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

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

@@ -153,7 +153,7 @@ code_s(c::LPCode) = size(c.repr(zero(c.GA)), 1) * (size(c.A, 1) * size(c.B, 1) +
 Two-block group algebra (2GBA) codes, which are a special case of lifted product codes
 from two group algebra elements `a` and `b`, used as `1x1` base matrices.
 
-See also: [`LPCode`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref)
+See also: [`LPCode`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref), [`bivariate_bicycle_codes`](@ref)
 """ # TODO doctest example
 function two_block_group_algebra_codes(a::GroupAlgebraElem, b::GroupAlgebraElem)
     A = reshape([a], (1, 1))
@@ -166,7 +166,7 @@ Generalized bicycle codes, which are a special case of 2GBA codes (and therefore
 Here the group is chosen as the cyclic group of order `l`,
 and the base matrices `a` and `b` are the sum of the group algebra elements corresponding to the shifts `a_shifts` and `b_shifts`.
 
-See also: [`two_block_group_algebra_codes`](@ref), [`bicycle_codes`](@ref).
+See also: [`two_block_group_algebra_codes`](@ref), [`bicycle_codes`](@ref), [`bivariate_bicycle_codes`](@ref)
 
 A [[254, 28, 14 ≤ d ≤ 20]] code from (A1) in Appendix B of [panteleev2021degenerate](@cite).
 
@@ -189,10 +189,48 @@ Bicycle codes are a special case of generalized bicycle codes,
 where `a` and `b` are conjugate to each other.
 The order of the cyclic group is `l`, and the shifts `a_shifts` and `b_shifts` are reverse to each other.
 
-See also: [`two_block_group_algebra_codes`](@ref), [`generalized_bicycle_codes`](@ref).
+See also: [`two_block_group_algebra_codes`](@ref), [`generalized_bicycle_codes`](@ref), [`bivariate_bicycle_codes`](@ref)
 """ # TODO doctest example
 function bicycle_codes(a_shifts::Array{Int}, l::Int)
     GA = group_algebra(GF(2), abelian_group(l))
     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).
+
+See also: [`two_block_group_algebra_codes`](@ref), [`generalized_bicycle_codes`](@ref),
+[`bicycle_codes`](@ref), [`LPCode`](@ref)
+
+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_base.jl

@@ -5,7 +5,7 @@ using InteractiveUtils
 
 import Nemo: GF
 import LinearAlgebra
-import Hecke: group_algebra, abelian_group, gens
+import Hecke: group_algebra, abelian_group, gens, one
 
 # generate instances of all implemented codes to make sure nothing skips being checked
 
@@ -56,14 +56,40 @@ A[LinearAlgebra.diagind(A, 5)] .= GA(1)
 B = reshape([1 + x + x^6], (1, 1))
 push!(other_lifted_product_codes, LPCode(A, B))
 
+# 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 = bivariate_bicycle_codes(A,B,GA)
+
+# 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 = [one(x), x^2 , x^7]
+bb2 = bivariate_bicycle_codes(A,B,GA)
+
+# 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 = bivariate_bicycle_codes(A,B,GA)
+
+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: 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