Haah's cubic code using abelian group ℤ₃ˣ³ via Hecke's Group Algebra (#…

…388)



Co-authored-by: Stefan Krastanov <[email protected]>

docs/src/references.bib

@@ -551,3 +551,13 @@ @article{bravyi2024high
   year={2024},
   publisher={Nature Publishing Group UK London}
 }
+
+@article{haah2011local,
+  title={Local stabilizer codes in three dimensions without string logical operators},
+  author={Haah, Jeongwan},
+  journal={Physical Review A?Atomic, Molecular, and Optical Physics},
+  volume={83},
+  number={4},
+  pages={042330},
+  year={2011},
+}

docs/src/references.md

@@ -44,6 +44,7 @@ For quantum code construction routines:
 - [voss2024multivariatebicyclecodes](@cite)
 - [lin2024quantum](@cite)
 - [bravyi2024high](@cite)
+- [haah2011local](@cite)
 
 For classical code construction routines:
 - [muller1954application](@cite)

ext/QuantumCliffordHeckeExt/QuantumCliffordHeckeExt.jl

@@ -11,7 +11,8 @@ 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
+    two_block_group_algebra_codes, generalized_bicycle_codes, bicycle_codes, check_repr_commutation_relation,
+    haah_cubic_codes
 
 include("util.jl")
 include("types.jl")

ext/QuantumCliffordHeckeExt/lifted_product.jl

@@ -77,7 +77,8 @@ The default representation, provided by `Hecke`, is the permutation representati
 
 We also accept a custom representation function as detailed in [`LiftedCode`](@ref).
 
-See also: [`LiftedCode`](@ref), [`two_block_group_algebra_codes`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref).
+See also: [`LiftedCode`](@ref), [`two_block_group_algebra_codes`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref),
+[`haah_cubic_codes`](@ref).
 
 $TYPEDFIELDS
 """
@@ -251,12 +252,10 @@ julia> 𝜋 = gens(GA)[1];
 julia> A = 𝜋^2 + 𝜋^5  + 𝜋^44;
 
 julia> B = 𝜋^8 + 𝜋^14 + 𝜋^47;
-
-
 (108, 12)
 ```
 
-See also: [`LPCode`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref).
+See also: [`LPCode`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref), [`haah_cubic_codes`](@ref).
 """
 function two_block_group_algebra_codes(a::GroupAlgebraElem, b::GroupAlgebraElem)
     LPCode([a;;], [b;;])
@@ -302,10 +301,33 @@ 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), [`haah_cubic_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
+
+"""
+Haah’s cubic codes [haah2011local](@cite) can be viewed as generalized bicycle (GB) codes
+with the group `G = Cₗ × Cₗ × Cₗ`, where `l` denotes the lattice size. In particular, a GB
+code with the group `G = ℤ₃ˣ³` corresponds to a cubic code.
+
+The ECC Zoo has an [entry for this family](https://errorcorrectionzoo.org/c/haah_cubic).
+
+```jldoctest
+julia> c = haah_cubic_codes([0, 15, 20, 28, 66], [0, 58, 59, 100, 121], 6);
+
+julia> code_n(c), code_k(c)
+(432, 8)
+```
+
+See also: [`bicycle_codes`](@ref), [`generalized_bicycle_codes`](@ref), [`two_block_group_algebra_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

src/ecc/ECC.jl

@@ -29,6 +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,
     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 haah_cubic_codes end

test/test_ecc_base.jl

@@ -42,6 +42,10 @@ test_gb_codes = [
     generalized_bicycle_codes([0, 1, 14, 16, 22], [0, 3, 13, 20, 42], 63), # (A2) [[126, 28, 8]]
 ]
 
+test_hcubic_codes = [
+    haah_cubic_codes([0, 15, 20, 28, 66], [0, 58, 59, 100, 121], 3)
+]
+
 other_lifted_product_codes = []
 
 # [[882, 24, d≤24]] code from (B1) in Appendix B of [panteleev2021degenerate](@cite)
@@ -150,7 +154,7 @@ 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, 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, other_lifted_product_codes)),
     :QuantumReedMuller => [3, 4, 5]
 )