noncliff: Cauchy Schwarz Inequality check for GeneralizedStabilizers

[Fe-r-oz]

This PR implements the Cauchy Schwarz Inequality for GeneralizedStabilizers. I have a test that check this for n =1:10

Inner product between two Generalized Stabilizers is the LHS: $Tr[sm'sm]$ without the absolute value.

Except about importance of inner product algorithm from TJ Yoder:

The inner product algorithm, allows us to determine whether two generalized stabilizer states are equal. This is not always a trivial thing to do, because two generalized stabilizers with different stabilizer bases and different χ-matrices may represent the same state.

Edit:

The inner product is implemented in #423 as well.

[Krastanov]

That would indeed be a usable check. But as mentioned in the PR that implements the dot product, this should not be relying on exponentially-expensive conversion into a density matrix.

This should not be exported either, as it is only for tests (generally it is important to be very conservative with exports, as otherwise you are promising to keep a stable API).

Given that it is internal and single-purpose, let's also rename it to something more detailed. E.g. generalizedstab_cauchyschwarz_check.

I will mark this as a draft as it can not be finished until the other related PR is finished.

[Fe-r-oz]

Indeed. Thank you for your comments!

[Fe-r-oz]

Thank you, I will keep these points in mind. I need to implement stabilizer union (theorem 16) first for the fast, in-place implementation of inner product. Details at: #423 (comment)