Lifted and lifted product codes via Hecke's GroupAlgebra (#356)

---------

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

Author: 3 people

CHANGELOG.md

@@ -11,6 +11,10 @@
 - **(fix)** Bug fix to the `parity_checks(ReedMuller(r, m))` of classical Reed-Muller code (it was returning generator matrix).
 - `RecursiveReedMuller` code implementation as an alternative implementation of `ReedMuller`.
 
+## v0.9.10 - 2024-09-26
+
+- **(fix)** `ECC.code_s` now gives the number of parity checks with redundancy. If you want the number of linearly independent parity checks, you can use `LinearAlgebra.rank`.
+
 ## v0.9.9 - 2024-08-05
 
 - `inv` is implemented for all Clifford operator types (symbolic, dense, sparse).

Project.toml

@@ -24,6 +24,7 @@ SumTypes = "8e1ec7a9-0e02-4297-b0fe-6433085c89f2"
 
 [weakdeps]
 CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba"
+Hecke = "3e1990a7-5d81-5526-99ce-9ba3ff248f21"
 LDPCDecoders = "3c486d74-64b9-4c60-8b1a-13a564e77efb"
 Makie = "ee78f7c6-11fb-53f2-987a-cfe4a2b5a57a"
 Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
@@ -33,6 +34,7 @@ QuantumOpticsBase = "4f57444f-1401-5e15-980d-4471b28d5678"
 
 [extensions]
 QuantumCliffordGPUExt = "CUDA"
+QuantumCliffordHeckeExt = "Hecke"
 QuantumCliffordLDPCDecodersExt = "LDPCDecoders"
 QuantumCliffordMakieExt = "Makie"
 QuantumCliffordPlotsExt = "Plots"
@@ -46,14 +48,15 @@ Combinatorics = "1.0"
 DataStructures = "0.18"
 DocStringExtensions = "0.9"
 Graphs = "1.9"
+Hecke = "0.28, 0.29, 0.30, 0.31, 0.32, 0.33"
 HostCPUFeatures = "0.1.6"
 ILog2 = "0.2.3"
 InteractiveUtils = "1.9"
 LDPCDecoders = "0.3.1"
 LinearAlgebra = "1.9"
 MacroTools = "0.5.9"
 Makie = "0.20, 0.21"
-Nemo = "0.42, 0.43, 0.44, 0.45, 0.46"
+Nemo = "^0.42.1, 0.43, 0.44, 0.45, 0.46"
 Plots = "1.38.0"
 PrecompileTools = "1.2"
 PyQDecoders = "0.2.1"

docs/Project.toml

@@ -5,6 +5,7 @@ Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
 DocumenterCitations = "daee34ce-89f3-4625-b898-19384cb65244"
 GraphMakie = "1ecd5474-83a3-4783-bb4f-06765db800d2"
 Graphs = "86223c79-3864-5bf0-83f7-82e725a168b6"
+Hecke = "3e1990a7-5d81-5526-99ce-9ba3ff248f21"
 LDPCDecoders = "3c486d74-64b9-4c60-8b1a-13a564e77efb"
 Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
 Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"

docs/make.jl

@@ -7,6 +7,11 @@ using QuantumClifford
 using QuantumClifford.Experimental.NoisyCircuits
 using QuantumInterface
 
+ENV["HECKE_PRINT_BANNER"] = "false"
+import Hecke
+
+const QuantumCliffordHeckeExt = Base.get_extension(QuantumClifford, :QuantumCliffordHeckeExt)
+
 #DocMeta.setdocmeta!(QuantumClifford, :DocTestSetup, :(using QuantumClifford); recursive=true)
 
 ENV["LINES"] = 80    # for forcing `displaysize(io)` to be big enough
@@ -20,8 +25,9 @@ doctest = false,
 clean = true,
 sitename = "QuantumClifford.jl",
 format = Documenter.HTML(size_threshold_ignore = ["API.md"]),
-modules = [QuantumClifford, QuantumClifford.Experimental.NoisyCircuits, QuantumClifford.ECC, QuantumInterface],
+modules = [QuantumClifford, QuantumClifford.Experimental.NoisyCircuits, QuantumClifford.ECC, QuantumInterface, QuantumCliffordHeckeExt],
 warnonly = [:missing_docs],
+linkcheck = true,
 authors = "Stefan Krastanov",
 pages = [
 "QuantumClifford.jl" => "index.md",

