implement Bivaraite Bicycle codes using 2BGA as parent

docs/src/references.bib

@@ -487,3 +487,14 @@ @article{anderson2014fault
   year={2014},
   publisher={APS}
 }
+
+@article{bravyi2024high,
+  title={High-threshold and low-overhead fault-tolerant quantum memory},
+  author={Bravyi, Sergey and Cross, Andrew W and Gambetta, Jay M and Maslov, Dmitri and Rall, Patrick and Yoder, Theodore J},
+  journal={Nature},
+  volume={627},
+  number={8005},
+  pages={778--782},
+  year={2024},
+  publisher={Nature Publishing Group UK London}
+}

ext/QuantumCliffordHeckeExt/QuantumCliffordHeckeExt.jl

@@ -5,13 +5,14 @@ using DocStringExtensions
 import QuantumClifford, LinearAlgebra
 import Hecke: Group, GroupElem, AdditiveGroup, AdditiveGroupElem,
     GroupAlgebra, GroupAlgebraElem, FqFieldElem, representation_matrix, dim, base_ring,
-    multiplication_table, coefficients, abelian_group, group_algebra
+    multiplication_table, coefficients, abelian_group, group_algebra,
+    FinGenAbGroup, FinGenAbGroupElem, one
 import Nemo
 import Nemo: characteristic, matrix_repr, GF, ZZ, lift
 
 import QuantumClifford.ECC: AbstractECC, CSS, ClassicalCode,
     hgp, code_k, code_n, code_s, iscss, parity_checks, parity_checks_x, parity_checks_z, parity_checks_xz,
-    two_block_group_algebra_codes, generalized_bicycle_codes, bicycle_codes
+    two_block_group_algebra_codes, generalized_bicycle_codes, bicycle_codes, bivariate_bicycle_codes
 
 include("types.jl")
 include("lifted.jl")

ext/QuantumCliffordHeckeExt/lifted_product.jl

@@ -196,3 +196,38 @@ function bicycle_codes(a_shifts::Array{Int}, l::Int)
     a = sum(GA[n÷l+1] for n in a_shifts)
     two_block_group_algebra_codes(a, group_algebra_conj(a))
 end
+
+"""
+Bivariate Bicycle codes are a class of Abelian 2BGA codes formed by the direct product
+of two cyclic groups `ℤₗ × ℤₘ`. The parameters `l` and `m` represent the orders of the
+first and second cyclic groups, respectively.
+
+The ECC Zoo has an [entry for this family](https://errorcorrectionzoo.org/c/q-ary_bch).
+
+A [[756, 16, ≤ 34]] code from Table 3 of [bravyi2024high](@cite).
+
+```jldoctest
+julia> import Hecke: group_algebra, GF, abelian_group, gens; # hide
+
+julia> l=21; m=18;
+
+julia> GA = group_algebra(GF(2), abelian_group([l, m]));
+
+julia> x, y = gens(GA);
+
+julia> A = [x^3 , y^10 , y^17];
+
+julia> B = [y^5 , x^3  , x^19];
+
+julia> c = bivariate_bicycle_codes(A,B,GA);
+
+julia> code_n(c), code_k(c)
+(756, 16)
+```
+"""
+function bivariate_bicycle_codes(A::Vector{GroupAlgebraElem{FqFieldElem, GroupAlgebra{FqFieldElem, FinGenAbGroup, FinGenAbGroupElem}}}, B::Vector{GroupAlgebraElem{FqFieldElem, GroupAlgebra{FqFieldElem, FinGenAbGroup, FinGenAbGroupElem}}}, GA::GroupAlgebra{FqFieldElem, FinGenAbGroup, FinGenAbGroupElem})
+    a = sum(GA(x) for x in A)
+    b = sum(GA(x) for x in B)
+    c = two_block_group_algebra_codes(a,b)
+    return c
+end

src/ecc/ECC.jl

@@ -22,6 +22,7 @@ export parity_checks, parity_checks_x, parity_checks_z, iscss,
     Shor9, Steane7, Cleve8, Perfect5, Bitflip3,
     Toric, Gottesman, Surface, Concat, CircuitCode, QuantumReedMuller,
     LPCode, two_block_group_algebra_codes, generalized_bicycle_codes, bicycle_codes,
+    bivariate_bicycle_codes,
     random_brickwork_circuit_code, random_all_to_all_circuit_code,
     evaluate_decoder,
     CommutationCheckECCSetup, NaiveSyndromeECCSetup, ShorSyndromeECCSetup,

