2D Local implementation of Bivariate Bicycle qLDPC codes: Toric Layouts

src/ecc/ECC.jl

@@ -7,7 +7,7 @@ using QuantumClifford: AbstractOperation, AbstractStabilizer, Stabilizer
 import QuantumClifford: Stabilizer, MixedDestabilizer, nqubits
 using DocStringExtensions
 using Combinatorics: combinations
-using SparseArrays: sparse
+using SparseArrays: SparseMatrixCSC, sparse, spzeros
 using Statistics: std
 using Nemo: ZZ, residue_ring, matrix, finite_field, GF, minpoly, coeff, lcm, FqPolyRingElem, FqFieldElem, is_zero, degree, defining_polynomial, is_irreducible, echelon_form
 
@@ -27,7 +27,8 @@ export parity_checks, parity_checks_x, parity_checks_z, iscss,
     CommutationCheckECCSetup, NaiveSyndromeECCSetup, ShorSyndromeECCSetup,
     TableDecoder,
     BeliefPropDecoder, BitFlipDecoder,
-    PyBeliefPropDecoder, PyBeliefPropOSDecoder, PyMatchingDecoder
+    PyBeliefPropDecoder, PyBeliefPropOSDecoder, PyMatchingDecoder,
+    bivariate_toric_layout
 
 """Parity check tableau of a code.
 
@@ -361,6 +362,7 @@ end
 
 include("circuits.jl")
 include("decoder_pipeline.jl")
+include("bivariate_toric_layout.jl")
 
 include("codes/util.jl")
 

src/ecc/bivariate_toric_layout.jl

@@ -0,0 +1,78 @@
+using LinearAlgebra
+using SparseArrays: SparseMatrixCSC, sparse, spzeros
+
+function cyclic_shift_matrix(l::Int)
+    arr = spzeros(Int,l,l)
+    for i in 1:l
+        arr[i, mod(i,l)+1] = 1
+    end  
+    return arr
+end
+
+pow²(mat::SparseMatrixCSC{Int, Int},exp::Int) = (mat^exp).%2
+
+function order(mat::SparseMatrixCSC{Int}, m::Int, l::Int)
+    n = size(mat,1)
+    I = sparse(LinearAlgebra.I(n))
+    p = mat
+    for i in 1:m*l
+        if p == I return i end
+        p *= mat
+    end
+    return -1
+end
+
+function _toric_layout₁(As::Vector{SparseMatrixCSC{Int64, Int64}}, Bs::Vector{SparseMatrixCSC{Int64, Int64}}, m::Int, l::Int)
+    A_ord = [order(AA,m,l) for AA in As]
+    B_ord = [order(BB,m,l) for BB in Bs]
+    valid_orders = []
+    for (i, Ao) in enumerate(A_ord)
+        for (j, Bo) in enumerate(B_ord)
+            Ao * Bo == m*l && push!(valid_orders, (Ao,Bo,i,j))
+        end
+    end
+    return valid_orders
+end
+
+function _toric_layout₂(As::Vector{SparseMatrixCSC{Int64, Int64}}, Bs::Vector{SparseMatrixCSC{Int64, Int64}}, m::Int, l::Int, codes::NTuple{4,Int64})
+    mₑ, lₑ, A_idx, B_idx = codes
+    (A_idx < 1 || A_idx > length(As)) && throw(DimensionMismatch("Invalid index A_idx: $A_idx"))
+    (B_idx < 1 || B_idx > length(Bs)) && throw(DimensionMismatch("Invalid index B_idx: $B_idx"))
+    visited = Set{Int}()                            
+    v = sparse(As[A_idx])                        
+    h = sparse(Bs[B_idx])                      
+    zero = zeros(Int, m*l)                    
+    zero[1] = 1
+    for i in 0:mₑ-1
+        tmp = (v^i)*zero
+        for j in 0:lₑ-1
+            visited = union(visited, Set(findall(j == 0 ? tmp .>0 : h^j*tmp .> 0)))
+        end
+    end
+    return length(visited) == l*m
+end
+
+"""Determines if a bivariate bicycle code has a toric layout on a `2ℓ × 2m` grid."""
+function bivariate_toric_layout(code::Vector{Int})
+    l = code[1]
+    m = code[2]
+    x = kron(cyclic_shift_matrix(l), LinearAlgebra.I(m))
+    y = kron(LinearAlgebra.I(l), cyclic_shift_matrix(m))
+    A₁ = pow²(x, code[3])
+    A₂ = pow²(y, code[4])
+    A₃ = pow²(y, code[5])
+    A = (A₁+A₂+A₃).%2
+    B₁ = pow²(y, code[6])
+    B₂ = pow²(x, code[7])
+    B₃ = pow²(x, code[8])
+    B = (B₁+B₂+B₃).%2
+    As = [A₁*A₂', A₂'*A₁, A₂*A₃', A₃'*A₂, A₁*A₃', A₃'*A₁]
+    Bs = [B₁*B₂', B₂'*B₁, B₂*B₃', B₃'*B₂, B₁*B₃', B₃'*B₁]
+    selected_codes = []
+    valid_codes = _toric_layout₁(As, Bs, m, l)
+    for valid_code in valid_codes
+        _toric_layout₂(As, Bs, m, l, valid_code) && push!(selected_codes, valid_code)
+    end
+    orig_layout = [(params[1], params[2], params[3]-1, params[4]-1) for params in selected_codes]
+    return orig_layout
+end

