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Fe-r-ozKrastanov
Fe-r-oz
and
Krastanov
authored
Multivariate Bicycle code via Hecke's Group Algebra (#381)
 Co-authored-by: Stefan Krastanov <[email protected]>
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‎docs/src/references.bib

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@@ -524,6 +524,16 @@ @article{lin2024quantum
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publisher={APS}
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}
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@misc{voss2024multivariatebicyclecodes,
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title={Multivariate Bicycle Codes},
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author={Lukas Voss and Sim Jian Xian and Tobias Haug and Kishor Bharti},
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year={2024},
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eprint={2406.19151},
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archivePrefix={arXiv},
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primaryClass={quant-ph},
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url={https://arxiv.org/abs/2406.19151},
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}
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@article{bravyi2024high,
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title={High-threshold and low-overhead fault-tolerant quantum memory},
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author={Bravyi, Sergey and Cross, Andrew W and Gambetta, Jay M and Maslov, Dmitri and Rall, Patrick and Yoder, Theodore J},

‎docs/src/references.md

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@@ -40,6 +40,7 @@ For quantum code construction routines:
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- [steane1999quantum](@cite)
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- [campbell2012magic](@cite)
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- [anderson2014fault](@cite)
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- [voss2024multivariatebicyclecodes](@cite)
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- [lin2024quantum](@cite)
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- [bravyi2024high](@cite)
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‎ext/QuantumCliffordHeckeExt/lifted_product.jl

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@@ -207,6 +207,29 @@ julia> c = two_block_group_algebra_codes(A,B);
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julia> code_n(c), code_k(c)
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(756, 16)
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```
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### Multivariate Bicycle code
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The group algebra of the qubit multivariate bicycle (MB) code with r variables is `𝔽₂[𝐺ᵣ]`,
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where `𝐺ᵣ = ℤ/l₁ × ℤ/l₂ × ... × ℤ/lᵣ`.
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A [[48, 4, 6]] Weight-6 TB-QLDPC code from Appendix A Table 2 of [voss2024multivariatebicyclecodes](@cite).
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```jldoctest
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julia> import Hecke: group_algebra, GF, abelian_group, gens; # hide
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julia> l=4; m=6;
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julia> z = x*y;
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julia> A = x^3 + y^5;
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julia> B = x + z^5 + y^5 + y^2;
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julia> c = two_block_group_algebra_codes(A, B);
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julia> code_n(c), code_k(c)
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(48, 4)
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```
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See also: [`LPCode`](@ref), [`generalized_bicycle_codes`](@ref), [`bicycle_codes`](@ref).
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"""

‎test/test_ecc_base.jl

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@@ -58,6 +58,45 @@ A[LinearAlgebra.diagind(A, 5)] .= GA(1)
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B = reshape([1 + x + x^6], (1, 1))
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push!(other_lifted_product_codes, LPCode(A, B))
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# Multivariate Bicycle codes taken from Table 1 of [voss2024multivariatebicyclecodes](@cite)
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# Weight-4 [[144, 2, 12]] MBB code
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l=8; m=9
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x^3 + y^7
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B = x + y^5
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weight4mbb = two_block_group_algebra_codes(A, B)
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# Weight-5 [96, 4, 8]] MBB code
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l=8; m=6
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x^6 + x^3
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B = z^5 + x^5 + y
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weight5mbb = two_block_group_algebra_codes(A, B)
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# Weight-6 [[48, 4, 6]] MBB code
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l=4; m=6
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x^3 + y^5
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B = x + z^5 + y^5 + y^2
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weight6mbb = two_block_group_algebra_codes(A, B)
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# Weight-7 [[30, 4, 5]] MBB code
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l=5; m=3
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GA = group_algebra(GF(2), abelian_group([l, m]));
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x, y = gens(GA)
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z = x*y
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A = x^4 + x^2
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B = x + x^2 + y + z^2 + z^3
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weight7mbb = two_block_group_algebra_codes(A, B)
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test_mbb_codes = [weight4mbb, weight5mbb, weight6mbb, weight7mbb]
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# Bivariate Bicycle codes
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# A [[72, 12, 6]] code from Table 3 of [bravyi2024high](@cite).
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l=6; m=6
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:CSS => (c -> (parity_checks_x(c), parity_checks_z(c))).([Shor9(), Steane7(), Toric(4, 4)]),
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:Concat => [(Perfect5(), Perfect5()), (Perfect5(), Steane7()), (Steane7(), Cleve8()), (Toric(2, 2), Shor9())],
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:CircuitCode => random_circuit_code_args,
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:LPCode => (c -> (c.A, c.B)).(vcat(LP04, LP118, test_gb_codes, test_bb_codes, other_lifted_product_codes)),
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:LPCode => (c -> (c.A, c.B)).(vcat(LP04, LP118, test_gb_codes, test_bb_codes, test_mbb_codes, other_lifted_product_codes)),
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:QuantumReedMuller => [3, 4, 5]
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)
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‎test/test_ecc_multivariate_bicycle.jl

