Circulant GF(2) permutation matrix-based construction of Bivariate Bicycle quantum LDPC code #352

Fe-r-oz · 2024-09-02T09:30:54Z

In this 2024 paper High-threshold and low-overhead fault-tolerant quantum memory, Bravyi et. al. has introduced a simple BBQLDPC code. In the Table 3, as seen below, these are the code constructions given in the paper.

All these constructions have been tested (especially at the moment, tests have been added to verify the number of logical qubits, k which is calculated on formula: k = n - rank(Hx) - rank(Hz). Please checkout the test section:

The interface is user-friendly:

julia> using QuantumClifford.ECC: BBQLDPC, parity_checks

julia> parity_checks(BBQLDPC(12, 6, [3, 1, 2], [3, 1, 2]))
# the stabilizer

The documentation is simple so it's easier to follow as well. Python implementation of this code is available at here.

Cross verification between python implementation and this implementation has been done. The matrices are huge, but we can export results from python, import them in julia and compare the results.

Would you like to have a dataset prepared for these given constructions, so we can import them and test for correctness comparison in the test section?

julia> using MAT
julia> using QuantumClifford
julia> using QuantumClifford.ECC: BBQLDPC, parity_checks, naive_encoding_circuit
julia> mat_file_path = "/home/Feroz/Desktop/matrices96.mat";
julia> mat_data = matopen(mat_file_path);
julia> AT = read(mat_data, "AT");
julia> BT = read(mat_data, "BT");
julia> BTT, ATT = parity_checks(BBQLDPC(9, 6, [3, 1, 2], [3, 1, 2]))
([0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0], [0 0 … 0 0; 1 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 1 0])
julia> BTT == BT
true
julia> ATT == AT
true
julia> mat_file_path = "/home/Feroz/Desktop/matrices2814.mat";
julia> mat_data = matopen(mat_file_path);
julia> AT = read(mat_data, "AT");
julia> BT = read(mat_data, "BT");
julia> BTT, ATT = parity_checks(BBQLDPC(28, 14, [26, 6, 8], [7, 9, 20]))
([0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0], [0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0])
julia> BTT == BT
true
julia> ATT == AT
true

There is bounds error as seen in #306.

Near-term use of this BBQLDPC: This pr will be helpful as BBLDPC code is a special case of LiftedProduct code as described the authors as follows: "We note that the code QC(A,B) can be viewed as a special case of the Lifted Product construction [54] based on the abelian group Z_ℓ × Z_m. Here Z_j denotes the cyclic group of order j." Maybe in the near-future, LiftedProduct code can be tested by using BBQLPC as a correctness check.

The aim and objective of the pr is small: to provide a correct and robust construction of BBQLDPC code. I will try to further improve it by adding more tests and incorprate your feedback.

I was very excited as I read the paper, so submitted it. I hope you find this plausible.

Thanks for your time.
...

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Fe-r-oz · 2024-09-04T14:31:59Z

Acknowledgement: I am very thankful to Claus from Oscar. He suggested a robust method to calculate the intersection of two kernels, ker(A) ∩ ker(B). This helped to construct a cross-reference check.

Downgrade error is occuring because hom is not present in Nemo's earlier versions. Probably, have to remove 0.42, 0.43, etc. Spell Error is occuring because of using Nemo: hom where hom is homomorphism

Fe-r-oz · 2024-09-04T17:32:55Z

@royess, two days ago, I completed the BBQLDPC code. This can serve as a correctness check for LiftedProduct code as stated in the Bravyi paper, as BBQLDPC is a special case of the LiftedProduct code. I have also added correctness tests based on techniques suggested by Oscar fellows for this code.

These might be interesting to you: A new QLDPC decoder released today: BPOTF (It's a C++ decoder) and this #14 (comment).

Fe-r-oz · 2024-09-05T05:25:57Z

@Krastanov, pinging for review. All the code constructions given in the paper I shared have been verified for BBQLDPC code. Furthermore, I have added cross-reference checks as well. To ensure robustness, I have also discussed (#352 (comment)) this with dear Oscar fellows whose guidance have been helpful in designing robust cross-verification test over the last 2 days. The code interface is user-friendly as well.

P.S Downgrade error is occuring because hom is not present in Nemo's earlier versions. Probably, have to remove 0.42, 0.43, etc. Spell Error is occuring because of using Nemo: hom where hom is homomorphism

Fe-r-oz · 2024-09-06T06:39:26Z

Referring to #353 to monitor progress.

Fe-r-oz · 2024-09-09T20:38:33Z

Referring to #356 as BB QLDPC can be viewed as a special case of LiftedProduct (please see comment: #352 (comment)) , so this PR serves as a correctness check for the latter codes, if desired.

