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Implementing Bivariate Bicycle Codes using 2BGA as the parent via Hecke's Group Algebra (#399)
 Co-authored-by: Stefan Krastanov <[email protected]> Co-authored-by: Stefan Krastanov <[email protected]>
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‎docs/src/references.bib

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@@ -523,3 +523,14 @@ @article{lin2024quantum
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year={2024},
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publisher={APS}
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}
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@article{bravyi2024high,
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title={High-threshold and low-overhead fault-tolerant quantum memory},
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author={Bravyi, Sergey and Cross, Andrew W and Gambetta, Jay M and Maslov, Dmitri and Rall, Patrick and Yoder, Theodore J},
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journal={Nature},
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volume={627},
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number={8005},
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pages={778--782},
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year={2024},
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publisher={Nature Publishing Group UK London}
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}

‎docs/src/references.md

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@@ -40,6 +40,8 @@ For quantum code construction routines:
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- [steane1999quantum](@cite)
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- [campbell2012magic](@cite)
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- [anderson2014fault](@cite)
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- [lin2024quantum](@cite)
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- [bravyi2024high](@cite)
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For classical code construction routines:
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- [muller1954application](@cite)

‎ext/QuantumCliffordHeckeExt/lifted_product.jl

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@@ -150,14 +150,18 @@ code_n(c::LPCode) = size(c.repr(zero(c.GA)), 2) * (size(c.A, 2) * size(c.B, 1) +
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code_s(c::LPCode) = size(c.repr(zero(c.GA)), 1) * (size(c.A, 1) * size(c.B, 1) + size(c.A, 2) * size(c.B, 2))
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"""
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Two-block group algebra (2GBA) codes, which are a special case of lifted product codes
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Two-block group algebra (2BGA) codes, which are a special case of lifted product codes
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from two group algebra elements `a` and `b`, used as `1x1` base matrices.
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## Examples of 2BGA code subfamilies
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### `C₄ x C₂`
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Here is an example of a [[56, 28, 2]] 2BGA code from Table 2 of [lin2024quantum](@cite)
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with direct product of `C₄ x C₂`.
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```jldoctest
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julia> import Hecke: group_algebra, GF, abelian_group, gens;
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julia> import Hecke: group_algebra, GF, abelian_group, gens
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julia> GA = group_algebra(GF(2), abelian_group([14,2]));
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(56, 28)
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```
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See also: [`LPCode`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref)
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### Bivariate Bicycle codes
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Bivariate Bicycle codes are a class of Abelian 2BGA codes formed by the direct product
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of two cyclic groups `ℤₗ × ℤₘ`. The parameters `l` and `m` represent the orders of the
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first and second cyclic groups, respectively.
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The ECC Zoo has an [entry for this family](https://errorcorrectionzoo.org/c/qcga).
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A [[756, 16, ≤ 34]] code from Table 3 of [bravyi2024high](@cite):
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```jldoctest
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julia> import Hecke: group_algebra, GF, abelian_group, gens
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julia> l=21; m=18;
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julia> GA = group_algebra(GF(2), abelian_group([l, m]));
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julia> x, y = gens(GA);
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julia> A = x^3 + y^10 + y^17;
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julia> B = y^5 + x^3 + x^19;
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julia> c = two_block_group_algebra_codes(A,B);
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julia> code_n(c), code_k(c)
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(756, 16)
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```
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See also: [`LPCode`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref).
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"""
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function two_block_group_algebra_codes(a::GroupAlgebraElem, b::GroupAlgebraElem)
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LPCode([a;;], [b;;])

