Product and BB Codes

These constructors return ordinary quantum-code objects, so accessors such as X_stabilizers, Z_stabilizers, stabilizer_weights, and minimum_distance apply uniformly.

Hypergraph product: [6]
Generalized Shor: [7]
Hyperbicycle: [8]
Generalized bicycle: [8], [9], [10]
Generalized hypergraph product: [10]
Bias-tailored lifted product: [11]
Bivariate bicycle (BB): [12]
Coprime bivariate bicycle: [12]

InfiniteBBCode and Generalized3DToricCode retain algebraic family data before a finite lattice is selected. BBCode and FiniteGeneralized3DToricCode are finite CSS codes and support the standard quantum-code accessors, distance metadata, and solvers.

For example, the coprime univariate BB form can be constructed over a polynomial ring:

using Oscar
using CodingTheory

F = GF(2)
R, z = polynomial_ring(F, :z)
a = one(R) + z
b = one(R) + z^2
S = BBCode(a, b, 7)

(S.n, S.k)
is_CSS(S)
X_stabilizers(S)

Finite BB matrices are evaluated lazily and cached on first access. Twisted two-dimensional lattices use the Laurent-polynomial constructor with two lattice vectors. The twisted property records which finite presentation was selected; use twist_vectors(S) to inspect those vectors.

Quantum Tanner codes are included here as well: like the product constructions, they build a quantum code out of classical ingredients placed on a graph, and their distance guarantees come from expansion of that graph.

CodingTheory.BBCode — Method

BBCode(
    a::AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}},
    b::AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}},
    l::Int64,
    m::Int64
) -> BBCode{AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}, AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}, AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}}

Return an untwisted finite BB code on an l by m torus.

CodingTheory.BBCode — Method

BBCode(
    a::AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}},
    b::AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}},
    a1::Tuple{Int64, Int64},
    a2::Tuple{Int64, Int64}
) -> BBCode{AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}, AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}, AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}}

Return a finite twisted BB code on the Laurent lattice generated by a1 and a2.

CodingTheory.BBCode — Method

BBCode(
    a::AbstractAlgebra.PolyRingElem{<:AbstractAlgebra.FinFieldElem},
    b::AbstractAlgebra.PolyRingElem{<:AbstractAlgebra.FinFieldElem},
    N::Int64
) -> BBCode{_A, U, V} where {_A, U<:(AbstractAlgebra.PolyRingElem{<:AbstractAlgebra.FinFieldElem}), V<:(AbstractAlgebra.PolyRingElem{<:AbstractAlgebra.FinFieldElem})}

Return the coprime univariate form modulo z^N - 1.

CodingTheory.BBCode — Method

BBCode(
    a::AbstractAlgebra.ResElem,
    b::AbstractAlgebra.ResElem
) -> BBCode{_A, U, V} where {_A, U<:AbstractAlgebra.ResElem, V<:AbstractAlgebra.ResElem}

Return the coprime univariate form from elements of F[z]/(z^N - 1).

CodingTheory.BBCode — Method

BBCode(
    a::Union{Oscar.MPolyQuoRingElem{Nemo.FqMPolyRingElem}, Oscar.MPolyQuoRingElem{Nemo.fpMPolyRingElem}},
    b::Union{Oscar.MPolyQuoRingElem{Nemo.FqMPolyRingElem}, Oscar.MPolyQuoRingElem{Nemo.fpMPolyRingElem}}
) -> BBCode{T, U, V} where {T<:Union{Oscar.MPolyQuoRing{Nemo.FqMPolyRingElem}, Oscar.MPolyQuoRing{Nemo.fpMPolyRingElem}}, U<:Union{Oscar.MPolyQuoRingElem{Nemo.FqMPolyRingElem}, Oscar.MPolyQuoRingElem{Nemo.fpMPolyRingElem}}, V<:Union{Oscar.MPolyQuoRingElem{Nemo.FqMPolyRingElem}, Oscar.MPolyQuoRingElem{Nemo.fpMPolyRingElem}}}

Return a standard finite BB code from elements of F[x,y]/(x^l - 1, y^m - 1).