…and the test base with many instances

Fe-r-oz · 2024-09-27T18:38:10Z

The PR is ready for review. I have added many instances of BBQLDPCs in the test_base_ecc so that their stabilizer correctness, encoding, decoding and syndrome properties are tested. Thank you!

P.S. There were a lot of merge conflicts, therefore had to merge branch into master many times.

Krastanov · 2024-09-27T18:56:58Z

@Fe-r-oz , thank you, this is a very useful addition.

@royess , @Fe-r-oz , do you know the proper way to represent the codes added here in terms of a lifted product? That would be a useful cross check.

Fe-r-oz · 2024-09-27T19:41:46Z

Indeed. Bravyi notes that BBQLDPC can be viewed as a special case of Lifted Product construction based on abelian group ℤⱼ x ℤₘ where ℤⱼ cyclic group of order j Page 7. This implementation can serve as a cross-check to the proper way which is by constructing the code via Lifted Product construction.

Krastanov · 2024-09-27T19:43:31Z

Do you know how to implement such a cross-check?

Fe-r-oz · 2024-09-27T19:57:07Z

I will have to see the literature on the LPCode construction method for these codes for the group theoretic route. The aforementioned paper used, let's say "matrix-based" approach, not the "group based" approach (LPCode construction).

side note: I did pondered about cross checking the matrix-based construction based on the formulas provided in the paper, especially calculating ker(A) ∩ ker(B) which is equal to k.

This manual correctness check uses Oscar hom (homomorphism) for Free Modules by taking intersection of two kernels, compute ker(A) ∩ ker(B) as suggested by Claus. This method for hom is not present in Hecke, Abstract Algebra or Nemo. It's actually presented in Oscar itself: https://github.com/oscar-system/Oscar.jl/blob/master/src/Modules/UngradedModules/FreeModuleHom.jl.

Without Oscar, one will get this error using hom for Free Modules: no method matching hom(::AbstractAlgebra.Generic.FreeModule{FqFieldElem}, ::AbstractAlgebra.Generic.FreeModule{FqFieldElem}, ::FqMatrix). It turns out that this specific method is present in Oscar FreeModuleElem.jl. Just some weeks ago, Nemo used to have this hom method as well, but they had breaking change quite often and have removed it.

Since we don't have Oscar as dependency, this is a manual check:

    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, BBQLDPC, parity_checks_x, parity_checks_z

    # 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)
        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

     @testset "Verify number of logical qubits `k` from  Table 3" begin
        # Refer to [bravyi2024high](@cite) for code constructions
        @test code_k(BBQLDPC(9 , 6 , [3 , 1 , 2] , [3 , 1 , 2]))   == 8   == _formula_k(BBQLDPC(9 , 6 , [3 , 1 , 2] , [3 , 1 , 2]))
        @test code_k(BBQLDPC(15, 3 , [9 , 1 , 2] , [0 , 2 , 7]))   == 8   == _formula_k(BBQLDPC(15, 3 , [9 , 1 , 2] , [0 , 2 , 7]))
        @test code_k(BBQLDPC(12, 12, [3 , 2 , 7] , [3 , 1 , 2]))   == 12  == _formula_k(BBQLDPC(12, 12, [3 , 2 , 7] , [3 , 1 , 2]))
        @test code_k(BBQLDPC(12, 6 , [3 , 1 , 2] , [3 , 1 , 2]))   == 12  == _formula_k(BBQLDPC(12, 6 , [3 , 1 , 2] , [3 , 1 , 2]))
        @test code_k(BBQLDPC(6 , 6 , [3 , 1 , 2] , [3 , 1 , 2]))   == 12  == _formula_k(BBQLDPC(6 , 6 , [3 , 1 , 2] , [3 , 1 , 2]))
        @test code_k(BBQLDPC(30, 6 , [9 , 1 , 2] , [3 , 25, 26]))  == 12  == _formula_k(BBQLDPC(30, 6 , [9 , 1 , 2] , [3 , 25, 26]))
        @test code_k(BBQLDPC(21, 18, [3 , 10, 17], [5 , 3 , 19]))  == 16  == _formula_k(BBQLDPC(21, 18, [3 , 10, 17], [5 , 3 , 19]))
        @test code_k(BBQLDPC(28, 14, [26, 6 , 8] , [7 , 9 , 20]))  == 24  == _formula_k(BBQLDPC(28, 14, [26, 6 , 8] , [7 , 9 , 20]))
    end