‎test/test_ecc_base.jl

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@@ -58,14 +58,41 @@ A[LinearAlgebra.diagind(A, 5)] .= GA(1)
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B = reshape([1 + x + x^6], (1, 1))
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push!(other_lifted_product_codes, LPCode(A, B))
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# Bivariate Bicycle codes
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# A [[72, 12, 6]] code from Table 3 of [bravyi2024high](@cite).
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l=6; m=6
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^3 + y + y^2
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B = y^3 + x + x^2
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bb1 = two_block_group_algebra_codes(A,B)
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# A [[90, 8, 10]] code from Table 3 of [bravyi2024high](@cite).
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l=15; m=3
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^9 + y + y^2
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B = 1 + x^2 + x^7
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bb2 = two_block_group_algebra_codes(A,B)
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# A [[360, 12, ≤ 24]] code from Table 3 of [bravyi2024high](@cite).
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l=30; m=6
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^9 + y + y^2
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B = y^3 + x^25 + x^26
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bb3 = two_block_group_algebra_codes(A,B)
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test_bb_codes = [bb1, bb2, bb3]
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const code_instance_args = Dict(
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:Toric => [(3,3), (4,4), (3,6), (4,3), (5,5)],
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:Surface => [(3,3), (4,4), (3,6), (4,3), (5,5)],
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:Gottesman => [3, 4, 5],
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:CSS => (c -> (parity_checks_x(c), parity_checks_z(c))).([Shor9(), Steane7(), Toric(4, 4)]),
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:Concat => [(Perfect5(), Perfect5()), (Perfect5(), Steane7()), (Steane7(), Cleve8()), (Toric(2, 2), Shor9())],
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:CircuitCode => random_circuit_code_args,
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:LPCode => (c -> (c.A, c.B)).(vcat(LP04, LP118, test_gb_codes, other_lifted_product_codes)),
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:LPCode => (c -> (c.A, c.B)).(vcat(LP04, LP118, test_gb_codes, test_bb_codes, other_lifted_product_codes)),
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:QuantumReedMuller => [3, 4, 5]
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)
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‎test/test_ecc_bivaraite_bicycle_as_twobga.jl

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@testitem "ECC Bivaraite Bicycle as 2BGA" begin
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using Hecke
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using Hecke: group_algebra, GF, abelian_group, gens, one
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using QuantumClifford.ECC: two_block_group_algebra_codes, code_k, code_n
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@testset "Reproduce Table 3 bravyi2024high" begin
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# [[72, 12, 6]]
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l=6; m=6
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^3 + y + y^2
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B = y^3 + x + x^2
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 72 && code_k(c) == 12
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# [[90, 8, 10]]
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l=15; m=3
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^9 + y + y^2
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B = 1 + x^2 + x^7
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 90 && code_k(c) == 8
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# [[108, 8, 10]]
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l=9; m=6
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^3 + y + y^2
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B = y^3 + x + x^2
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 108 && code_k(c) == 8
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# [[144, 12, 12]]
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l=12; m=6
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^3 + y + y^2
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B = y^3 + x + x^2
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 144 && code_k(c) == 12
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# [[288, 12, 12]]
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l=12; m=12
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^3 + y^2 + y^7
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B = y^3 + x + x^2
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 288 && code_k(c) == 12
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# [[360, 12, ≤ 24]]
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l=30; m=6
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^9 + y + y^2
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B = y^3 + x^25 + x^26
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 360 && code_k(c) == 12
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# [[756, 16, ≤ 34]]
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l=21; m=18
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^3 + y^10 + y^17
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B = y^5 + x^3 + x^19
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 756 && code_k(c) == 16
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end
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@testset "Reproduce Table 1 berthusen2024toward" begin
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# [[72, 8, 6]]
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l=12; m=3
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^9 + y + y^2
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B = 1 + x + x^11
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 72 && code_k(c) == 8
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# [[90, 8, 6]]
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l=9; m=5
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^8 + y^4 + y
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B = y^5 + x^8 + x^7
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 90 && code_k(c) == 8
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# [[120, 8, 8]]
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l=12; m=5
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^10 + y^4 + y
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B = 1 + x + x^2
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 120 && code_k(c) == 8
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# [[150, 8, 8]]
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l=15; m=5
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^5 + y^2 + y^3
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B = y^2 + x^7 + x^6
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 150 && code_k(c) == 8
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# [[196, 12, 8]]
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l=14; m=7
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^6 + y^5 + y^6
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B = 1 + x^4 + x^13
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 196 && code_k(c) == 12
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end
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@testset "Reproduce Table 1 wang2024coprime" begin
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# [[54, 8, 6]]
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l=3; m=9
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = 1 + y^2 + y^4
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B = y^3 + x + x^2
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 54 && code_k(c) == 8
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# [[98, 6, 12]]
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l=7; m=7
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^3 + y^5 + y^6
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B = y^2 + x^3 + x^5
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 98 && code_k(c) == 6
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# [[126, 8, 10]]
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l=3; m=21
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = 1 + y^2 + y^10
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B = y^3 + x + x^2
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 126 && code_k(c) == 8
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# [[150, 16, 8]]
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l=5; m=15
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = 1 + y^6 + y^8
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B = y^5 + x + x^4
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 150 && code_k(c) == 16
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# [[162, 8, 14]]
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l=3; m=27
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = 1 + y^10 + y^14
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B = y^12 + x + x^2
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 162 && code_k(c) == 8
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# [[180, 8, 16]]
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l=6; m=15
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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A = x^3 + y + y^2
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B = y^6 + x^4 + x^5
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c = two_block_group_algebra_codes(A,B)
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@test code_n(c) == 180 && code_k(c) == 8
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end
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end

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