src/ecc/codes/lifted_product.jl

@@ -17,3 +17,6 @@ function generalized_bicycle_codes end
 
 """Implemented in a package extension with Hecke."""
 function bicycle_codes end
+
+"""Implemented in a package extension with Hecke."""
+function bivariate_bicycle_codes end

test/test_ecc_bivaraite_bicycle_as_twobga.jl

@@ -0,0 +1,173 @@
+@testitem "ECC Bivaraite Bicycle as 2BGA" begin
+    using Hecke
+    using Hecke: group_algebra, GF, abelian_group, gens, one
+    using QuantumClifford.ECC: bivariate_bicycle_codes, code_k, code_n
+
+    @testset "Reproduce Table 3 bravyi2024high" begin
+        # [[72, 12, 6]]
+        l=6; m=6
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [x^3, y, y^2]
+        B = [y^3, x, x^2]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 72 &&  code_k(c) == 12
+
+        # [[90, 8, 10]]
+        l=15; m=3
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [x^9   , y   , y^2]
+        B = [one(x), x^2 , x^7]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 90 &&  code_k(c) == 8
+
+        # [[108, 8, 10]]
+        l=9; m=6
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [x^3 , y , y^2]
+        B = [y^3 , x , x^2]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 108 &&  code_k(c) == 8
+
+        # [[144, 12, 12]]
+        l=12; m=6
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [x^3 , y , y^2]
+        B = [y^3 , x , x^2]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 144 &&  code_k(c) == 12
+
+        # [[288, 12, 12]]
+        l=12; m=12
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [x^3 , y^2, y^7]
+        B = [y^3 , x  , x^2]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 288 &&  code_k(c) == 12
+
+        # [[360, 12, ≤ 24]]
+        l=30; m=6
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [x^9 , y    ,  y^2]
+        B = [y^3 , x^25 , x^26]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 360 &&  code_k(c) == 12
+
+        # [[756, 16, ≤ 34]]
+        l=21; m=18
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [x^3 , y^10 , y^17]
+        B = [y^5 , x^3  , x^19]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 756 &&  code_k(c) == 16
+    end
+
+    @testset "Reproduce Table 1 berthusen2024toward" begin
+        # [[72, 8, 6]]
+        l=12; m=3
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [x^9   , y ,  y^2]
+        B = [one(x), x , x^11]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 72 &&  code_k(c) == 8
+
+        # [[90, 8, 6]]
+        l=9; m=5
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [x^8 , y^4 ,  y]
+        B = [y^5 , x^8 , x^7]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 90 &&  code_k(c) == 8
+
+        # [[120, 8, 8]]
+        l=12; m=5
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [x^10  , y^4,   y]
+        B = [one(x), x  , x^2]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 120 &&  code_k(c) == 8
+
+        # [[150, 8, 8]]
+        l=15; m=5
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [x^5 , y^2 , y^3]
+        B = [y^2 , x^7 , x^6]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 150 &&  code_k(c) == 8
+
+        # [[196, 12, 8]]
+        l=14; m=7
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [x^6   , y^5 ,  y^6]
+        B = [one(x), x^4 , x^13]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 196 &&  code_k(c) == 12
+    end
+
+    @testset "Reproduce Table 1 wang2024coprime" begin
+        # [[54, 8, 6]]
+        l=3; m=9
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [one(x), y^2, y^4]
+        B = [y^3   , x  , x^2]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 54 &&  code_k(c) == 8
+
+        # [[98, 6, 12]]
+        l=7; m=7
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [x^3 , y^5 , y^6]
+        B = [y^2 , x^3 , x^5]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 98 &&  code_k(c) == 6
+
+        # [[126, 8, 10]]
+        l=3; m=21
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [one(x), y^2, y^10]
+        B = [y^3   , x  ,  x^2]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 126 &&  code_k(c) == 8
+
+        # [[150, 16, 8]]
+        l=5; m=15
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [one(x), y^6, y^8]
+        B = [y^5   , x  , x^4]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 150 &&  code_k(c) == 16
+
+        # [[162, 8, 14]]
+        l=3; m=27
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [one(x), y^10, y^14]
+        B = [y^12  , x   ,  x^2]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 162 &&  code_k(c) == 8
+
+        # [[180, 8, 16]]
+        l=6; m=15
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x, y = gens(GA)
+        A = [x^3 , y   , y^2]
+        B = [y^6 , x^4 , x^5]
+        c = bivariate_bicycle_codes(A,B,GA)
+        @test code_n(c) == 180 &&  code_k(c) == 8
+    end
+end