    @testset "Verify number of logical qubits `k` from  Table 1" begin
        # Refer to [berthusen2024toward](cite for code constructions
        @test code_k(BBQLDPC(12, 3 , [9 , 1 , 2] , [0 , 1 , 11]))  == 8   == _formula_k(BBQLDPC(12, 3 , [9 , 1 , 2] , [0 , 1 , 11]))
        @test code_k(BBQLDPC(9 , 5 , [8 , 4 , 1] , [5 , 8 , 7]))   == 8   == _formula_k(BBQLDPC(9 , 5 , [8 , 4 , 1],  [5 , 8 , 7]))
        @test code_k(BBQLDPC(12, 5 , [10, 4 , 1] , [0 , 1 , 2]))   == 8   == _formula_k(BBQLDPC(12, 5 , [10, 4 , 1] , [0 , 1 , 2]))
        @test code_k(BBQLDPC(15, 5 , [5 , 2 , 3] , [2 , 7 , 6]))   == 8   == _formula_k(BBQLDPC(15, 5 , [5 , 2 , 3] , [2 , 7 , 6]))
        @test code_k(BBQLDPC(14, 7 , [6 , 5 , 6] , [0 , 4, 13]))   == 12  == _formula_k(BBQLDPC(14, 7 , [6 , 5 , 6] , [0 , 4, 13]))
    end

Krastanov · 2024-09-27T20:16:40Z

Oscar can be added to the test-runner without being added as a dependency to the package itself, if it can provide useful testing capabilities

Fe-r-oz · 2024-10-02T23:24:03Z

Pardon for the delay, Added Oscar to the test-runner. There will be some CI errors which are unrelated:

ERROR: LoadError: InitError: GAP.jl currently does not support multithreaded garbage collection. Please run julia with `--gcthreads=1` for now.

ERROR: LoadError: Windows is not supported, please try the Windows Subsystem for Linux.
For details please check: https://www.oscar-system.org/install/

I didn't add Oscar to deps, only in test-runner but Downgrade gives error. Spell check error is occurring because it see hom as error. Also, reintroducing piracy in 371 introduced 30 broken errors in test summary which are unrelated to this PR: https://github.com/QuantumSavory/QuantumClifford.jl/runs/30774357893?check_suite_focus=true#step:7:373

Plots:

julia> codes = [circulant_bivariate_bicycle(7,7,[3,5,6],[2,3,5]),
                circulant_bivariate_bicycle(15,3,[9,1,2],[0,2,7]),
                circulant_bivariate_bicycle(9,5,[8,4,1],[5,8,7])]

julia>  mem_errors = 0.00001:0.00005:0.001;

julia> julia> results = zeros(length(codes), length(mem_errors), 2);

julia> for (ic, c) in pairs(codes)
                 for (i,m) in pairs(mem_errors)
                     setup = CommutationCheckECCSetup(m)
                     decoder = TableDecoder(c)
                     r = evaluate_decoder(decoder, setup, 100000)
                     results[ic,i,:] .= r
                 end
        end


julia> for (ic, c) in pairs(codes)
                 for (i,m) in pairs(mem_errors)
                     setup = ShorSyndromeECCSetup(m,0)
                     decoder = TableDecoder(c)
                     r = evaluate_decoder(decoder, setup, 100000)
                     results[ic,i,:] .= r
                 end
         end

Fe-r-oz · 2024-10-05T07:24:29Z

Do you know how to implement such a cross-check?

@Krastanov, Thank you for your suggestion. The cross check (#372) has been implemented as well. Code_n and Code_k values match even though the two constructions used vastly different formulas and techniques. I also plotted their Table Decoding graphs which are also similar. Their canonical forms are nonequivalent because this construction applies a circulant permutation matrix to the column space. Permutating the rowspace does not change the canonical forms of matrices as we have seen and tested in the past, but permutating the column space leads to nonequivalent canonical form. Hopefully, this is satisfactory and reasonable.

In this construction, x = Sₗ ⊗ Iₘ permutes the rowspace and y = Iₗ ⊗ Sₘ permutes the column space where Sₘ is the circulant permutation matrix. Also, Added all of the code instances in literature (to my knowledge) to test section, testing their code_nand code_k values based on the formulas given in Bravyi paper: code_k is n - rank(Hx) - rank(Hz), formula_k is 2 * dim(kernel(A) ∩ kernel(B)) == k. Cross checked it with original python implementation as well.

