add check_repr_commutation_relation to test the CSS orthogonality condition for 2BGA's Group Algebra with a General Group G

[Fe-r-oz]

This PR implements the CSS orthogonality condition given by left and right representation matrix commutation relation of group algebra of the 2BGA code. This consistency check will be used in the tests PRs. This is inspired from insights from @thofma!

Given that we are use a rich set of groups in the tests PRs , this consistency test will be provide an additional crosscheck.

Edit:

Some details from literature

• In a group algebra $\mathbb{F}[G]$, the left $L(a)$ and right $R(b)$ representations of elements commute:

  $$L(a) \cdot R(b) = R(b) \cdot L(a)$$

  This results from the associativity of multiplication.

The snapshot below talks about this in the following general way: Reference

• In an algebra $R$, the left $\lambda_r$ and right $\rho_r$ regular matrix representations commute:

  $$\rho_a \lambda_b = \lambda_b \rho_a$$

  This follows from the associativity of multiplication in $R$.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 84.82%. Comparing base (6a8dbb4) to head (dda1502).
Report is 14 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #403      +/-   ##
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+ Coverage   83.08%   84.82%   +1.74%     
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  Files          70       71       +1     
  Lines        4410     5120     +710     
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+ Hits         3664     4343     +679     
- Misses        746      777      +31     

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[Fe-r-oz]

@Krastanov, This PR is ready for review. Thanks! check_repr_commutation_relation will be used by the tests PRs to check the consistency of Group Algebra for 2BGA codes.

[Krastanov]

This looks useful, thanks! I left a few questions below

[Krastanov]

let's not export it for now, just keep it as an internal check

[Krastanov]

Is this check expected to be used anywhere else if merged?

Also, you do not need to do == true -- the result is already a boolean

[Fe-r-oz]

This check will be used by all the tests PRs. Since we are extending the functionality to a finite general group via group presentation, usually non-abelian, this checks whether this commutation relation holds for such groups.

[Fe-r-oz]

I have incorporated your suggestions, Thanks!

P.S A few misc cleanups were done: removed typo and moved the consistency check to ECC\codes\util.jl

[Krastanov]

thanks, this will be of great use throughout the tests