Coprime Bivariate Bicycle Codes via Univariate Polynomial Ring

Project.toml

@@ -13,6 +13,7 @@ Graphs = "86223c79-3864-5bf0-83f7-82e725a168b6"
 JLD2 = "033835bb-8acc-5ee8-8aae-3f567f8a3819"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Oscar = "f1435218-dba5-11e9-1e4d-f1a5fab5fc13"
+Hecke = "3e1990a7-5d81-5526-99ce-9ba3ff248f21"
 ProgressMeter = "92933f4c-e287-5a05-a399-4b506db050ca"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

docs/src/Quantum/product_codes.md

@@ -6,6 +6,8 @@
 - [Generealized bicycle](https://errorcorrectionzoo.org/c/generalized_bicycle): [pryadko2013quantum](@cite), [Kovalev_2013](@cite), [panteleev2021degenerate](@cite)
 - Generalized hypergraph product: [panteleev2021degenerate](@cite)
 - Bias-tailored lifted product: [roffe2023bias](@cite)
+- Bivariate bicycle: [wang2024coprime](@cite)
+- Coprime bivarate bicycle: [wang2024coprime](@cite)
 
 
 ```@autodocs

docs/src/references.bib

@@ -183,3 +183,21 @@ @inproceedings{yao2024belief
   year={2024},
   organization={IEEE}
 }
+
+@article{bravyi2024high,
+  title={High-threshold and low-overhead fault-tolerant quantum memory},
+  author={Bravyi, Sergey and Cross, Andrew W and Gambetta, Jay M and Maslov, Dmitri and Rall, Patrick and Yoder, Theodore J},
+  journal={Nature},
+  volume={627},
+  number={8005},
+  pages={778--782},
+  year={2024},
+  publisher={Nature Publishing Group UK London}
+}
+
+@article{wang2024coprime,
+  title={Coprime Bivariate Bicycle Codes and their Properties},
+  author={Wang, Ming and Mueller, Frank},
+  journal={arXiv preprint arXiv:2408.10001},
+  year={2024}
+}

src/CodingTheory.jl

@@ -376,7 +376,7 @@ export HypergraphProductCode, GeneralizedShorCode, BaconCasaccinoConstruction,
     GeneralizedHypergraphProductCode, LiftedProductCode, bias_tailored_lifted_product_matrices,
     BiasTailoredLiftedProductCode, SPCDFoldProductCode, SingleParityCheckDFoldProductCode,
     Quintavalle_basis, asymmetric_product, symmetric_product, random_homological_product_code,
-    homological_product, ⊠, BivariateBicycleCode
+    homological_product, ⊠, BivariateBicycleCode, CoprimeBivariateBicycleCode
 
 #############################
 #   Quantum/simulation.jl

src/Quantum/product_codes.jl

@@ -917,7 +917,7 @@ function BivariateBicycleCode(a::T, b::T) where T <: Union{MPolyQuoRingElem{FqMP
             l = exps[1][1]
         end
     end
-    
+
     x = matrix(F, [mod1(i + 1, l) == j ? 1 : 0 for i in 1:l, j in 1:l]) ⊗ identity_matrix(F, m)
     y = identity_matrix(F, l) ⊗ matrix(F, [mod1(i + 1, m) == j ? 1 : 0 for i in 1:m, j in 1:m])
 
@@ -943,5 +943,56 @@ function BivariateBicycleCode(a::T, b::T) where T <: Union{MPolyQuoRingElem{FqMP
         end
     end
 
+
+    return CSSCode(hcat(A, B), hcat(transpose(B), transpose(A)))
+end
+
+"""
+    CoPrimeBivariateBicycleCode(a::MPolyQuoRingElem{FqMPolyRingElem}, b::MPolyQuoRingElem{FqMPolyRingElem})
+
+Return the coprime bivariate bicycle code defined by the residue ring elements `a` and `b`.
+
+# Note
+- This is defined in https://arxiv.org/pdf/2408.10001v1.
+"""
+function CoprimeBivariateBicycleCode(a::T, b::T) where T <: Union{MPolyQuoRingElem{FqMPolyRingElem}, MPolyQuoRingElem{fpMPolyRingElem}}
+    R = parent(a)
+    R == parent(b) || throw(DomainError("Polynomials must have the same parent."))
+    F = base_ring(base_ring(a))