A small test which shows permutating the column space leads to nonequivalent canonical forms:

using LinearAlgebra, Nemo, Test

H = [1 1 0 1 0 0 0;
     1 0 1 0 1 0 0;
     0 1 1 0 0 1 0;
     0 0 0 1 1 1 1]

# permutations over column space
function permute_columns(matrix, perm)
    return matrix[:, perm]
end

col_permutations = [
    [1, 2, 3, 4, 5, 6, 7],
    [2, 1, 3, 4, 5, 6, 7],
    [1, 3, 2, 4, 5, 6, 7],
    [3, 1, 2, 4, 5, 6, 7]
    ]

for (i, perm) in enumerate(col_permutations)
    permuted_H = permute_columns(H, perm)
    rref_H = echelon_form(Nemo.matrix(Nemo.GF(2), permuted_H))
    rref_without_permut = echelon_form(Nemo.matrix(Nemo.GF(2), H))
    @test rref_H == rref_without_permut # gives error because of permutation of column space
end


# permutations over row space: Canonical form is equivalent as expected
row_permutations = [
           [1, 2, 3, 4],
           [2, 1, 3, 4],
           [1, 3, 2, 4],
           [4, 3, 2, 1]
       ]

function permute_rows(matrix, perm)
    return matrix[perm, :]
end

for (i, perm) in enumerate(row_permutations)
    permuted_H = permute_rows(H, perm)
    rref_H = echelon_form(Nemo.matrix(Nemo.GF(2), permuted_H))
    rref_without_permut = echelon_form(Nemo.matrix(Nemo.GF(2), H))
    @test rref_H == rref_without_permut
end

P.S. Thank you for your help is resolving the weird CI errors when adding Oscar to test-runner. That is much appreciated.

Fe-r-oz · 2024-10-07T02:30:20Z

Experimental WIP: Stacked Model and Bilayered Architecture

Future work: The papers on these codes talk about Toric Layout and Embeddings for these codes for 2D Implementation. Thus, two future features, namely - bivaraite_toric_layout and bivaraite_embedding are also being worked on at the moment for later PR. I have cross checked these features with the original implementation, and implemented these two functionalities as well. But I think the plotting indexing conventions needs to be much polished. I have constructed it using Plots, not Makie/CaireMakie though. I have checked the embedding and toric layout, it matches in calculation, but in plotting, the ancilla seems to be upside down. I think these two features will be useful w.r.t to circuit-level simulation, for these codes.

bivaraite_toric_layout : Toric Layout of Bivariate bicycle codes (lemma 3)
bivaraite_embedding: 2D Embeddings of the Bivariate Bicycle codes

Krastanov

This is awesome! Left a few comments in.

Krastanov · 2024-11-05T13:22:22Z

Project.toml

@@ -48,15 +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, 0.34"
+Hecke = "0.28, 0.29, 0.30, 0.31, 0.32, 0.33, 0.34, 0.34.3"


why is this needed?

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

Krastanov · 2024-11-05T13:22:30Z

Project.toml

 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 = "0.42.1, 0.43, 0.44, 0.45, 0.46, 0.47"
+Nemo = "0.42.1, 0.43, 0.44, 0.45, 0.46, 0.47, 0.47.1"


why is this needed?

Krastanov · 2024-11-05T13:27:00Z

src/ecc/codes/circulant_bivariate_bicycle.jl

+
+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.
+
+The ECC Zoo has an [entry for this family](https://errorcorrectionzoo.org/c/qcga).


This also needs:

explanation of what the l,m,A,B entries are

doctest example

comparison to using directly the other ways in which this code can already be created in this library

"see also" for other codes and constructions

a remark in the "see also" that mentions alternative way to construct these same codes

Krastanov · 2024-11-05T13:27:34Z

src/ecc/codes/circulant_bivariate_bicycle.jl

+
+The ECC Zoo has an [entry for this family](https://errorcorrectionzoo.org/c/qcga).
+"""
+struct circulant_bivariate_bicycle <: AbstractECC


this should be named like a struct, not like a function (e.g. CirculantBivariateBicycle)

Krastanov · 2024-11-05T13:28:19Z

src/ecc/codes/circulant_bivariate_bicycle.jl

+        (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))].")


could you use an explicit throw of an argument error or domain error instead of a generic error

Krastanov · 2024-11-05T13:28:54Z

test/Project.toml

@@ -16,12 +17,15 @@ LDPCDecoders = "3c486d74-64b9-4c60-8b1a-13a564e77efb"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 MacroTools = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09"
 Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
+Oscar = "f1435218-dba5-11e9-1e4d-f1a5fab5fc13"
+Polymake = "d720cf60-89b5-51f5-aff5-213f193123e7"


why is polymake needed?

