Concatenated quantum code (QuantumSavory#289)

---------

Co-authored-by: Stefan Krastanov <[email protected]>

CHANGELOG.md

@@ -5,8 +5,10 @@
 
 # News
 
-## v0.9.4 - dev
+## v0.9.4 - 2024-06-14
 
+- Addition of a constructor for concatenated quantum codes -- `Concat`.
+- Addition of multiple unexported classical code constructors.
 - Gate errors are now conveniently supported by the various ECC benchmark setups in the `ECC` module.
 - Remove printing of spurious debug info from the PyBP decoder. 
 - Significant improvements to the low-level circuit compiler (the sumtype compactifier), leading to faster Pauli frame simulation of noisy circuits.

docs/src/references.bib

@@ -378,4 +378,11 @@ @book{error2024lin
   author={Lin, Shu and Costello, Daniel},
   year={2024},
   publisher={Pearson}
-}
+}
+
+@article{knill1996concatenated,
+  title={Concatenated quantum codes},
+  author={Knill, Emanuel and Laflamme, Raymond},
+  journal={arXiv preprint quant-ph/9608012},
+  year={1996}
+}

docs/src/references.md

@@ -36,6 +36,7 @@ For quantum code construction routines:
 - [chao2018quantum](@cite)
 - [kitaev2003fault](@cite)
 - [fowler2012surface](@cite)
+- [knill1996concatenated](@cite)
 
 For classical code construction routines:
 - [muller1954application](@cite)

src/ecc/ECC.jl

@@ -8,7 +8,7 @@ using DocStringExtensions
 using Combinatorics: combinations
 using SparseArrays: sparse
 using Statistics: std
-using Nemo: ZZ, residue_ring, matrix, finite_field, GF, minpoly, coeff, lcm, FqPolyRingElem, FqFieldElem, is_zero, degree, defining_polynomial, is_irreducible 
+using Nemo: ZZ, residue_ring, matrix, finite_field, GF, minpoly, coeff, lcm, FqPolyRingElem, FqFieldElem, is_zero, degree, defining_polynomial, is_irreducible
 
 abstract type AbstractECC end
 
@@ -19,7 +19,7 @@ export parity_checks, parity_checks_x, parity_checks_z, iscss,
     RepCode,
     CSS,
     Shor9, Steane7, Cleve8, Perfect5, Bitflip3,
-    Toric, Gottesman, Surface,
+    Toric, Gottesman, Surface, Concat,
     evaluate_decoder,
     CommutationCheckECCSetup, NaiveSyndromeECCSetup, ShorSyndromeECCSetup,
     TableDecoder,
@@ -49,6 +49,11 @@ function parity_checks_z(code::AbstractECC)
     throw(lazy"Codes of type $(typeof(code)) do not have separate X and Z parity checks, either because they are not a CSS code and thus inherently do not have separate checks, or because its separate checks are not yet implemented in this library.")
 end
 
+
+"""Check if the code is CSS.
+
+Return `nothing` if unknown from the type.
+"""
 function iscss(::Type{T}) where T<:AbstractECC
     return false
 end
@@ -353,6 +358,7 @@ include("codes/clevecode.jl")
 include("codes/toric.jl")
 include("codes/gottesman.jl")
 include("codes/surface.jl")
+include("codes/concat.jl")
 include("codes/classical/reedmuller.jl")
 include("codes/classical/bch.jl")
 end #module

src/ecc/codes/concat.jl

@@ -0,0 +1,47 @@
+"""
+`Concat(c₁, c₂)` is a code concatenation of two quantum codes [knill1996concatenated](@cite).
+
+The inner code c₁ and the outer code c₂.
+The construction is the following: replace each qubit in code c₂ with logical qubits encoded by code c₁.
+The resulting code will have `n = n₁ × n₂` qubits and `k = k₁ × k₂` logical qubits.
+"""
+struct Concat <: AbstractECC
+    c₁::AbstractECC
+    c₂::AbstractECC
+end
+
+function parity_checks(c::Concat)
+    c₁ = c.c₁
+    c₂ = c.c₂
+    k₁ = code_k(c₁)
+    n₁ = code_n(c₁)
+    n₂ = code_n(c₂)
+    s₁ = code_s(c₁)
+    s₂ = code_s(c₂)
+    inner_checks = Stabilizer(vcat([embed(n₁ * n₂, 1+(i-1)*n₁:i*n₁, parity_checks(c₁)[j]) for i in 1:n₂ for j in 1:s₁])) # parity checks of c₁ on each qubit of c₂
+    h₂ = parity_matrix(c₂)
+    phases₂ = phases(parity_checks(c₂))
+    h_logx₁ = stab_to_gf2(logx_ops(c₁))
+    phases_logx₁ = phases(logx_ops(c₁))
+    h_logz₁ = stab_to_gf2(logz_ops(c₁))
+    phases_logz₁ = phases(logz_ops(c₁))
+    # parity checks of c₂ with qubits repalced with logical qubits of c₁
+    outer_check_h = transpose(hcat([vcat(
+        kron(h₂[i, 1:end÷2], h_logx₁[j, 1:end÷2]) .⊻ kron(h₂[i, end÷2+1:end], h_logz₁[j, 1:end÷2]), # X part
+        kron(h₂[i, 1:end÷2], h_logx₁[j, end÷2+1:end]) .⊻ kron(h₂[i, end÷2+1:end], h_logz₁[j, end÷2+1:end]) # Z part
+    ) for i in 1:s₂ for j in 1:k₁]...))
+    outer_check_phase = [UInt8(sum(h₂[i, 1:end÷2] * phases_logx₁[j]) + sum(h₂[i, end÷2+1:end] * phases_logz₁[j]) + phases₂[i]) & 0x3 for i in 1:s₂ for j in 1:k₁]
+    outer_checks = Stabilizer(outer_check_phase, outer_check_h)
+    vcat(inner_checks, outer_checks)
+end
+
+code_n(c::Concat) = code_n(c.c₁) * code_n(c.c₂)
+
+code_k(c::Concat) = code_k(c.c₁) * code_k(c.c₂)
+
+function iscss(c::Concat)
+    if iscss(c.c₁) && iscss(c.c₂)
+        true
+    end
+    return nothing # if c.c₁ or c.c₂ are non-CSS; in this case, `Concat(c₁, c₂)` can still be CSS
+end

test/test_ecc_base.jl

@@ -9,7 +9,8 @@ const code_instance_args = Dict(
     Toric => [(3,3), (4,4), (3,6), (4,3), (5,5)],
     Surface => [(3,3), (4,4), (3,6), (4,3), (5,5)],
     Gottesman => [3, 4, 5],
-    CSS => (c -> (parity_checks_x(c), parity_checks_z(c))).([Shor9(), Steane7(), Toric(4,4)])
+    CSS => (c -> (parity_checks_x(c), parity_checks_z(c))).([Shor9(), Steane7(), Toric(4,4)]),
+    Concat => [(Perfect5(), Perfect5()), (Perfect5(), Steane7()), (Steane7(), Cleve8()), (Toric(2,2), Shor9())],
 )
 
 function all_testablable_code_instances(;maxn=nothing)

test/test_ecc_codeproperties.jl

@@ -22,7 +22,7 @@ end
 @testset "is CSS" begin
     for code in all_testablable_code_instances()
         H = parity_checks(code)
-        @test iscss(code) == is_css_matrix(H)
+        @test iscss(code) in (is_css_matrix(H), nothing)
     end
 end