Extending the capabilities of 2BGA codes via Oscar's Group Algebra

[Fe-r-oz]

Extending the capabilities of 2BGA codes via Oscar's Group Algebra

Introduction

The PR #356 introduces Lifted Product codes with "commutative" group algebra using Hecke that works with abelian groups. Currently, constructing groups via group presentations and non-abelian groups are not supported. The paper Quantum two-block group algebra codes introduced the 2BGA codes incorporating the functionalities such as working with very specific group presentations, non-abelian groups, direct or semidirect products of groups which are currently not supported. The specific group presentations are the key ingredient for construction of Group Algebra of 2BGA with abelian and non-abelian groups.

The goal is to extend the capabilities of 2BGA codes in three verticals:

graph TD
    A[Extend 2BGA Capabilities] --> B[Table 1: Specific Group Presentations ⟨S∣R⟩]
    A --> C[Table 2: Direct Product of Cyclic Groups Cₗ x Cₘ]
    A --> D[Table 3: Group Presentation for Dihedral Groups Dₘ]

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Objectives

1. Direct Product of Groups $C_l \times C_m$
2. Specific Group Presentations $\langle S \mid R \rangle$
3. Non-abelian groups such as Dihedral Group,Symmetric Group, Alternating Groups
4. Group presentation of non-abelian Dihedral group, $D_m$
5. Small Groups with Multiplication Table via Free Groups
6. Semidirect Product of two Cyclic groups $C_l \ltimes C_m$

Implementation

• 1) Direct Product of Groups Cₗ x Cₘ: add tests and examples of Cₘ × C₂ 2BGA codes #392
• 2), 3), 5) Non-abelian groups via Small Groups: tests: Specific Group Presentations Using Oscar's Free Group for Non-Abelian and Abelian Groups #390
• Discrepancy in 2BGA code_k values when reproducing some results from paper by Lin et al via Hecke/Oscar.small_group #391. Many Thanks to Tommy Hofmann for his wonderful insights in resolving this issue.
• Direct Product of non-abelian group with Cyclic group, Dₗ x Cₘ tests: dihedral group based LPCode via group presentation #397
• Reproduce Table 3 using Dₗ x Cₘ tests: dihedral group based LPCode via group presentation #397
• Group Presentation for Cyclic Group ⟨x,s|xᵐ=s²=xsx⁻¹s⁻¹=1⟩ tests: Group Presentation for Cyclic Group Cₘₕ = Cₘ × C₂ #398
• twobga_from_fp_group API, documentation, and testing via test_ecc_base.jl: add QuantumCliffordOscarExt to provide convenient API for finitely presented groups via specific group presentation #400
• Add documentation of using LPCode via Hecke's small_group #406

Outcome

For correctness, the goal is to reproduce the results (Table 1, Table 2 , Table 3) from the paper.

• Reproduce results from Table 1 of Lin et al.
• """"""""""""""""""""""""""" Table 2 of Lin et al.
• """"""""""""""""""""""""""" Table 3 of Lin et al.
• API, docs, and tests via test_ecc_base.jl:QuantumCliffordOscarExt
• """"""""""""""""""""""""""" Table 3 of Bravyi et al.
• """"""""""""""""""""""""""" Table 1 of Berthusen et al.
• """"""""""""""""""""""""""" Table 1 of Wang et al.

Table 2 has been reproduced using two approaches: direct products of cyclic groups and group presentation, as a cross-check for consistency. Similarly, Table 3 has been cross-checked using semidirect products of cyclic groups and group presentation.

Reviewer: Stefan Krastanov

Additional Details

Edit: P.S. After some time, I've revisited and cleaned up this issue request. Removed outdated/filler details and refocused on clearly outlining the features to implement.

[Fe-r-oz]

Hi, @Krastanov,

I wanted to check with you about a potential Bug Bounty for some recent work. I've been collaborating with @thofma to fully reproduce the results from the Lin and Pryadko paper, extending Group Algebra-based functionality, and building significant new features on top of #356. Tommy’s contributions were invaluable, particularly with the group presentation, and he helped resolve issue #391. Initially, I pursued this to explore the paper and add new functionality, without the intention of turning it into a Bug Bounty.

However, given the scope and impact of this work, I was wondering whether it might qualify for the Quantum Savory Bug Bounty program. It seems like the type of contribution that could be a good fit, though I understand that you're best placed to assess its relevance and whether it aligns with the program.

Additionally, I believe Tommy could quickly address the downgrade and the specific CI errors we've encountered with Oscar. I wanted to explore whether this could also be structured as a Bug Bounty, with shared credit, though I’ll leave the details to your judgment and am happy to accept whatever you decide.

Thank you for your time and guidance, and I look forward to hearing your thoughts.

Best Regards.

[Krastanov]

Indeed, this is a pretty significant contribution, and a very valuable set of tests verifying the consistency of @royess 's work, thank you for looking into it. It will be appropriate to provide a bounty for its completion. It will take me some time to go over the PRs and review them and to write down the exact parameters of the bounty (we will probably need a convenient API for accessing these codes, having some decoder benchmarks, etc, not just having them as examples in the test runner). I will try to write this up over the next week.

@thofma , thank you for the guidance you have provided to Feroz, it is much appreciated.

[Fe-r-oz]

Indeed. I think that implementing a convenient API, namely twobga_from_fp_group, similar to LPCode(A,B), within a new QuantumCliffordOscarExt would be helpful. The twobga_from_fp_group method would utilize the 2bga method from HeckeExt and further extend its functionalities through group presentations. For example, in the first doctest, the user defines the elements A and B using group_algebra, GA, which are then passed to LPCode(A,B). Similarly, users will be able to define GA based on group presentation for non-abelian/abelian groups, specify the elements a and b, and input them into twobga_from_fp_group.

In addition, QuantumCliffordOscarExt will be helpful in the future, particularly if we need specific methods from Oscar, as we can define those methods within this Ext. It will enable clear differentiation in the functionalities we require from Oscar that are not available in AA, Hecke, or Nemo and should be used exclusively for those purposes.

Please refine and enhance this along with the other requirements during the writing process. Thank you.