like a case-by-case basis? One fascinating pathological example was given in Sec B.2 of this paper (starts on page 14), also related in Sec IX.A in this paper. Turns out that although you can partition the expectation into two measurements, it will still require more shots (33% more) to achieve the same precision as the seemingly sub-optimal three measurement partitioning. Anyway, don't really have a suggestion here other than to be aware and maybe in the future we can support computing variances given a QWC group in post-processing.

[ValentinS4t1qbit]

Thank you for contributing this feature to Tangelo Josh, we really appreciate it.
Congratulations on your 1st PR to this project !