Krastanov · 2024-11-05T13:29:02Z

test/Project.toml

 PyQDecoders = "17f5de1a-9b79-4409-a58d-4d45812840f7"
 Quantikz = "b0d11df0-eea3-4d79-b4a5-421488cbf74b"
 QuantumInterface = "5717a53b-5d69-4fa3-b976-0bf2f97ca1e5"
 QuantumOpticsBase = "4f57444f-1401-5e15-980d-4471b28d5678"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 RecipesBase = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
+Singular = "bcd08a7b-43d2-5ff7-b6d4-c458787f915c"


why is Singular needed?

Krastanov · 2024-11-05T13:29:51Z

test/test_ecc_circulantbivariatebicycle.jl

@@ -0,0 +1,53 @@
+@testitem "ECC circulant_bivariate_bicycle" begin


these tests probable also need a subset of tests that verify correctness against the other construction method you have already contributed

Krastanov · 2024-11-05T13:30:23Z

test/test_ecc_circulantbivariatebicycle.jl

+        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)


hom needs to be added to the spellchecker (the .typos file)

Krastanov · 2024-11-05T13:30:42Z

docs/src/references.bib

+
+@article{berthusen2024toward,
+  title={Toward a 2D local implementation of quantum LDPC codes},
+  author={Berthusen, Noah and Devulapalli, Dhruv and Schoute, Eddie and Childs, Andrew M and Gullans, Michael J and Gorshkov, Alexey V and Gottesman, Daniel},


"Childs" needs to be added to the spellchecker (the .typos file)

Fe-r-oz · 2024-11-05T13:41:24Z

Thank you for your comments. This PR was done before Tommy's CI Injection Oscar approach which greatly simplified things for us.

Before the CI Injection, I was trying to add Oscar into test runner and meeting requirements which was not the right approach.

The tests require hom method for Oscar, this hom method provides homomorphism between free modules for kernel calculation.

So, let's turn this into draft till we have Oscar into the system.

bivariate bicycle quantum ldpc code

b680589

Fe-r-oz force-pushed the BBQLDPC branch from 24840d4 to b680589 Compare September 2, 2024 09:55

Fe-r-oz changed the title ~~bivariate bicycle quantum LDPC code~~ bivariate bicycle quantum ldpc code Sep 2, 2024

Fe-r-oz marked this pull request as ready for review September 3, 2024 20:23

improve code and test quality

7eeb82c

Fe-r-oz mentioned this pull request Sep 5, 2024

Make an interface to QEC libraries, specifically so that you have access to predefined codes #14

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Fe-r-oz added 3 commits September 24, 2024 06:30

Merge branch 'master' into BBQLDPC

8c4d158

Merge branch 'master' into BBQLDPC

e77e1ca

remove AbstractAlgebra hom conflicts with Hecke and significantly exp…

a575cbf

…and the test base with many instances

Fe-r-oz force-pushed the BBQLDPC branch from 9dd9398 to a575cbf Compare September 27, 2024 12:08

Fe-r-oz added 4 commits September 27, 2024 20:03

Merge branch 'master' into BBQLDPC

3977d80

misc cleanup after merging master

d4e8669

Merge branch 'master' into BBQLDPC

f6ff830

resolve any change after merge conflicts

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Krastanov self-requested a review September 27, 2024 18:57

Fe-r-oz mentioned this pull request Sep 29, 2024

tests: test consistency of Bivariate Bicycle code using LPCode constructor #372

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Fe-r-oz changed the title ~~bivariate bicycle quantum ldpc code~~ Circulant GF(2)-matrix-based construction of Bivariate Bicycle quantum LDPC code Sep 29, 2024

add Oscar to test-runner - improve code quality

254768e

Fe-r-oz changed the title ~~Circulant GF(2)-matrix-based construction of Bivariate Bicycle quantum LDPC code~~ Circulant GF(2) permutation matrix-based construction of Bivariate Bicycle quantum LDPC code Oct 3, 2024

Fe-r-oz mentioned this pull request Oct 10, 2024

2D Local implementation of Bivariate Bicycle qLDPC codes: Toric Layouts #387

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Circulant GF(2) permutation matrix-based construction of Bivariate Bicycle quantum LDPC code #352

Circulant GF(2) permutation matrix-based construction of Bivariate Bicycle quantum LDPC code #352

Fe-r-oz commented Sep 2, 2024 •

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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.

		The ECC Zoo has an [entry for this family](https://errorcorrectionzoo.org/c/qcga).

		@@ -0,0 +1,53 @@
		@testitem "ECC circulant_bivariate_bicycle" begin

Circulant GF(2) permutation matrix-based construction of Bivariate Bicycle quantum LDPC code #352

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Circulant GF(2) permutation matrix-based construction of Bivariate Bicycle quantum LDPC code #352

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Experimental WIP: Stacked Model and Bilayered Architecture

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