Honeycomb color code via Hecke's Group Algebra

docs/src/references.bib

@@ -561,3 +561,10 @@ @article{haah2011local
   pages={042330},
   year={2011},
 }
+
+@article{eberhardt2024logical,
+  title={Logical operators and fold-transversal gates of bivariate bicycle codes},
+  author={Eberhardt, Jens Niklas and Steffan, Vincent},
+  journal={arXiv preprint arXiv:2407.03973},
+  year={2024}
+}

docs/src/references.md

@@ -45,6 +45,7 @@ For quantum code construction routines:
 - [lin2024quantum](@cite)
 - [bravyi2024high](@cite)
 - [haah2011local](@cite)
+- [eberhardt2024logical](@cite)
 
 For classical code construction routines:
 - [muller1954application](@cite)

ext/QuantumCliffordHeckeExt/QuantumCliffordHeckeExt.jl

@@ -5,14 +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, rand
+    multiplication_table, coefficients, abelian_group, group_algebra, rand, gens
 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, check_repr_commutation_relation,
-    haah_cubic_codes
+    haah_cubic_codes, honeycomb_color_codes
 
 include("util.jl")
 include("types.jl")

ext/QuantumCliffordHeckeExt/lifted_product.jl

@@ -255,7 +255,8 @@ julia> B = 𝜋^8 + 𝜋^14 + 𝜋^47;
 (108, 12)
 ```
 
-See also: [`LPCode`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref), [`haah_cubic_codes`](@ref).
+See also: [`LPCode`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref), [`haah_cubic_codes`](@ref),
+[`honeycomb_color_codes`](@ref).
 """
 function two_block_group_algebra_codes(a::GroupAlgebraElem, b::GroupAlgebraElem)
     LPCode([a;;], [b;;])
@@ -301,7 +302,8 @@ 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), [`haah_cubic_codes`](@ref).
+See also: [`two_block_group_algebra_codes`](@ref), [`generalized_bicycle_codes`](@ref), [`haah_cubic_codes`](@ref),
+[`honeycomb_color_codes`](@ref).
 """ # TODO doctest example
 function bicycle_codes(a_shifts::Array{Int}, l::Int)
     GA = group_algebra(GF(2), abelian_group(l))
@@ -323,11 +325,40 @@ julia> code_n(c), code_k(c)
 (432, 8)
 ```
 
-See also: [`bicycle_codes`](@ref), [`generalized_bicycle_codes`](@ref), [`two_block_group_algebra_codes`](@ref).
+See also: [`bicycle_codes`](@ref), [`generalized_bicycle_codes`](@ref), [`two_block_group_algebra_codes`](@ref),
+[`honeycomb_color_codes`](@ref).
 """
 function haah_cubic_codes(a_shifts::Array{Int}, b_shifts::Array{Int}, l::Int)
     GA = group_algebra(GF(2), abelian_group([l,l,l]))
     a = sum(GA[n%dim(GA)+1] for n in a_shifts)
     b = sum(GA[n%dim(GA)+1] for n in b_shifts)
     two_block_group_algebra_codes(a, b)
 end
+
+"""
+The honeycomb color codes [eberhardt2024logical](@cite) are exactly the Bivariate
+Bicycle (BB) codes defined by the polynomials `c = 1 + x + xy` and `d = 1 + y + xy`,
+provided that both `ℓ` and `m` are divisible by three.
+
+The ECC Zoo has an [entry for this family](https://errorcorrectionzoo.org/c/triangular_color).
+
+```jldoctest
+julia> ℓ = 9; m = 6;
+
+julia> c = honeycomb_color_codes(ℓ, m);
+
+julia> code_n(c), code_k(c)
+(108, 4)
+```
+
+See also: [`bicycle_codes`](@ref), [`generalized_bicycle_codes`](@ref), [`two_block_group_algebra_codes`](@ref),
+[`honeycomb_color_codes`](@ref).
+"""
+function honeycomb_color_codes(ℓ::Int, m::Int)
+    (ℓ % 3 == 0 && m % 3 == 0) || throw(ArgumentError("Both ℓ and m must be divisible by 3"))
+    GA = group_algebra(GF(2), abelian_group([ℓ, m]))
+    x, y = gens(GA)
+    c = 1 + x + x*y
+    d = 1 + y + x*y
+    two_block_group_algebra_codes(c, d)
+end

src/ecc/ECC.jl

@@ -29,7 +29,7 @@ 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,
-    haah_cubic_codes,
+    haah_cubic_codes, honeycomb_color_codes,
     random_brickwork_circuit_code, random_all_to_all_circuit_code,
     evaluate_decoder,
     CommutationCheckECCSetup, NaiveSyndromeECCSetup, ShorSyndromeECCSetup,

src/ecc/codes/lifted_product.jl

@@ -20,3 +20,6 @@ function bicycle_codes end
 
 """Implemented in a package extension with Hecke."""
 function haah_cubic_codes end
+
+"""Implemented in a package extension with Hecke."""
+function honeycomb_color_codes end

test/test_ecc_base.jl

@@ -46,6 +46,12 @@ test_hcubic_codes = [
     haah_cubic_codes([0, 15, 20, 28, 66], [0, 58, 59, 100, 121], 3)
 ]
 
+# honeycomb color codes from [eberhardt2024logical](@cite).
+test_honeycomb_color_codes = [
+    honeycomb_color_codes(6 , 6), honeycomb_color_codes(9 , 6),
+    honeycomb_color_codes(12, 6), honeycomb_color_codes(12, 9),
+]
+
 other_lifted_product_codes = []
 
 # [[882, 24, d≤24]] code from (B1) in Appendix B of [panteleev2021degenerate](@cite)
@@ -154,7 +160,7 @@ const code_instance_args = Dict(
     :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, test_bb_codes, test_mbb_codes, test_coprimeBB_codes, test_hcubic_codes, other_lifted_product_codes)),
+    :LPCode => (c -> (c.A, c.B)).(vcat(LP04, LP118, test_gb_codes, test_bb_codes, test_mbb_codes, test_coprimeBB_codes, test_hcubic_codes, test_honeycomb_color_codes, other_lifted_product_codes)),
     :QuantumReedMuller => [3, 4, 5]
 )