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Generalized Toric Codes
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‎src/CodingTheory.jl‎

Lines changed: 12 additions & 1 deletion
Original file line numberDiff line numberDiff line change
@@ -54,6 +54,8 @@ const CTPolyRingElem = PolyRingElem{<:CTFieldElem}
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const CTGroupAlgebra = GroupAlgebraElem{fpFieldElem, GroupAlgebra{fpFieldElem, FinGenAbGroup, FinGenAbGroupElem}}
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const CTChainComplex = Union{ComplexOfMorphisms{AbstractAlgebra.FPModule{fpFieldElem}}} # residue and group algebras later
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const CTPolyMatrix = Union{AbstractAlgebra.Generic.MatSpaceElem{fpPolyRingElem}, AbstractAlgebra.Generic.MatSpaceElem{FqPolyRingElem}}
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const CTLRPolyElem = AbstractAlgebra.Generic.LaurentMPolyWrap{fpFieldElem, fpMPolyRingElem,
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AbstractAlgebra.Generic.LaurentMPolyWrapRing{fpFieldElem, fpMPolyRing}}
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include("Classical/types.jl")
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export AbstractCode, AbstractNonadditiveCode, AbstractNonlinearCode, AbstractAdditiveCode,
@@ -72,7 +74,8 @@ include("Quantum/types.jl")
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export AbstractSubsystemCode, AbstractSubsystemCodeCSS, AbstractStabilizerCode, AbstractStabilizerCodeCSS,
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AbstractGraphStateSubsystem, AbstractGraphStateSubsystemCSS, AbstractGraphStateStabilizer,
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AbstractGraphStateStabilizerCSS, AbstractHypergraphProductCode, AbstractEASubsystemCode,
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AbstractEASubsystemCodeCSS, AbstractEAStabilizerCode, AbstractEAStabilizerCodeCSS
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AbstractEASubsystemCodeCSS, AbstractEAStabilizerCode, AbstractEAStabilizerCodeCSS, AbstractGeneralizedToricCode
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# misc
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export LogicalTrait, GaugeTrait, CSSTrait, HasLogicals, HasNoLogicals, HasGauges, HasNoGauges,
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IsCSS, IsNotCSS, copy, ChainComplex
@@ -473,4 +476,12 @@ export copying, gauging, thickening_and_choose_heights, coning, quantum_weight_r
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include("Quantum/homological_measurements.jl")
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export homological_measurement, Cheeger_constant
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#############################
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# Quantum/GeneralizedToricCode.jl
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#############################
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include("Quantum/GeneralizedToricCode.jl")
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export GeneralizedToricCode, FiniteGeneralizedToricCode, maximum_dimension,
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Laurent_polynomial_ring, defining_polynomials, twist_vectors
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end