docs/src/ECC_API.md

@@ -4,3 +4,10 @@
 Modules = [QuantumClifford.ECC]
 Private = false
 ```
+
+## Implemented in an extension requiring `Hecke.jl`
+
+```@autodocs
+Modules = [QuantumCliffordHeckeExt]
+Private = true
+```

docs/src/references.bib

@@ -402,3 +402,55 @@ @inproceedings{brown2013short
   pages = {346--350},
   doi = {10.1109/ISIT.2013.6620245}
 }
+
+@article{panteleev2021degenerate,
+  title = {Degenerate {{Quantum LDPC Codes With Good Finite Length Performance}}},
+  author = {Panteleev, Pavel and Kalachev, Gleb},
+  year = {2021},
+  month = nov,
+  journal = {Quantum},
+  volume = {5},
+  eprint = {1904.02703},
+  primaryclass = {quant-ph},
+  pages = {585},
+  issn = {2521-327X},
+  doi = {10.22331/q-2021-11-22-585},
+  archiveprefix = {arXiv}
+}
+
+
+@inproceedings{panteleev2022asymptotically,
+  title = {Asymptotically Good {{Quantum}} and Locally Testable Classical {{LDPC}} Codes},
+  booktitle = {Proceedings of the 54th {{Annual ACM SIGACT Symposium}} on {{Theory}} of {{Computing}}},
+  author = {Panteleev, Pavel and Kalachev, Gleb},
+  year = {2022},
+  month = jun,
+  pages = {375--388},
+  publisher = {ACM},
+  address = {Rome Italy},
+  doi = {10.1145/3519935.3520017},
+  isbn = {978-1-4503-9264-8}
+}
+
+@article{roffe2023bias,
+  title = {Bias-Tailored Quantum {{LDPC}} Codes},
+  author = {Roffe, Joschka and Cohen, Lawrence Z. and Quintavalle, Armanda O. and Chandra, Daryus and Campbell, Earl T.},
+  year = {2023},
+  month = may,
+  journal = {Quantum},
+  volume = {7},
+  pages = {1005},
+  doi = {10.22331/q-2023-05-15-1005}
+}
+
+@article{raveendran2022finite,
+  title = {Finite {{Rate QLDPC-GKP Coding Scheme}} That {{Surpasses}} the {{CSS Hamming Bound}}},
+  author = {Raveendran, Nithin and Rengaswamy, Narayanan and Rozp{\k e}dek, Filip and Raina, Ankur and Jiang, Liang and Vasi{\'c}, Bane},
+  year = {2022},
+  month = jul,
+  journal = {Quantum},
+  volume = {6},
+  pages = {767},
+  issn = {2521-327X},
+  doi = {10.22331/q-2022-07-20-767},
+}

ext/QuantumCliffordHeckeExt/QuantumCliffordHeckeExt.jl

@@ -0,0 +1,19 @@
+module QuantumCliffordHeckeExt
+
+using DocStringExtensions
+
+import QuantumClifford, LinearAlgebra
+import Hecke: Group, GroupElem, AdditiveGroup, AdditiveGroupElem,
+    GroupAlgebra, GroupAlgebraElem, FqFieldElem, representation_matrix, dim, base_ring,
+    multiplication_table, coefficients, abelian_group, group_algebra
+import Nemo: characteristic, matrix_repr, GF, ZZ
+
+import QuantumClifford.ECC: AbstractECC, CSS, ClassicalCode,
+    hgp, code_k, code_n, code_s, iscss, parity_checks, parity_checks_x, parity_checks_z, parity_checks_xz,
+    two_block_group_algebra_codes, generalized_bicycle_codes, bicycle_codes
+
+include("types.jl")
+include("lifted.jl")
+include("lifted_product.jl")
+
+end # module

ext/QuantumCliffordHeckeExt/lifted.jl

@@ -0,0 +1,101 @@
+"""
+$TYPEDEF
+
+Classical codes lifted over a group algebra, used for lifted product code construction ([panteleev2021degenerate](@cite), [panteleev2022asymptotically](@cite))
+
+The parity-check matrix is constructed by applying `repr` to each element of `A`,
+which is mathematically a linear map from a group algebra element to a binary matrix.
+The size of the parity check matrix will enlarged with each element of `A` being inflated into a matrix.
+The procedure is called a lift [panteleev2022asymptotically](@cite).
+
+## Constructors
+
+A lifted code can be constructed via the following approaches:
+
+1. A matrix of group algebra elements.
+
+2. A matrix of group elements, where a group element will be considered as a group algebra element by assigning a unit coefficient.
+
+3. A matrix of integers, where each integer represent the shift of a cyclic permutation. The order of the cyclic permutation should be specified.
