fix random statevector distribution

[kevinsung]

Summary

Fixes #9619

Details and comments

[qiskit-bot]

One or more of the the following people are requested to review this:

• @Qiskit/terra-core
• @ikkoham

[jakelishman]

Thanks for this. I think the output from your fixed version is Haar random, isn't it? - in the sense of the result is distributed the same as if we applied a Haar-random unitary to some fixed statevector. I'm not certain of that, but if so, it might be good to keep some text describing that in the documentation. If not, then I think we still might need to define what we mean "uniform distribution".

Could we add a bugfix note about fixing the distribution? I'm feeling about iffy about backporting this, even if we call it a bugfix, just because of the (intentional!) RNG incompatibility - I think personally I'd prefer to wait til a minor with it.

[kevinsung]

I don't think we need to mention the Haar measure on unitary matrices. This is the uniform distribution. I've added a citation for it.

[jakelishman]

Ah super thanks - this reads well to me. Julien, can you tag for merge if you're happy as well?