+    order(F) == 2 || throw(DomainError("This code family is currently only defined over binary fields."))
+    length(symbols(parent(a))) == 1 || throw(DomainError("Polynomials must be over one variable."))
+    g = gens(modulus(R))
+
+    m = -1
+    l = -1
+
+    exps = collect(exponents(g[1]))[1][1]
+    length(exps) == 1 || throw(ArgumentError("Moduli of the incorrect form."))
+    facs = factor(ZZ(exps)).fac
+    pfacs = collect(keys(facs))
+    pexps = collect(values(facs))
+
+    l = Int(pfacs[1]^pexps[1])
+    m = Int(pfacs[2]^pexps[2])
+
+    length(pfacs) == 2 ? (gcd([l, m]) == 1 ?
+        nothing : throw(ArgumentError("l and m must be coprime numbers."))) : throw(ArgumentError("Moduli of the incorrect form."))
+
+    x = matrix(F, [mod1(i + 1, l) == j ? 1 : 0 for i in 1:l, j in 1:l]) ⊗ identity_matrix(F, m)
+    y = identity_matrix(F, l) ⊗ matrix(F, [mod1(i + 1, m) == j ? 1 : 0 for i in 1:m, j in 1:m])
+
+    P = x*y
+    A = zero_matrix(F, exps, exps)
+    for ex in exponents(lift(a))
+        power, _ = findmax(ex)
+        A += P^power
+    end
+
+    B = zero_matrix(F, exps, exps)
+    for ex in exponents(lift(b))
+        power, _ = findmax(ex)
+        B += P^power
+    end
+
     return CSSCode(hcat(A, B), hcat(transpose(B), transpose(A)))
 end

test/Quantum/product_codes_test.jl

@@ -606,5 +606,62 @@
         @test length(Q) == 196
         @test dimension(Q) == 12
         @test_broken minimum_distance(S) == 8
-    end
+
+        # Coprime Bivariate Bicycle codes
+        # Table 2 of https://arxiv.org/pdf/2408.10001v1
+        # [[30, 4, 6]]
+        S, (P) = polynomial_ring(Oscar.Nemo.Native.GF(2), [:P])
+        l = 3
+        m = 5
+        R, _ = quo(S, ideal(S, [P[1]^(l*m)]))
+        a = R(1 + P[1] + P[1]^2)
+        b = R(P[1] + P[1]^3 + P[1]^8)
+        Q = CoprimeBivariateBicycleCode(a, b)
+        @test length(Q) == 30
+        @test dimension(Q) == 4
+        @test_broken minimum_distance(Q) == 6
+
+        # [[42, 6, 6]]
+        l = 3
+        m = 7
+        R, _ = quo(S, ideal(S, [P[1]^(l*m)]))
+        a = R(1 + P[1]^2 + P[1]^3)
+        b = R(P[1] + P[1]^3 + P[1]^11)
+        Q = CoprimeBivariateBicycleCode(a, b)
+        @test length(Q) == 42
+        @test dimension(Q) == 6
+        @test_broken minimum_distance(Q) == 6
+
+        # [[70, 6, 8]]
+        l = 5
+        m = 7
+        R, _ = quo(S, ideal(S, [P[1]^(l*m)]))
+        a = R(1 + P[1] + P[1]^5)
+        b = R(1 + P[1] + P[1]^12)
+        Q = CoprimeBivariateBicycleCode(a, b)
+        @test length(Q) == 70
+        @test dimension(Q) == 6
+        @test_broken minimum_distance(Q) == 8
+
+        # [[108, 12, 6]]
+        l = 2
+        m = 27
+        R, _ = quo(S, ideal(S, [P[1]^(l*m)]))
+        a = R(P[1]^2 + P[1]^5 + P[1]^44)
+        b = R(P[1]^8 + P[1]^14 + P[1]^47)
+        Q = CoprimeBivariateBicycleCode(a, b)
+        @test length(Q) == 108
+        @test dimension(Q) == 12
+        @test_broken minimum_distance(Q) == 6
+
+        # [126, 12, 10]]
+        l = 7
+        m = 9
+        R, _ = quo(S, ideal(S, [P[1]^(l*m)]))
+        a = R(1 + P[1] + P[1]^58)
+        b = R(P[1]^3 + P[1]^16 + P[1]^44)
+        Q = CoprimeBivariateBicycleCode(a, b)
+        @test length(Q) == 126
+        @test dimension(Q) == 12
+        @test_broken minimum_distance(Q) == 10
 end