+
+The default `GA` is the group algebra of `A[1, 1]`, the default representation `repr` is the permutation representation.
+
+## The representation function `repr`
+
+In this struct, we use the default representation function `default_repr` to convert a `GF(2)`-group algebra element to a binary matrix.
+The default representation, provided by `Hecke`, is the permutation representation.
+
+We also accept a custom representation function.
+Such a customization would be useful to reduce the number of bits required by the code construction.
+
+For example, if we use a D4 group for lifting, our default representation will be `8×8` permutation matrices,
+where 8 is the group's order.
+However, we can find a `4×4` matrix representation for the group,
+e.g. by using the typical [`2×2` representation](https://en.wikipedia.org/wiki/Dihedral_group)
+and converting it into binary representation by replacing "1" with the Pauli I, and "-1" with the Pauli X matrix.
+
+See also: [`LPCode`](@ref).
+
+$TYPEDFIELDS
+"""
+struct LiftedCode <: ClassicalCode
+    """the base matrix of the code, whose elements are in a group algebra."""
+    A::GroupAlgebraElemMatrix
+    """the group algebra for which elements in `A` are from."""
+    GA::GroupAlgebra
+    """
+    a function that converts a group algebra element to a binary matrix;
+    default to be the permutation representation for GF(2)-algebra."""
+    repr::Function
+
+    function LiftedCode(A::GroupAlgebraElemMatrix; GA::GroupAlgebra=parent(A[1, 1]), repr::Function)
+        all(elem.parent == GA for elem in A) || error("The base ring of all elements in the code must be the same as the group algebra")
+        new(A, GA, repr)
+    end
+end
+
+default_repr(y::GroupAlgebraElem{FqFieldElem, <: GroupAlgebra}) = Matrix((x -> Bool(Int(lift(ZZ, x)))).(representation_matrix(y)))
+
+"""
+`LiftedCode` constructor using the default `GF(2)` representation (coefficients converted to a permutation matrix by `representation_matrix` provided by Hecke).
+""" # TODO doctest example
+function LiftedCode(A::Matrix{GroupAlgebraElem{FqFieldElem, <: GroupAlgebra}}; GA::GroupAlgebra=parent(A[1,1]))
+    !(characteristic(base_ring(A[1, 1])) == 2) && error("The default permutation representation applies only to GF(2) group algebra; otherwise, a custom representation function should be provided")
+    LiftedCode(A; GA=GA, repr=default_repr)
+end
+
+# TODO document and doctest example
+function LiftedCode(group_elem_array::Matrix{<: GroupOrAdditiveGroupElem}; GA::GroupAlgebra=group_algebra(GF(2), parent(group_elem_array[1,1])), repr::Union{Function, Nothing}=nothing)
+    A = zeros(GA, size(group_elem_array)...)
+    for i in axes(group_elem_array, 1), j in axes(group_elem_array, 2)
+        A[i, j] = GA[A[i, j]]
+    end
+    if repr === nothing
+        return LiftedCode(A; GA=GA, repr=default_repr)
+    else
+        return LiftedCode(A; GA=GA, repr=repr)
+    end
+end
+
+# TODO document and doctest example
+function LiftedCode(shift_array::Matrix{Int}, l::Int; GA::GroupAlgebra=group_algebra(GF(2), abelian_group(l)))
+    A = zeros(GA, size(shift_array)...)
+    for i in 1:size(shift_array, 1)
+        for j in 1:size(shift_array, 2)
+            A[i, j] = GA[shift_array[i, j]%l+1]
+        end
+    end
+    return LiftedCode(A; GA=GA, repr=default_repr)
+end
+
+function lift(repr::Function, mat::GroupAlgebraElemMatrix)
+    vcat([hcat([repr(mat[i, j]) for j in axes(mat, 2)]...) for i in axes(mat, 1)]...)
+end
+
+function parity_checks(c::LiftedCode)
+    return lift(c.repr, c.A)
+end
+
+code_n(c::LiftedCode) = size(c.A, 2) * size(zero(c.GA), 2)
+
+code_s(c::LiftedCode) = size(c.A, 1) * size(zero(c.GA), 1)