PSL

[michaelontiveros]

No description provided.

[perlinm]

Thanks for opening a PR! And apologies in advance that I am a bit bandwidth-limited at the moment, so it might take me some time to review this.

Some immediate questions:

1. Why are the generator matrices for PSL the same as for SL? I can define a group from the set of its generator matrices (namely: the subspace of matrices spanned by arbitrary products of the generators), so equal generator matrices should mean equal groups, no?
2. It looks like this assertion that PSL(2, 2).generators == SL(2, 2).generators is now broken --- which may be fine, but might warrant investigation to understand why. More worryingly, this assertion that PSL(2, 3).order == 24 is now broken. Why is that? Was the previous assertion wrong?

[perlinm]

I can answer my last question: it looks like PSL(2, 3).order should indeed be 12 and not 24. The GAP command Size(PSL(2, 3)); prints 12.

[michaelontiveros]

The finite groups SL and PSL are this big:

SL(n, q).order = q**( n*(n-1)/2 ) * ( q**2 - 1 )*( q**3 - 1 ) *...* ( q**n - 1 )
PSL(n, q).order = SL(n, q).order / gcd(n, q-1)

[michaelontiveros]

PSL(2, 2).generators and SL(2, 2).generators are different, but conjugate. This is because the elements of the target spaces are ordered differently.

[michaelontiveros]

Why are the generator matrices for PSL the same as for SL? I can define a group from the set of its generator matrices (namely: the subspace of matrices spanned by arbitrary products of the generators), so equal generator matrices should mean equal groups, no?

You are right, I don't actually construct a linear representation of PSL(n, q). As things stand, if linear_rep == False, then PSL constructs SL. All I do is quotient SL(n, q) by the action of the center. This maps the generators of the standard representation of SL(n, q) to equivalence classes of matrices, and I work out the multiplication table for PSL using coset representatives and the multiplication table for SL. The result is a permutation representation of PSL in Sym( Fq^n / (nth roots in Fq) ). To get a linear representation, we can compose with the standard linear representation of the symmetric group.

[michaelontiveros]

Okay now PSL.get_generator_mats() returns SL.get_generator_mats() in the n**2dimensional adjoint representation of SL. The adjoint representation extends to a faithful representation of the quotient group PSL(n) = SL(n)/center, so the PSL constructor should work even when linear_rep == false.

[perlinm]

Sorry again for the delay.

To clarify: is it currently the case that

Group.from_generating_mats(*PSL(dim, field).get_generator_mats())

essentially returns PSL(dim, field) (albeit with a different lift/representation)?

(Incidentally, I should probably make the naming of Group.from_generating_mats and Group.get_generator_mats consistent with each other. I'll probably switch to Group.get_generating_mats)

[michaelontiveros]

To clarify: is it currently the case that

Group.from_generating_mats(*PSL(dim, field).get_generator_mats())

returns PSL(dim, field) (albeit with a different lift/representation)?

Yes, that is the case :)

[perlinm]

Great! There is a bit of cleanup (formatting, typing, etc.) that would be required before merging into main, but I'm happy to merge this into a PSL branch for now and take care of the cleanup myself 🙂

Thanks again for the contribution!

[michaelontiveros]

Welcome!
Thanks for the help

[perlinm]

@michaelontiveros can you help me understand why we use roots of unity here, rather than all non-zero scalars (1, 2, ..., self.field.order - 1)?

PSL should be SL mod scalars, no? I guess I'm missing what that implies for the target space of PSL (or rather, I'm missing why it implies that the target space of PSL is equal to [target space of SL] mod [roots of unity of order dimension]).

[perlinm]

Also, I cleaned up this contribution a bit on the PSL branch. Do you want to open a new PR from PSL to main to have an "official" contribution into main? 🙂

I would also a appreciate a quick review (if you have the time) to make sure that I did not make any incorrect modifications to the code or comments.

[michaelontiveros]

We mod out by roots of unity of order dimension because the scalar matrix rI is in the group SL(dimension) if and only if the determinant is 1, and the determinant is r**dimension, which is 1 if and only if r is a root of unity of order dimension.

[michaelontiveros]

There is a typo in test_PSL():

qLDPC/qldpc/abstract_test.py

Line 185 in 3ce4ca2

order = order_SL // math.gcd(2, field - 1)

The 2 should be dimension.

Otherwise looks great, thanks!

[perlinm]

Awesome, that makes sense. Thanks!

[michaelontiveros]

Super, I think it's ready.
I will draft a PR to merge PSL to main :)

[michaelontiveros]

I get an error

Pull request creation failed. Validation failed: must be a collaborator

[perlinm]

Hmm... can you maybe fork the repo and make a PR from michaelontiveros:PSL into Infleqtion:main (similarly to how you did with this PR)?