tests: test consistency of Bivariate Bicycle code using LPCode constructor

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}
+}

docs/src/references.md

@@ -40,6 +40,7 @@ For quantum code construction routines:
 - [steane1999quantum](@cite)
 - [campbell2012magic](@cite)
 - [anderson2014fault](@cite)
+- [bravyi2024high](@cite)
 
 For classical code construction routines:
 - [muller1954application](@cite)

ext/QuantumCliffordHeckeExt/lifted_product.jl

@@ -70,6 +70,36 @@ julia> code_n(c2), code_k(c2)
 - When the base matrices of the `LPCode` are 1×1 and their elements are sums of cyclic permutations, the code is called a generalized bicycle code [`generalized_bicycle_codes`](@ref).
 - When the two matrices are adjoint to each other, the code is called a bicycle code [`bicycle_codes`](@ref).
 
+# Examples
+
+Bivariate Bicycle codes belong to a wider class of generalized bicycle (GB) codes, which
+are further generalized into of two block group algebra (2BGA) codes. They can be viewed
+as a special case of Lifted Product construction based on abelian group `ℤₗ x ℤₘ` where `ℤⱼ`
+cyclic group of order `j`.
+
+A [[756, 16, ≤ 34]] code from Table 3 of [bravyi2024high](@cite).
+
+```jldoctest
+julia> import Hecke: group_algebra, GF, abelian_group, gens;
+
+julia> l=21; m=18;
+
+julia> GA = group_algebra(GF(2), abelian_group([l, m]));
+
+julia> x = gens(GA)[1];
+
+julia> y = gens(GA)[2];
+
+julia> A = reshape([x^3 + y^10 + y^17], (1, 1));
+
+julia> B = reshape([y^5 + x^3 + x^19], (1, 1));
+
+julia> c1 = LPCode(A, B);
+
+julia> code_n(c1), code_k(c1)
+(756, 16)
+```
+
 ## The representation function
 
 We use the default representation function `Hecke.representation_matrix` to convert a `GF(2)`-group algebra element to a binary matrix.

test/test_ecc_bivaraite_bicycle.jl

@@ -0,0 +1,191 @@
+@testitem "ECC Bivaraite Bicycle" begin
+    using Hecke
+    using Hecke: group_algebra, GF, abelian_group, gens
+    using QuantumClifford.ECC: LPCode, 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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([x^3 + y + y^2], (1, 1))
+        B = reshape([y^3 + x + x^2], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([x^9 + y + y^2], (1, 1))
+        B = reshape([1 + x^2 + x^7], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([x^3 + y + y^2], (1, 1))
+        B = reshape([y^3 + x + x^2], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([x^3 + y + y^2], (1, 1))
+        B = reshape([y^3 + x + x^2], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([x^3 + y^2 + y^7], (1, 1))
+        B = reshape([y^3 + x + x^2], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([x^9 + y + y^2], (1, 1))
+        B = reshape([y^3 + x^25 + x^26], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([x^3 + y^10 + y^17], (1, 1))
+        B = reshape([y^5 + x^3 + x^19], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([x^9 + y + y^2], (1, 1))
+        B = reshape([1 + x + x^11], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([x^8 + y^4 + y], (1, 1))
+        B = reshape([y^5 + x^8 + x^7], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([x^10 + y^4 + y], (1, 1))
+        B = reshape([1 + x + x^2], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([x^5 + y^2 + y^3], (1, 1))
+        B = reshape([y^2 + x^7 + x^6], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([x^6 + y^5 + y^6], (1, 1))
+        B = reshape([1 + x^4 + x^13], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([1 + y^2 + y^4], (1, 1))
+        B = reshape([y^3 + x + x^2], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([x^3 + y^5 + y^6], (1, 1))
+        B = reshape([y^2 + x^3 + x^5], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([1 + y^2 + y^10], (1, 1))
+        B = reshape([y^3 + x + x^2], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([1 + y^6 + y^8], (1, 1))
+        B = reshape([y^5 + x + x^4], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([1 + y^10 + y^14], (1, 1))
+        B = reshape([y^12 + x + x^2], (1, 1))
+        c = LPCode(A, B)
+        @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 = gens(GA)[1]
+        y = gens(GA)[2]
+        A = reshape([x^3 + y + y^2], (1, 1))
+        B = reshape([y^6+ x^4 + x^5], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 180 &&  code_k(c) == 8
+    end
+end