CodingTheory.InfiniteBBCode — Method

InfiniteBBCode(
    a::AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}},
    b::AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}
) -> InfiniteBBCode{AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}, AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}, AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}}

Return the algebraic bivariate-bicycle datum defined by Laurent polynomials a and b. A finite lattice must be supplied to BBCode before stabilizer matrices or finite code parameters exist.

CodingTheory.FiniteGeneralized3DToricCode — Method

FiniteGeneralized3DToricCode(
    a::AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}},
    b::AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}},
    l_x::Int64,
    l_y::Int64,
    l_z::Int64
) -> FiniteGeneralized3DToricCode{AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}, AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}, AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}}

Return a finite, untwisted three-dimensional generalized toric code.

CodingTheory.FiniteGeneralized3DToricCode — Method

FiniteGeneralized3DToricCode(
    a::AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}},
    b::AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}},
    a1::Tuple{Int64, Int64},
    a2::Tuple{Int64, Int64},
    l_z::Int64
) -> FiniteGeneralized3DToricCode{AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}, AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}, AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}}

Return a finite, twisted three-dimensional generalized toric code.

CodingTheory.Generalized3DToricCode — Method

Generalized3DToricCode(
    a::AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}},
    b::AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}
) -> Generalized3DToricCode{AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}, AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}, AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}}

Return the algebraic three-dimensional generalized toric-code datum defined by a and b. This object is not a finite stabilizer code.

CodingTheory.BBCode3D — Method

BBCode3D(
    a::AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}},
    b::AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}
) -> Generalized3DToricCode{AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}, AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}, AbstractAlgebra.Generic.LaurentMPolyWrap{Nemo.fpFieldElem, Nemo.fpMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{Nemo.fpFieldElem, Nemo.fpMPolyRing}}}

Return the corresponding member of the generalized three-dimensional toric-code family. Two polynomial arguments construct the algebraic object; lattice arguments construct a FiniteGeneralized3DToricCode.

CodingTheory.Laurent_polynomial_ring — Method

Laurent_polynomial_ring(
    S::Union{FiniteGeneralized3DToricCode, Generalized3DToricCode}
) -> Any

Return the three-variable Laurent polynomial ring of S.

CodingTheory.defining_polynomials — Method

defining_polynomials(
    S::Union{FiniteGeneralized3DToricCode, Generalized3DToricCode}
) -> Tuple{Any, Any}

Return the pair of Laurent polynomials defining the generalized three-dimensional toric code S.

CodingTheory.maximum_dimension — Method

maximum_dimension(S::Generalized3DToricCode) -> Any

Return twice the vector-space dimension of the quotient by the defining polynomials, the maximum dimension of a finite member of the family. For a finite code, return its dimension.

CodingTheory.twist_vectors — Method

twist_vectors(
    S::FiniteGeneralized3DToricCode
) -> Tuple{Tuple{Int64, Int64}, Tuple{Int64, Int64}}

Return the two integer vectors generating the finite code's periodic lattice in the $xy$ plane.

CodingTheory.HypergraphProductCode — Method

HypergraphProductCode(
    C1::AbstractLinearCode,
    C2::AbstractLinearCode;
    char_vec,
    logs_alg
) -> HypergraphProductCode

Return a lazy HypergraphProductCode. Computes parameters n, k, and bounds instantly without generating the quantum parity check matrices.

CodingTheory.Quintavalle_basis — Method

Quintavalle_basis(
    C::HypergraphProductCode
) -> Tuple{Any, Any}

Return a symplectic canonical basis for the logical operators of C.

Note

This implements https://doi.org/10.48550/arXiv.2204.10812.

CodingTheory.BaconCasaccinoConstruction — Method

BaconCasaccinoConstruction(
    C1::AbstractLinearCode,
    C2::AbstractLinearCode;
    kwargs...
) -> GeneralizedShorCode

Return the Bacon–Casaccino subsystem code obtained from the classical codes C1 and C2, requiring $C1^\perp \subseteq C2$. Its $X$ gauge generators replicate checks of C1 across columns, and its $Z$ gauge generators replicate checks of C2 across rows.

CodingTheory.GeneralizedBicycleCode — Method

GeneralizedBicycleCode(
    a::AbstractAlgebra.ResElem,
    b::AbstractAlgebra.ResElem;
    kwargs...
) -> GeneralizedBicycleCode{T} where T<:Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat}

Return the generealized bicycle code determined by a and b.