test/test_ecc_bivariate_toric_layout.jl

@@ -0,0 +1,20 @@
+@testitem "ECC Bivaraite Toric Layout" begin
+    using QuantumClifford.ECC
+    using QuantumClifford.ECC: AbstractECC, bivariate_toric_layout
+
+    codes = [[6 ,6 ,3 ,1 ,2 ,3 ,1 ,2 ],
+             [15,3 ,9 ,1 ,2 ,0 ,2 ,7 ],
+             [6 ,9 ,3 ,1 ,2 ,3 ,1 ,2 ],
+             [12,6 ,3 ,1 ,2 ,3 ,1 ,2 ],
+             [12,12,3 ,2 ,7 ,3 ,1 ,2 ],
+             [30,6 ,9 ,1 ,2 ,3 ,25,26],
+             [21,18,3 ,10,17,5 ,3 ,19]]
+    # cross checks taken from https://arxiv.org/pdf/2404.17676.
+    @test bivariate_toric_layout(codes[1]) == [(6, 6, 0, 2), (6, 6, 0, 3), (6, 6, 0, 4), (6, 6, 0, 5), (6, 6, 1, 2), (6, 6, 1, 3), (6, 6, 1, 4), (6, 6, 1, 5), (6, 6, 2, 0), (6, 6, 2, 1), (6, 6, 2, 2), (6, 6, 2, 3), (6, 6, 3, 0), (6, 6, 3, 1), (6, 6, 3, 2), (6, 6, 3, 3), (6, 6, 4, 0), (6, 6, 4, 1), (6, 6, 4, 4), (6, 6, 4, 5), (6, 6, 5, 0), (6, 6, 5, 1), (6, 6, 5, 4), (6, 6, 5, 5)]
+    @test  bivariate_toric_layout(codes[2]) == [(15, 3, 0, 2), (15, 3, 0, 3), (15, 3, 1, 2), (15, 3, 1, 3), (3, 15, 2, 0), (3, 15, 2, 1), (3, 15, 2, 4), (3, 15, 2, 5), (3, 15, 3, 0), (3, 15, 3, 1), (3, 15, 3, 4), (3, 15, 3, 5), (15, 3, 4, 2), (15, 3, 4, 3), (15, 3, 5, 2), (15, 3, 5, 3)]
+    @test bivariate_toric_layout(codes[3]) == [(18, 3, 0, 4), (18, 3, 0, 5), (18, 3, 1, 4), (18, 3, 1, 5), (9, 6, 2, 0), (9, 6, 2, 1), (9, 6, 2, 2), (9, 6, 2, 3), (9, 6, 3, 0), (9, 6, 3, 1), (9, 6, 3, 2), (9, 6, 3, 3), (18, 3, 4, 4), (18, 3, 4, 5), (18, 3, 5, 4), (18, 3, 5, 5)]
+    @test bivariate_toric_layout(codes[4]) == [(12, 6, 0, 4), (12, 6, 0, 5), (12, 6, 1, 4), (12, 6, 1, 5), (6, 12, 2, 0), (6, 12, 2, 1), (6, 12, 2, 2), (6, 12, 2, 3), (6, 12, 3, 0), (6, 12, 3, 1), (6, 12, 3, 2), (6, 12, 3, 3), (12, 6, 4, 4), (12, 6, 4, 5), (12, 6, 5, 4), (12, 6, 5, 5)]
+    @test bivariate_toric_layout(codes[5]) == [(12, 12, 0, 0), (12, 12, 0, 1), (12, 12, 0, 4), (12, 12, 0, 5), (12, 12, 1, 0), (12, 12, 1, 1), (12, 12, 1, 4), (12, 12, 1, 5), (12, 12, 2, 0), (12, 12, 2, 1), (12, 12, 2, 2), (12, 12, 2, 3), (12, 12, 3, 0), (12, 12, 3, 1), (12, 12, 3, 2), (12, 12, 3, 3), (12, 12, 4, 2), (12, 12, 4, 3), (12, 12, 4, 4), (12, 12, 4, 5), (12, 12, 5, 2), (12, 12, 5, 3), (12, 12, 5, 4), (12, 12, 5, 5)]
+    @test bivariate_toric_layout(codes[6]) == [(6, 30, 2, 2), (6, 30, 2, 3), (6, 30, 3, 2), (6, 30, 3, 3), (30, 6, 4, 0), (30, 6, 4, 1), (30, 6, 5, 0), (30, 6, 5, 1)]
+    @test bivariate_toric_layout(codes[7]) == [(18, 21, 2, 2), (18, 21, 2, 3), (18, 21, 3, 2), (18, 21, 3, 3)]
+end