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add check_repr_commutation_relation to test the CSS orthogonality condition for 2BGA's Group Algebra with a General Group G #403

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merged 5 commits into from
Nov 5, 2024
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add check_repr_commutation_relation to test the CSS orthogonality condition for 2BGA's Group Algebra with a General Group G #403

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429f21c
Check representation matrix commutation relation for the CSS orthogon…
Fe-r-oz Oct 23, 2024
fb86505
code review suggestions
Fe-r-oz Oct 25, 2024
a0a1ec6
fix typo
Fe-r-oz Oct 29, 2024
2ff2d3e
move consistency check to util.l
Fe-r-oz Oct 29, 2024
dda1502
minor documentation notes
Krastanov Nov 5, 2024
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5 changes: 3 additions & 2 deletions ext/QuantumCliffordHeckeExt/QuantumCliffordHeckeExt.jl
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Expand Up @@ -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")
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15 changes: 15 additions & 0 deletions ext/QuantumCliffordHeckeExt/util.jl
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@@ -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)
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return L_a * R_a == R_a * L_a
end
2 changes: 1 addition & 1 deletion src/ecc/ECC.jl
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Expand Up @@ -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,
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let's not export it for now, just keep it as an internal check

naive_syndrome_circuit, shor_syndrome_circuit, naive_encoding_circuit,
RepCode, LiftedCode,
CSS,
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3 changes: 3 additions & 0 deletions src/ecc/codes/lifted_product.jl
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Expand Up @@ -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
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1 change: 1 addition & 0 deletions test/test_ecc_base.jl
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Expand Up @@ -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
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Is this check expected to be used anywhere else if merged?

Also, you do not need to do == true -- the result is already a boolean

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This check will be used by all the tests PRs. Since we are extending the functionality to a finite general group via group presentation, usually non-abelian, this checks whether this commutation relation holds for such groups.

A = zeros(GA, 7, 7)
x = gens(GA)[]
A[LinearAlgebra.diagind(A)] .= x^27
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