Notes

l x l circulant matrices are constructed using the coefficients of the polynomials a and b in F_q[x]/(x^l - 1) (gcd(q, l) = 1) as the first column

CodingTheory.GeneralizedBicycleCode — Method

GeneralizedBicycleCode(
    a::Hecke.GroupAlgebraElem{Nemo.fpFieldElem, Hecke.GroupAlgebra{Nemo.fpFieldElem, Hecke.FinGenAbGroup, Hecke.FinGenAbGroupElem}},
    b::Hecke.GroupAlgebraElem{Nemo.fpFieldElem, Hecke.GroupAlgebra{Nemo.fpFieldElem, Hecke.FinGenAbGroup, Hecke.FinGenAbGroupElem}};
    kwargs...
) -> GeneralizedBicycleCode{Nemo.fpMatrix}

Return the generealized bicycle code determined by a and b.

Notes

|G| x |G| circulant matrices are constructed using the coefficients of the elements in the group algebra FG as` the first column

CodingTheory.GeneralizedBicycleCode — Method

GeneralizedBicycleCode(
    A::Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat, SparseArrays.SparseMatrixCSC},
    B::Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat, SparseArrays.SparseMatrixCSC};
    char_vec,
    logs_alg
) -> GeneralizedBicycleCode{T} where T<:Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat}

Return the generealized bicycle code given by A and B.

Example

[[254, 28, 14 ≤ d ≤ 20]] Generalized Bicycle Code from Appendix B, Example A1 of [10].

julia> using CodingTheory, Oscar;

julia> F = Oscar.Nemo.Native.GF(2);

julia> S, x = polynomial_ring(F, :x);

julia> l = 127;

julia> R, _ = residue_ring(S, x^l - 1);

julia> a = 1 + x^15 + x^20 + x^28 + x^66;

julia> b = 1 + x^58 + x^59 + x^100 + x^121;

julia> code = GeneralizedBicycleCode(R(a), R(b));

julia> length(code), dimension(code)
(254, 28)

CodingTheory.BicycleCode — Method

BicycleCode(
    a::AbstractAlgebra.ResElem;
    char_vec,
    logs_alg
) -> Union{GeneralizedBicycleCode{Nemo.FqMatrix}, GeneralizedBicycleCode{Nemo.fpMatrix}}

Return the lazy Bicycle code determined by the residue ring element a.

CodingTheory.BicycleCode — Method

BicycleCode(
    a::Hecke.GroupAlgebraElem{Nemo.fpFieldElem, Hecke.GroupAlgebra{Nemo.fpFieldElem, Hecke.FinGenAbGroup, Hecke.FinGenAbGroupElem}};
    char_vec,
    logs_alg
) -> GeneralizedBicycleCode{Nemo.fpMatrix}

Return the lazy Bicycle code determined by the group algebra element a.

CodingTheory.BicycleCode — Method

BicycleCode(
    A::Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat, SparseArrays.SparseMatrixCSC};
    char_vec,
    logs_alg
) -> Union{GeneralizedBicycleCode{Nemo.FqMatrix}, GeneralizedBicycleCode{Nemo.fpMatrix}}

Return a lazy Bicycle code given by the square matrix A. This is equivalent to a Generalized Bicycle Code where B = A^T.

CodingTheory.HyperBicycleCode — Method

HyperBicycleCode(
    a::Array{T<:Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat, SparseArrays.SparseMatrixCSC}, 1},
    b::Array{T<:Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat, SparseArrays.SparseMatrixCSC}, 1},
    χ::Int64;
    char_vec,
    logs_alg
) -> HyperBicycleCode

Return the hyperbicycle non-CSS code of a and b given χ.

Arguments

a: A vector of length c of binary matrices of the same dimensions.
b: A vector of length c of binary matrices of the same dimensions, potentially different from those of a.
χ: A strictly positive integer coprime with c.