‎src/Quantum/GeneralizedToricCode.jl‎

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,130 @@
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# Copyright (c) 2025 Eric Sabo
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# All rights reserved.
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#
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# This source code is licensed under the BSD-style license found in the
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# LICENSE file in the root directory of this source tree.
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#############################
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# constructors
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#############################
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function GeneralizedToricCode(f::CTLRPolyElem, g::CTLRPolyElem)
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LR = parent(f)
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LR == parent(g) || throw(ArgumentError("The polynomials must be over the same ring."))
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length(symbols(LR)) == 2 || throw(ArgumentError("The polynomials must be over a Laurent polynomial ring in two variables."))
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return GeneralizedToricCode(LR, base_ring(LR), f, g)
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end
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function FiniteGeneralizedToricCode(f::CTLRPolyElem, g::CTLRPolyElem, a1::Tuple{Int, Int},
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a2::Tuple{Int, Int})
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LR = parent(f)
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LR == parent(g) || throw(ArgumentError("The polynomials must be over the same ring."))
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length(symbols(LR)) == 2 || throw(ArgumentError("The polynomials must be over a Laurent polynomial ring in two variables."))
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return FiniteGeneralizedToricCode(LR, base_ring(LR), f, g, a1, a2)
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end
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#############################
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# getter functions
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#############################
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Laurent_polynomial_ring(S::AbstractGeneralizedToricCode) = S.LR
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field(S::AbstractGeneralizedToricCode) = S.F
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defining_polynomials(S::AbstractGeneralizedToricCode) = S.f, S.g
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twist_vectors(S::FiniteGeneralizedToricCode) = S.a1, S.a2
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#############################
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# setter functions
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#############################
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#############################
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# general functions
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#############################
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function maximum_dimension(S::AbstractGeneralizedToricCode)
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R2 = Oscar._polyringquo(S.LR)
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R = codomain(R2)
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(x, y) = gens(S.LR)
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if isa(S, FiniteGeneralizedToricCode)
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I = ideal(S.LR, [S.f, S.g, x^S.a1[1] * y^S.a1[2] - 1, x^S.a2[1] * y^S.a2[2] - 1])
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else
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I = ideal(S.LR, [S.f, S.g])
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end
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II = ideal(R, R2.(gens(I)))
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Q, ϕ = quo(R, II)
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return 2 * vector_space_dimension(Q)
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end
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function CSSCode(S::FiniteGeneralizedToricCode)
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R2 = Oscar._polyringquo(S.LR)
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R = codomain(R2)
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(x, y) = gens(S.LR)
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swap = hom(S.LR, S.LR, [x^-1, y^-1])
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f_anti = swap(S.f)
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g_anti = swap(S.g)
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f_R2 = R2(S.f)
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g_R2 = R2(S.g)
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f_anti_R2 = R2(f_anti)
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g_anti_R2 = R2(g_anti)
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I = ideal(S.LR, [x^S.a1[1] * y^S.a1[2] - 1, x^S.a2[1] * y^S.a2[2] - 1])
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II = ideal(R, R2.(gens(I)))
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Q, ϕ = quo(R, II)
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mono = monomial_basis(Q)
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len_mon = length(mono)
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n = 2 * len_mon
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LR_edge_index = Dict{fpMPolyRingElem, Int}(mono[i] => i for i in 1:len_mon)
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TB_edge_index = Dict{fpMPolyRingElem, Int}(mono[i] => i + len_mon for i in 1:len_mon)
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Fone = S.F(1)
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row = 1
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X_stabs = zero_matrix(S.F, len_mon, n)
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Z_stabs = zero_matrix(S.F, len_mon, n)
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for edge in mono
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f_shift = simplify(ϕ(edge * f_R2))
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for term in terms(f_shift.f)
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# X_12
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X_stabs[row, TB_edge_index[term]] = Fone
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end
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g_shift = simplify(ϕ(edge * g_R2))
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for term in terms(g_shift.f)
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# X_14
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X_stabs[row, LR_edge_index[term]] = Fone
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end
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g_shift = simplify(ϕ(edge * g_anti_R2))
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for term in terms(g_shift.f)
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# Z_12
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Z_stabs[row, TB_edge_index[term]] = Fone
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end
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f_shift = simplify(ϕ(edge * f_anti_R2))
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for term in terms(f_shift.f)
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# Z_14
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Z_stabs[row, LR_edge_index[term]] = Fone
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end
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row += 1
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end
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# println("X stabs")
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# display(X_stabs)
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# println("Z stabs")
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# display(Z_stabs)
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return CSSCode(X_stabs, Z_stabs)
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end
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function show(io::IO, S::AbstractGeneralizedToricCode)
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if isa(S, FiniteGeneralizedToricCode)
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println(io, "Finite Generalized Toric Code:")
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println(io, "\tf: $(S.f)")
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println(io, "\tg: $(S.g)")
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println(io, "\ta1: $(S.a1)")
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println(io, "\ta2: $(S.a2)")
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else
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println(io, "Generalized Toric Code:")
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println(io, "\tf: $(S.f)")
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println(io, "\tg: $(S.g)")
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end
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end

‎src/Quantum/types.jl‎

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Original file line numberDiff line numberDiff line change
@@ -21,6 +21,7 @@ abstract type AbstractEASubsystemCode <: AbstractSubsystemCode end
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abstract type AbstractEASubsystemCodeCSS <: AbstractEASubsystemCode end
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abstract type AbstractEAStabilizerCode <: AbstractStabilizerCode end
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abstract type AbstractEAStabilizerCodeCSS <: AbstractEAStabilizerCode end
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abstract type AbstractGeneralizedToricCode <: AbstractStabilizerCodeCSS end
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# AbstractQuantumLDPCCode, AbstractQuantumLDPCCSSCode?
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@@ -336,6 +337,26 @@ mutable struct HypergraphProductCode <: AbstractHypergraphProductCode
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Z_metacheck::Union{CTMatrixTypes, Missing}
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end
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#############################
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# Quantum/GeneralizedToricCode.jl
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#############################
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struct GeneralizedToricCode <: AbstractGeneralizedToricCode
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LR::AbstractAlgebra.Generic.LaurentMPolyWrapRing{fpFieldElem, fpMPolyRing}
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F::CTFieldTypes
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f::CTLRPolyElem
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g::CTLRPolyElem
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end
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struct FiniteGeneralizedToricCode <: AbstractGeneralizedToricCode
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LR::AbstractAlgebra.Generic.LaurentMPolyWrapRing{fpFieldElem, fpMPolyRing}
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F::CTFieldTypes
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f::CTLRPolyElem
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g::CTLRPolyElem
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a1::Tuple{Int, Int}
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a2::Tuple{Int, Int}
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end
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#############################
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# traits
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#############################

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