bivariate bicycle code using polynomial quotient ring

Author: Fe-r-oz

ext/QuantumCliffordOscarExt/QuantumCliffordOscarExt.jl

@@ -17,7 +17,8 @@ import Oscar: free_group, small_group_identification, describe, order, FPGroupEl
     base_ring, ComplexOfMorphisms, coefficients, zero_matrix, hcat, circshift, size, zeros, enumerate,
     kronecker_product, FqMatrix, identity_matrix, iszero, FqPolyRingElem, laurent_polynomial_ring,
     hnf_with_transform, ideal, intersect, ==, is_coprime, quo, groebner_basis, length, FqMPolyRingElem,
-    first, length, MPolyQuoRingElem, FqMPolyRingElem, modulus, ideal, monomials, terms, coeff, degree, mod
+    first, MPolyQuoRingElem, FqMPolyRingElem, modulus, ideal, monomials, terms, coeff, degree, mod,
+    monomial, exponent_vector, nvars
 import Oscar.Generic.MatSpaceElem
 import Oscar.Generic.DirectSumModule
 import Oscar.Generic.LaurentMPolyWrap
@@ -37,6 +38,7 @@ export twobga_from_direct_product, twobga_from_fp_group, DDimensionalSurfaceCode
 
 include("types.jl")
 include("direct_product.jl")
+include("bivariate_bicycle.jl")
 include("generalized_toric.jl")
 include("group_presentation.jl")
 include("d_dimensional_codes.jl")

ext/QuantumCliffordOscarExt/bivariate_bicycle.jl

@@ -0,0 +1,79 @@
+"""
+Bivariate Bicycle code using *polynomial quotient ring formalism*: The set of
+monomials ``\\{x^i y^j \\mid 0 \\leq i < \\ell, 0 \\leq j < m}`` forms a basis
+for the matrix algebra generated by ``x`` and ``y``, establishing a [bijection](https://en.wikipedia.org/wiki/Bijection)
+between these monomials and the resulting ``(\\ell m) \\times (\\ell m)`` matrices.
+This correspondence allows us to work equivalently with either the polynomial or
+matrix formulation [wang2024coprime](@cite).
+"""
+struct BivariateBicycleCode <: AbstractCSSCode
+    """Order of the first abelian group in ``\\mathbb{F}_2[\\mathbb{Z}_\\ell \\times \\mathbb{Z}_m]``"""
+    ℓ::Int
+    """Order of the second abelian group in ``\\mathbb{F}_2[\\mathbb{Z}_\\ell \\times \\mathbb{Z}_m]``"""
+    m::Int
+    """First bivariate polynomial in quotient ring ``\\frac{\\mathbb{F}_2[x, y]}{\\langle x^\\ell-1, y^m-1 \\rangle}``"""
+    c::MPolyQuoRingElem
+    """Second bivariate polynomial in quotient ring ``\\frac{\\mathbb{F}_2[x, y]}{\\langle x^\\ell-1, y^m-1 \\rangle}``"""
+    d::MPolyQuoRingElem
+    
+    function BivariateBicycleCode(ℓ::Int, m::Int, c::MPolyQuoRingElem, d::MPolyQuoRingElem)
+        ℓ > 0 || throw(ArgumentError("ℓ must be positive"))
+        m > 0 || throw(ArgumentError("m must be positive"))
+        parent(c) == parent(d) || throw(ArgumentError("Polynomials must be in the same quotient ring"))
+        Q = parent(c)
+        R = base_ring(Q)
+        base_ring(R) == GF(2) || throw(ArgumentError("Base ring must be GF(2)"))
+        nvars(R) == 2 || throw(ArgumentError("Must be bivariate polynomials"))
+        new(ℓ, m, c, d)
+    end
+end
+
+function poly_to_coeff_mat(poly::MPolyQuoRingElem, ℓ::Int, m::Int)
+    mat = zeros(Int, ℓ, m)
+    poly_lift = lift(poly)
+    for term in terms(poly_lift)
+        coeffᵥₐₗ = coeff(term, 1)
+        coeffᵢₙₜ = iszero(coeffᵥₐₗ) ? 0 : 1
+        monom = monomial(term, 1)
+        exps = exponent_vector(monom, 1)
+        i = length(exps) >= 1 ? exps[1] : 0
+        j = length(exps) >= 2 ? exps[2] : 0
+        i = mod(i, ℓ)
+        j = mod(j, m)
+        if i < ℓ && j < m
+            mat[i+1, j+1] = coeffᵢₙₜ
+        end
+    end
+    return mat
+end
+
+function build_bivariate_circulant(coeff_mat::Matrix{Int})
+    ℓ, m = size(coeff_mat)
+    n = ℓ*m
+    circ_mat = zeros(Int, n, n)
+    for i₁ in 0:ℓ-1, j₁ in 0:m-1
+        row_idx = i₁*m+j₁+1
+        for i₂ in 0:ℓ-1, j₂ in 0:m-1
+            i_res = mod(i₁ + i₂, ℓ)
+            j_res = mod(j₁ + j₂, m)
+            col_idx = i_res*m + j_res + 1
+            circ_mat[row_idx, col_idx] += coeff_mat[i₂+1, j₂+1]
+        end
+    end
+    return circ_mat
+end
+
+function parity_matrix_xz(c::BivariateBicycleCode)
+    A_coeff = poly_to_coeff_mat(c.c, c.ℓ, c.m)
+    B_coeff = poly_to_coeff_mat(c.d, c.ℓ, c.m)
+    A = build_bivariate_circulant(A_coeff)
+    B = build_bivariate_circulant(B_coeff)
+    hx, hz = hcat(A, B), hcat(transpose(B), transpose(A))
+    return hx, hz
+end
+
+parity_matrix_x(c::BivariateBicycleCode) = parity_matrix_xz(c)[1]
+
+parity_matrix_z(c::BivariateBicycleCode) = parity_matrix_xz(c)[2]
+
+code_n(c::BivariateBicycleCode) = 2*c.ℓ*c.m