Example

[[289, 81, 5]] non-CSS Hyperbicycle Code from Example 13 of [9].

julia> using CodingTheory, Oscar;

julia> S, x = polynomial_ring(Oscar.Nemo.Native.GF(2), :x);

julia> l = 17; χ = 1;

julia> R, = residue_ring(S, x^l - 1);

julia> h = R(x^4 * (1 + x + x^3 + x^6 + x^8 + x^9));

julia> A = residue_polynomial_to_circulant_matrix(h);

julia> code = HyperBicycleCode([A], [A], χ);

julia> length(code), dimension(code)
(289, 81)

CodingTheory.HyperBicycleCodeCSS — Method

HyperBicycleCodeCSS(
    a::Array{T<:Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat, SparseArrays.SparseMatrixCSC}, 1},
    b::Array{T<:Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat, SparseArrays.SparseMatrixCSC}, 1},
    χ::Int64;
    char_vec,
    logs_alg
) -> HyperBicycleCodeCSS

Return the hyperbicycle CSS code of a and b given χ.

Arguments

a: A vector of length c of binary matrices of the same dimensions.
b: A vector of length c of binary matrices of the same dimensions, potentially different from those of a.
χ: A strictly positive integer coprime with c.

Example

[[900, 50, 14]] CSS Hyperbicycle Code from Example 6 of [9].

julia> S, x = polynomial_ring(Oscar.Nemo.Native.GF(2), :x);

julia> l = 30; χ = 1;

julia> R, = residue_ring(S, x^l - 1);

julia> h = R(1 + x + x^3 + x^5);

julia> A = residue_polynomial_to_circulant_matrix(h);

julia> a1 = A[1:15, 1:15];

julia> a2 = A[1:15, 16:30];

julia> code = HyperBicycleCodeCSS([a1, a2], [a1, a2], χ);

julia> length(code), dimension(code)
(900, 50)

CodingTheory.GeneralizedHypergraphProductCode — Method

GeneralizedHypergraphProductCode(
    A,
    b;
    kwargs...
) -> LiftedProductCode

Return the generalized hypergraph-product code obtained as the lifted-product code whose second matrix is the $1 \times 1$ matrix with entry b.

CodingTheory.SPCDFoldProductCode — Function

SPCDFoldProductCode(
    D::Int64
) -> CodingTheory.SymmetricProductCode
SPCDFoldProductCode(
    D::Int64,
    s::Int64
) -> CodingTheory.SymmetricProductCode

Return the single-parity-check D-fold product code.

Note

This is defined in https://arxiv.org/abs/2209.13474

Example

[512, 174, 8]] Symmetric 2-fold product CSS code from [13]

julia> using CodingTheory, Oscar;

julia> F = Oscar.Nemo.Native.GF(2);

julia> h = matrix(F, [1 1]);

julia> id = identity_matrix(F, 2);

julia> H_X = vcat(
             h ⊗ h ⊗ h ⊗ id ⊗ id ⊗ id ⊗ id ⊗ id ⊗ id,
             id ⊗ id ⊗ id ⊗ h ⊗ h ⊗ h ⊗ id ⊗ id ⊗ id,
             id ⊗ id ⊗ id ⊗ id ⊗ id ⊗ id ⊗ h ⊗ h ⊗ h);

julia> H_Z = vcat(
             h ⊗ id ⊗ id ⊗ h ⊗ id ⊗ id ⊗ h ⊗ id ⊗ id,
             id ⊗ h ⊗ id ⊗ id ⊗ h ⊗ id ⊗ id ⊗ h ⊗ id,
             id ⊗ id ⊗ h ⊗ id ⊗ id ⊗ h ⊗ id ⊗ id ⊗ h);

julia> code = SPCDFoldProductCode(3);

julia> length(code), dimension(code)
(512, 174)

CodingTheory.SingleParityCheckDFoldProductCode — Function

SingleParityCheckDFoldProductCode(
    D::Int64
) -> CodingTheory.SymmetricProductCode
SingleParityCheckDFoldProductCode(
    D::Int64,
    s::Int64
) -> CodingTheory.SymmetricProductCode

Return the single-parity-check $D$-fold product code with scale s.

CodingTheory.asymmetric_product — Method

asymmetric_product(
    ::IsCSS,
    S1::AbstractSubsystemCode,
    S2::AbstractSubsystemCode;
    char_vec,
    logs_alg
) -> CodingTheory.AsymmetricProductCode

Return the asymmetric 2-fold product quantum CSS code of the CSS codes S1 and S2.

