Implementing Bivariate Bicycle Codes using 2BGA as the parent via Hecke's Group Algebra

docs/src/references.bib

@@ -523,3 +523,14 @@ @article{lin2024quantum
   year={2024},
   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 +40,8 @@ For quantum code construction routines:
 - [steane1999quantum](@cite)
 - [campbell2012magic](@cite)
 - [anderson2014fault](@cite)
+- [lin2024quantum](@cite)
+- [bravyi2024high](@cite)
 
 For classical code construction routines:
 - [muller1954application](@cite)

ext/QuantumCliffordHeckeExt/lifted_product.jl

@@ -150,14 +150,18 @@ code_n(c::LPCode) = size(c.repr(zero(c.GA)), 2) * (size(c.A, 2) * size(c.B, 1) +
 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))
 
 """
-Two-block group algebra (2GBA) codes, which are a special case of lifted product codes
+Two-block group algebra (2BGA) codes, which are a special case of lifted product codes
 from two group algebra elements `a` and `b`, used as `1x1` base matrices.
 
+## Examples of 2BGA code subfamilies
+
+### `C₄ x C₂`
+
 Here is an example of a [[56, 28, 2]] 2BGA code from Table 2 of [lin2024quantum](@cite)
 with direct product of `C₄ x C₂`.
 
 ```jldoctest
-julia> import Hecke: group_algebra, GF, abelian_group, gens;
+julia> import Hecke: group_algebra, GF, abelian_group, gens
 
 julia> GA = group_algebra(GF(2), abelian_group([14,2]));
 
@@ -175,7 +179,36 @@ julia> code_n(c), code_k(c)
 (56, 28)
 ```
 
-See also: [`LPCode`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref)
+### Bivariate Bicycle codes
+
+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/qcga).
+
+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, y = gens(GA);
+
+julia> A = x^3 + y^10 + y^17;
+
+julia> B = y^5 + x^3  + x^19;
+
+julia> c = two_block_group_algebra_codes(A,B);
+
+julia> code_n(c), code_k(c)
+(756, 16)
+```
+
+See also: [`LPCode`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref).
 """
 function two_block_group_algebra_codes(a::GroupAlgebraElem, b::GroupAlgebraElem)
     LPCode([a;;], [b;;])

test/test_ecc_base.jl

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

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: two_block_group_algebra_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 = two_block_group_algebra_codes(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, y = gens(GA)
+        A = x^9 + y   + y^2
+        B = 1   + x^2 + x^7
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = x^3 + y + y^2
+        B = y^3 + x + x^2
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = x^3 + y + y^2
+        B = y^3 + x + x^2
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = x^3 + y^2 + y^7
+        B = y^3 + x   + x^2
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = x^9 + y    + y^2
+        B = y^3 + x^25 + x^26
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = x^3 + y^10 + y^17
+        B = y^5 + x^3  + x^19
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = x^9 + y + y^2
+        B = 1   + x + x^11
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = x^8 + y^4 + y
+        B = y^5 + x^8 + x^7
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = x^10 + y^4 + y
+        B = 1    + x   + x^2
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = x^5 + y^2 + y^3
+        B = y^2 + x^7 + x^6
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = x^6 + y^5 + y^6
+        B = 1   + x^4 + x^13
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = 1   + y^2 + y^4
+        B = y^3 + x   + x^2
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = x^3 + y^5 + y^6
+        B = y^2 + x^3 + x^5
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = 1   + y^2 + y^10
+        B = y^3 + x  +  x^2
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = 1   + y^6 + y^8
+        B = y^5 + x   + x^4
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = 1    + y^10 + y^14
+        B = y^12 + x    + x^2
+        c = two_block_group_algebra_codes(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, y = gens(GA)
+        A = x^3 + y   + y^2
+        B = y^6 + x^4 + x^5
+        c = two_block_group_algebra_codes(A,B)
+        @test code_n(c) == 180 && code_k(c) == 8
+    end
+end