Circulant GF(2) permutation matrix-based construction of Bivariate Bicycle quantum LDPC code #352
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| """ | ||
| A bivariate bicycle (BB) quantum LDPC code was introduced by Bravyi et al. in their 2024 paper [bravyi2024high](@cite). This code uses identity and cyclic shift matrices. Define `Iₗ` as the `l × l` identity matrix and `Sₗ` as the cyclic shift matrix of the same size, where each row of `Sₗ` has a single '1' at the column `(i + 1) mod l`. | ||
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| The matrices `x = Sₗ ⊗ Iₘ` and `y = Iₗ ⊗ Sₘ` are used. The BB code is represented by matrices `A` and `B`, defined as: `A = A₁ + A₂ + A₃` and `B = B₁ + B₂ + B₃`. The addition and multiplication operations on binary matrices are performed modulo 2. The check matrices are: `Hx = [A|B]` and `Hz = [B'|A']`. Both `Hx` and `Hz` are `(n/2)×n` matrices. | ||
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| The ECC Zoo has an [entry for this family](https://errorcorrectionzoo.org/c/qcga). | ||
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| """ | ||
| struct circulant_bivariate_bicycle <: AbstractECC | ||
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There was a problem hiding this comment. this should be named like a struct, not like a function (e.g. |
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| l::Int | ||
| m::Int | ||
| A::Vector{Int} | ||
| B::Vector{Int} | ||
| function circulant_bivariate_bicycle(l,m,A,B) | ||
| (l >= 0 && m >= 0) || error("l and m must be non-negative") | ||
| (length(A) == 3 && length(B) == 3) || error("A and B must each have exactly 3 entries") | ||
| (all(x -> x >= 0, A) && all(x -> x >= 0, B)) || error("A and B must contain only non-negative integers") | ||
| (all(x -> x in 0:max(l, m), A) && all(x -> x in 0:max(l, m), B)) || error("Each element in A and B must be in the range [0, $(max(l, m))].") | ||
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| new(l,m,A,B) | ||
| end | ||
| end | ||
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| function iscss(::Type{circulant_bivariate_bicycle}) | ||
| return true | ||
| end | ||
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| function parity_checks(c::circulant_bivariate_bicycle) | ||
| a₁,a₂,a₃ = c.A[1],c.A[2],c.A[3] | ||
| b₁,b₂,b₃ = c.B[1],c.B[2],c.B[3] | ||
| Iₗ = Matrix{Bool}(LinearAlgebra.I,c.l,c.l) | ||
| Iₘ = Matrix{Bool}(LinearAlgebra.I,c.m,c.m) | ||
| x = Dict{Bool, Matrix{Bool}}() | ||
| y = Dict{Bool, Matrix{Bool}}() | ||
| x = Dict(i => kron(circshift(Iₗ,(0,i)),Iₘ) for i in 0:(c.l)) | ||
| y = Dict(i => kron(Iₗ,circshift(Iₘ,(0,i))) for i in 0:(c.m)) | ||
| A = mod.(x[a₁]+y[a₂]+y[a₃],2) | ||
| B = mod.(y[b₁]+x[b₂]+x[b₃],2) | ||
| Hx = hcat(A,B) | ||
| Hz = hcat(B',A') | ||
| H = CSS(Hx,Hz) | ||
| Stabilizer(H) | ||
| end | ||
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| code_n(c::circulant_bivariate_bicycle) = 2*c.l*c.m | ||
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| code_k(c::circulant_bivariate_bicycle) = code_n(c) - LinearAlgebra.rank(matrix(GF(2), parity_checks_x(c))) - LinearAlgebra.rank(matrix(GF(2), parity_checks_z(c))) | ||
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| parity_checks_x(c::circulant_bivariate_bicycle) = stab_to_gf2(parity_checks(circulant_bivariate_bicycle(c.l, c.m, c.A, c.B)))[1:end÷2,:] | ||
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| parity_checks_z(c::circulant_bivariate_bicycle) = stab_to_gf2(parity_checks(circulant_bivariate_bicycle(c.l, c.m, c.A, c.B)))[end÷2+1:end,:] | ||
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| @testitem "ECC circulant_bivariate_bicycle" begin | ||
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There was a problem hiding this comment. these tests probable also need a subset of tests that verify correctness against the other construction method you have already contributed |
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| using Test | ||
| using QuantumClifford: stab_to_gf2 | ||
| using QuantumClifford.ECC | ||
| using Nemo: nullspace, GF, matrix | ||
| using Oscar: hom, free_module, kernel, domain, map, gens | ||
| using QuantumClifford.ECC: AbstractECC, circulant_bivariate_bicycle, parity_checks_x, parity_checks_z | ||
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| # According to Lemma 1 from [bravyi2024high](@cite), k = 2·dim(ker(A)∩ker(B)). | ||
| function _formula_k(stab) | ||
| Hx = parity_checks_x(stab) | ||
| n = size(Hx,2)÷2 | ||
| A = matrix(GF(2), Hx[:,1:n]) | ||
| B = matrix(GF(2), Hx[:,n+1:end]) | ||
| k = GF(2) | ||