Note

This is defined in https://arxiv.org/abs/2209.13474

CodingTheory.symmetric_product — Method

symmetric_product(
    ::IsCSS,
    vec_S::Array{T<:AbstractSubsystemCode, 1};
    char_vec,
    logs_alg
) -> CodingTheory.SymmetricProductCode

Return the symmetric D-fold product quantum CSS code, where D is the square-root of the length of the vector of CSS codes vec_S.

Note

This is defined in https://arxiv.org/abs/2209.13474

CodingTheory.:⊠ — Method

homological_product(
    S1::CodingTheory.AbstractStabilizerCode,
    S2::CodingTheory.AbstractStabilizerCode;
    ...
) -> Any
homological_product(
    S1::CodingTheory.AbstractStabilizerCode,
    S2::CodingTheory.AbstractStabilizerCode,
    U::Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat, SparseArrays.SparseMatrixCSC};
    ...
) -> CodingTheory.HomologicalProductCode
homological_product(
    S1::CodingTheory.AbstractStabilizerCode,
    S2::CodingTheory.AbstractStabilizerCode,
    U::Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat, SparseArrays.SparseMatrixCSC},
    V::Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat, SparseArrays.SparseMatrixCSC};
    char_vec,
    logs_alg
) -> Any

⊠(S1::AbstractStabilizerCode, S2::AbstractStabilizerCode) = homological_product(S1, S2)

Return the single-sector homological product code of S1 and S2.

Note

This is the single-sector homological product. Use ⊗ for the more general product.

CodingTheory.homological_product — Function

homological_product(
    S1::CodingTheory.AbstractStabilizerCode,
    S2::CodingTheory.AbstractStabilizerCode;
    ...
) -> Any
homological_product(
    S1::CodingTheory.AbstractStabilizerCode,
    S2::CodingTheory.AbstractStabilizerCode,
    U::Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat, SparseArrays.SparseMatrixCSC};
    ...
) -> CodingTheory.HomologicalProductCode
homological_product(
    S1::CodingTheory.AbstractStabilizerCode,
    S2::CodingTheory.AbstractStabilizerCode,
    U::Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat, SparseArrays.SparseMatrixCSC},
    V::Union{Nemo.FqMatrix, Nemo.fpMatrix, Hecke.SMat, SparseArrays.SparseMatrixCSC};
    char_vec,
    logs_alg
) -> Any

⊠(S1::AbstractStabilizerCode, S2::AbstractStabilizerCode) = homological_product(S1, S2)

Return the single-sector homological product code of S1 and S2.

Note

This is the single-sector homological product. Use ⊗ for the more general product.

CodingTheory.random_homological_product_code — Method

random_homological_product_code(
    n1::Int64,
    k1::Int64,
    n2::Int64,
    k2::Int64
) -> StabilizerCodeCSS

Return a random homological product code.

Note

This implements the construction in https://arxiv.org/abs/1311.0885.

CodingTheory.Tanner_graph — Method

Tanner_graph(S::AbstractSubsystemCode) -> Any

Return the SimpleGraph object representing the Tanner graph of the code S. Automatically generates a tripartite graph (4-tuple) for CSS codes and a bipartite graph (3-tuple) for non-CSS codes. Results are cached in S.cache[:Tanner_graph].

CodingTheory.Tanner_graph_X — Method

Tanner_graph_X(S::AbstractSubsystemCode) -> Any

Return (G, qubits, checks) for the $X$ sector of the CSS code S, where G is the bipartite SimpleGraph on the qubits and the $X$ checks and the other two entries are the vertex indices of each side.

Notes

Throws an ArgumentError for a non-CSS code.
The result is cached on the code.

CodingTheory.Tanner_graph_Z — Method

Tanner_graph_Z(S::AbstractSubsystemCode) -> Any

Return (G, qubits, checks) for the $Z$ sector of the CSS code S, where G is the bipartite SimpleGraph on the qubits and the $Z$ checks and the other two entries are the vertex indices of each side.

Notes

Throws an ArgumentError for a non-CSS code.
The result is cached on the code.