src/ecc/ECC.jl

@@ -41,7 +41,7 @@ export parity_checks, parity_matrix_x, parity_matrix_z, iscss,
     GeneralizedCirculantBivariateBicycle, GeneralizedHyperGraphProductCode,
     GeneralizedBicycleCode, ExtendedGeneralizedBicycleCode,
     HomologicalProductCode, DoubleHomologicalProductCode,
-    GeneralizedToricCode, TrivariateTricycleCode,
+    GeneralizedToricCode, TrivariateTricycleCode, BivariateBicycleCode,
     evaluate_decoder,
     CommutationCheckECCSetup, NaiveSyndromeECCSetup, ShorSyndromeECCSetup,
     TableDecoder,

src/ecc/codes/qeccs_using_oscar.jl

@@ -74,3 +74,13 @@ function TrivariateTricycleCode(args...; kwargs...)
     end
     return ext.TrivariateTricycleCode(args...; kwargs...)
 end
+
+"""Bivariate Bicycle codes ([jacob2025singleshotdecodingfaulttolerantgates](@cite))
+Implemented as a package extension with `Oscar`. Check the [QuantumClifford documentation](http://qc.quantumsavory.org/stable/ECC_API/) for more details on that extension."""
+function BivariateBicycleCode(args...; kwargs...)
+    ext = Base.get_extension(QuantumClifford, :QuantumCliffordOscarExt)
+    if isnothing(ext)
+        throw("The `BivariateBicycleCode` depends on the package `Oscar` but you have not installed or imported it yet. Immediately after you import `Oscar`, the `BivariateBicycleCode` will be available.")
+    end
+    return ext.BivariateBicycleCode(args...; kwargs...)
+end