| hA = hom(free_module(k, size(A, 1)), free_module(k, size(A, 2)), A) | ||
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There was a problem hiding this comment. hom needs to be added to the spellchecker (the |
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| hB = hom(free_module(k, size(B, 1)), free_module(k, size(B, 2)), B) | ||
| ans = kernel(hA)[1] ∩ kernel(hB)[1] | ||
| k = 2*size(map(domain(hA), gens(ans[1])), 1) | ||
| return k | ||
| end | ||
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| @testset "Verify number of logical qubits `k` from Table 3: bravyi2024high" begin | ||
| # Refer to [bravyi2024high](@cite) for code constructions | ||
| @test code_k(circulant_bivariate_bicycle(9 , 6 , [3 , 1 , 2] , [3 , 1 , 2])) == 8 == _formula_k(circulant_bivariate_bicycle(9 , 6 , [3 , 1 , 2] , [3 , 1 , 2])) | ||
| @test code_k(circulant_bivariate_bicycle(15, 3 , [9 , 1 , 2] , [0 , 2 , 7])) == 8 == _formula_k(circulant_bivariate_bicycle(15, 3 , [9 , 1 , 2] , [0 , 2 , 7])) | ||
| @test code_k(circulant_bivariate_bicycle(12, 12, [3 , 2 , 7] , [3 , 1 , 2])) == 12 == _formula_k(circulant_bivariate_bicycle(12, 12, [3 , 2 , 7] , [3 , 1 , 2])) | ||
| @test code_k(circulant_bivariate_bicycle(12, 6 , [3 , 1 , 2] , [3 , 1 , 2])) == 12 == _formula_k(circulant_bivariate_bicycle(12, 6 , [3 , 1 , 2] , [3 , 1 , 2])) | ||
| @test code_k(circulant_bivariate_bicycle(6 , 6 , [3 , 1 , 2] , [3 , 1 , 2])) == 12 == _formula_k(circulant_bivariate_bicycle(6 , 6 , [3 , 1 , 2] , [3 , 1 , 2])) | ||
| @test code_k(circulant_bivariate_bicycle(30, 6 , [9 , 1 , 2] , [3 , 25, 26])) == 12 == _formula_k(circulant_bivariate_bicycle(30, 6 , [9 , 1 , 2] , [3 , 25, 26])) | ||
| @test code_k(circulant_bivariate_bicycle(21, 18, [3 , 10, 17], [5 , 3 , 19])) == 16 == _formula_k(circulant_bivariate_bicycle(21, 18, [3 , 10, 17], [5 , 3 , 19])) | ||
| @test code_k(circulant_bivariate_bicycle(28, 14, [26, 6 , 8] , [7 , 9 , 20])) == 24 == _formula_k(circulant_bivariate_bicycle(28, 14, [26, 6 , 8] , [7 , 9 , 20])) | ||
| end | ||
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| @testset "Verify number of logical qubits `k` from Table 1: berthusen2024toward" begin | ||
| # Refer to [berthusen2024toward](@cite) for code constructions | ||
| @test code_k(circulant_bivariate_bicycle(12, 3 , [9 , 1 , 2] , [0 , 1 , 11])) == 8 == _formula_k(circulant_bivariate_bicycle(12, 3 , [9 , 1 , 2] , [0 , 1 ,11])) | ||
| @test code_k(circulant_bivariate_bicycle(9 , 5 , [8 , 4 , 1] , [5 , 8 , 7])) == 8 == _formula_k(circulant_bivariate_bicycle(9 , 5 , [8 , 4 , 1], [5 , 8 , 7])) | ||
| @test code_k(circulant_bivariate_bicycle(12, 5 , [10, 4 , 1] , [0 , 1 , 2])) == 8 == _formula_k(circulant_bivariate_bicycle(12, 5 , [10, 4 , 1] , [0 , 1 , 2])) | ||
| @test code_k(circulant_bivariate_bicycle(15, 5 , [5 , 2 , 3] , [2 , 7 , 6])) == 8 == _formula_k(circulant_bivariate_bicycle(15, 5 , [5 , 2 , 3] , [2 , 7 , 6])) | ||
| @test code_k(circulant_bivariate_bicycle(14, 7 , [6 , 5 , 6] , [0 , 4, 13])) == 12 == _formula_k(circulant_bivariate_bicycle(14, 7 , [6 , 5 , 6] , [0 , 4, 13])) | ||
| end | ||
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| @testset "Verify number of logical qubits `k` from Table 1: wang2024coprime" begin | ||
| # # Refer to [wang2024coprime](@cite) for code constructions | ||
| @test code_k(circulant_bivariate_bicycle(3, 9 , [0 , 2 , 4] , [3 , 1 , 2])) == 8 == _formula_k(circulant_bivariate_bicycle(3, 9 , [0 , 2 , 4] , [3 , 1 , 2])) | ||
| @test code_k(circulant_bivariate_bicycle(7, 7 , [3 , 5 , 6] , [2 , 3 , 5])) == 6 == _formula_k(circulant_bivariate_bicycle(7, 7 , [3 , 5 , 6] , [2 , 3 , 5])) | ||
| @test code_k(circulant_bivariate_bicycle(3, 21 , [0 , 2 ,10] , [3 , 1 , 2])) == 8 == _formula_k(circulant_bivariate_bicycle(3, 21 , [0 , 2 ,10] , [3 , 1 , 2])) | ||
| @test code_k(circulant_bivariate_bicycle(5, 15 , [0 , 6 , 8] , [5 , 1 , 4])) == 16 == _formula_k(circulant_bivariate_bicycle(5, 15 , [0 , 6 , 8] , [5 , 1 , 4])) | ||
| @test code_k(circulant_bivariate_bicycle(3, 27 , [0 , 10,14] , [12, 1 , 2])) == 8 == _formula_k(circulant_bivariate_bicycle(3, 27 , [0 , 10,14] , [12, 1 , 2])) | ||
| @test code_k(circulant_bivariate_bicycle(6, 15 , [3 , 1 , 2] , [6 , 4 , 5])) == 8 == _formula_k(circulant_bivariate_bicycle(6, 15 , [3 , 1 , 2] , [6 , 4 , 5])) | ||
| end | ||
| end | ||
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why is this needed?
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This is not needed now thanks to Tommy CI Oscar Injection!
This PR was done earlier, why I was trying to add Oscar to test runner, so that's why the weird additions