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@testitem "ECC Multivaraite Bicycle" begin
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using Hecke
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using Hecke: group_algebra, GF, abelian_group, gens
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using QuantumClifford.ECC: two_block_group_algebra_codes, code_k, code_n
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# Multivariate Bicycle codes taken from Table 1 of [voss2024multivariatebicyclecodes](@cite)
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@testset "Weight-4 QLDPC Codes" begin
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# [[112, 8, 5]]
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l=7; m=8
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = z^2 + z^6
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B = x + x^6
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c = two_block_group_algebra_codes(A, B)
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@test code_n(c) == 112 && code_k(c) == 8
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# [[64, 2, 8]]
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l=8; m=4
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x + x^2
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B = x^3 + y
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c = two_block_group_algebra_codes(A, B)
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@test code_n(c) == 64 && code_k(c) == 2
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# [[72, 2, 8]]
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l=4; m=9
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x + y^2
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B = x^2 + y^2
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c = two_block_group_algebra_codes(A, B)
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@test code_n(c) == 72 && code_k(c) == 2
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# [[96, 2, 8]]
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l=6; m=8
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x^5 + y^6
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B = z + z^4
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c = two_block_group_algebra_codes(A, B)
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@test code_n(c) == 96 && code_k(c) == 2
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# [[112, 2, 10]]
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l=7; m=8
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = z^6 + x^5
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B = z^2 + y^5
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c = two_block_group_algebra_codes(A, B)
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@test code_n(c) == 112 && code_k(c) == 2
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# [[144, 2, 12]]
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l=8; m=9
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x^3 + y^7
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B = x + y^5
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c = two_block_group_algebra_codes(A, B)
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@test code_n(c) == 144 && code_k(c) == 2
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end
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@testset "Weight-5 QLDPC Codes" begin
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# [[30, 4, 5]]
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l=3; m=5
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x + z^4
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B = x + y^2 + z^2
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c = two_block_group_algebra_codes(A, B)
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@test code_n(c) == 30 && code_k(c) == 4
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# [[72, 4, 8]]
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l=4; m=9
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x + y^3
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B = x^2 + y + y^2
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c = two_block_group_algebra_codes(A, B)
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@test code_n(c) == 72 && code_k(c) == 4
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# [96, 4, 8]]
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l=8; m=6
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x^6 + x^3
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B = z^5 + x^5 + y
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c = two_block_group_algebra_codes(A, B)
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@test code_n(c) == 96 && code_k(c) == 4
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end
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@testset "Weight-6 QLDPC codes" begin
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# [[30, 6, 4]]
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l=5; m=3
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x^4 + z^3
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B = x^4 + x + z^4 + y
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c = two_block_group_algebra_codes(A, B)
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@test code_n(c) == 30 && code_k(c) == 6
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# [[48, 6, 6]]
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l=4; m=6
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x^2 + y^4
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B = x^3 + z^3 + y^2 + y
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c = two_block_group_algebra_codes(A, B)
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@test code_n(c) == 48 && code_k(c) == 6
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# [[40, 4, 6]]
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l=4; m=5
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x^2 + y
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B = y^4 + y^2 + x^3 + x
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c = two_block_group_algebra_codes(A, B)
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@test code_n(c) == 40 && code_k(c) == 4
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# [[48, 4, 6]]
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l=4; m=6
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x^3 + y^5
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B = x + z^5 + y^5 + y^2
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c = two_block_group_algebra_codes(A, B)
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@test code_n(c) == 48 && code_k(c) == 4
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end
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@testset "Weight-7 QLDPC codes" begin
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# [[30, 4, 5]]
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l=5; m=3
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GA = group_algebra(GF(2), abelian_group([l, m]))
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x, y = gens(GA)
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z = x*y
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A = x^4 + x^2
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B = x + x^2 + y + z^2 + z^3
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c = two_block_group_algebra_codes(A, B)
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@test code_n(c) == 30 && code_k(c) == 4
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end
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end

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