Check representation matrix commutation relation for the CSS orthogon…

…ality condition of the 2BGA code

ext/QuantumCliffordHeckeExt/QuantumCliffordHeckeExt.jl

@@ -5,14 +5,15 @@ 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, rand
 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, check_repr_commutation_relation
 
+include("util.jl")
 include("types.jl")
 include("lifted.jl")
 include("lifted_product.jl")

ext/QuantumCliffordHeckeExt/util.jl

@@ -0,0 +1,15 @@
+"""
+Checks the commutation relation between the left and right representation matrices
+for two elements `a` and `b` in the group algebra `ℱ[G]` with a general group `G`.
+It verifies the commutation relation that states, `L(a) R(b) = R(b) L(a)`. This
+property shows that matrices from the left and right representation sets commute
+with each other, which is an important property related to the CSS orthogonality
+condition.
+"""
+function check_repr_commutation_relation(GA::GroupAlgebra)
+    a, b = rand(GA), rand(GA)
+    # Check commutation relation: L(a) R(b) == R(b) L(a)
+    L_a = representation_matrix(a)
+    R_a = representation_matrix(b, :right)
+    return L_a * R_a == R_a * L_a
+end

src/ecc/ECC.jl

@@ -15,7 +15,7 @@ abstract type AbstractECC end
 
 export parity_checks, parity_checks_x, parity_checks_z, iscss,
     code_n, code_s, code_k, rate, distance,
-    isdegenerate, faults_matrix,
+    isdegenerate, faults_matrix, check_repr_commutation_relation,
     naive_syndrome_circuit, shor_syndrome_circuit, naive_encoding_circuit,
     RepCode, LiftedCode,
     CSS,

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 check_repr_commutation_relation end

test/test_ecc_base.jl

@@ -46,6 +46,7 @@ other_lifted_product_codes = []
 # [[882, 24, d≤24]] code from (B1) in Appendix B of [panteleev2021degenerate](@cite)
 l = 63
 GA = group_algebra(GF(2), abelian_group(l))
+@test check_repr_commutation_relation(GA) == true
 A = zeros(GA, 7, 7)
 x = gens(GA)[]
 A[LinearAlgebra.diagind(A)] .= x^27