Multivariate Bicycle code via Hecke's Group Algebra

docs/src/references.bib

@@ -487,3 +487,13 @@ @article{anderson2014fault
   year={2014},
   publisher={APS}
 }
+
+@misc{voss2024multivariatebicyclecodes,
+  title={Multivariate Bicycle Codes}, 
+  author={Lukas Voss and Sim Jian Xian and Tobias Haug and Kishor Bharti},
+  year={2024},
+  eprint={2406.19151},
+  archivePrefix={arXiv},
+  primaryClass={quant-ph},
+  url={https://arxiv.org/abs/2406.19151},
+}

docs/src/references.md

@@ -40,6 +40,7 @@ For quantum code construction routines:
 - [steane1999quantum](@cite)
 - [campbell2012magic](@cite)
 - [anderson2014fault](@cite)
+- [voss2024multivariatebicyclecodes](@cite)
 
 For classical code construction routines:
 - [muller1954application](@cite)

ext/QuantumCliffordHeckeExt/lifted_product.jl

@@ -70,6 +70,60 @@ julia> code_n(c2), code_k(c2)
 - When the base matrices of the `LPCode` are 1×1 and their elements are sums of cyclic permutations, the code is called a generalized bicycle code [`generalized_bicycle_codes`](@ref).
 - When the two matrices are adjoint to each other, the code is called a bicycle code [`bicycle_codes`](@ref).
 
+# Examples
+
+The group algebra of the qubit multivariate bicycle (MB) code with r variables is `𝔽₂[𝐺ᵣ]`, where `𝐺ᵣ = ℤ/l₁ × ℤ/l₂ × ... × ℤ/lᵣ`.
+
+[[48, 4, 6]] Weight-6 TB-QLDPC code from Appendix A Table 2 of [voss2024multivariatebicyclecodes](@cite).
+
+```jldoctest
+julia> import Hecke: group_algebra, GF, abelian_group, gens;
+
+julia> l=4; m=6;
+
+julia> GA = group_algebra(GF(2), abelian_group([l, m]));
+
+julia> x = gens(GA)[1];
+
+julia> y = gens(GA)[2];
+
+julia> z = x*y;
+
+julia> A = reshape([x^3 + y^5], (1, 1));
+
+julia> B = reshape([x + z^5 + y^5 + y^2], (1, 1));
+
+julia> c1 = LPCode(A, B);
+
+julia> code_n(c1), code_k(c1)
+(48, 4)
+```
+
+[[30, 4, 5]] Weight-7 TB-QLDPC code from Appendix A Table 2 of [voss2024multivariatebicyclecodes](@cite).
+
+```jldoctest
+julia> import Hecke: group_algebra, GF, abelian_group, gens;
+
+julia> l=5; m=3;
+
+julia> GA = group_algebra(GF(2), abelian_group([l, m]));
+
+julia> x = gens(GA)[1];
+
+julia> y = gens(GA)[2];
+
+julia> z = x*y;
+
+julia> A = reshape([x^4 + x^2], (1, 1));
+
+julia> B = reshape([x + x^2 + y + z^2 + z^3], (1, 1));
+
+julia> c1 = LPCode(A, B);
+
+julia> code_n(c1), code_k(c1)
+(30, 4)
+```
+
 ## The representation function
 
 We use the default representation function `Hecke.representation_matrix` to convert a `GF(2)`-group algebra element to a binary matrix.

test/test_ecc_multivariate_bicycle.jl

@@ -0,0 +1,167 @@
+@testitem "ECC Multivaraite Bicycle" begin
+    using Hecke
+    using Hecke: group_algebra, GF, abelian_group, gens
+    using QuantumClifford.ECC: LPCode, code_k, code_n
+
+    @testset "Weight-4 QLDPC Codes" begin
+        # [[112, 8, 5]]
+        l=7; m=8
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x = gens(GA)[1]
+        y = gens(GA)[2]
+        z = x*y
+        A = reshape([z^2 + z^6], (1, 1))
+        B = reshape([x + x^6], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 112 &&  code_k(c) == 8
+
+        # [[64, 2, 8]]
+        l=8; m=4
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x = gens(GA)[1]
+        y = gens(GA)[2]
+        z = x*y
+        A = reshape([x + x^2], (1, 1))
+        B = reshape([x^3 + y], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 64 &&  code_k(c) == 2
+
+        # [[72, 2, 8]]
+        l=4; m=9
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x = gens(GA)[1]
+        y = gens(GA)[2]
+        z = x*y
+        A = reshape([x + y^2], (1, 1))
+        B = reshape([x^2 + y^2], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 72 &&  code_k(c) == 2
+
+        # [[96, 2, 8]]
+        l=6; m=8
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x = gens(GA)[1]
+        y = gens(GA)[2]
+        z = x*y
+        A = reshape([x^5 + y^6], (1, 1))
+        B = reshape([z + z^4], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 96 &&  code_k(c) == 2
+
+        # [[112, 2, 10]]
+        l=7; m=8
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x = gens(GA)[1]
+        y = gens(GA)[2]
+        z = x*y
+        A = reshape([z^6 + x^5], (1, 1))
+        B = reshape([z^2 + y^5], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 112 &&  code_k(c) == 2
+
+        # [[144, 2, 12]]
+        l=8; m=9
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x = gens(GA)[1]
+        y = gens(GA)[2]
+        z = x*y
+        A = reshape([x^3 + y^7], (1, 1))
+        B = reshape([x + y^5], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 144 &&  code_k(c) == 2
+    end
+
+    @testset "Weight-5 QLDPC Codes" begin
+        # [[30, 4, 5]]
+        l=3; m=5
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x = gens(GA)[1]
+        y = gens(GA)[2]
+        z = x*y
+        A = reshape([x + z^4], (1, 1))
+        B = reshape([x + y^2 + z^2 ], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 30 &&  code_k(c) == 4
+
+        # [[72, 4, 8]]
+        l=4; m=9
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x = gens(GA)[1]
+        y = gens(GA)[2]
+        z = x*y
+        A = reshape([x + y^3], (1, 1))
+        B = reshape([x^2 + y + y^2 ], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 72 &&  code_k(c) == 4
+
+        # [96, 4, 8]]
+        l=8; m=6
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x = gens(GA)[1]
+        y = gens(GA)[2]
+        z = x*y
+        A = reshape([x^6 + x^3], (1, 1))
+        B = reshape([z^5 + x^5 + y ], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 96 &&  code_k(c) == 4
+    end
+
+    @testset "Weight-6 QLDPC codes" begin
+        # [[30, 6, 4]]
+        l=5; m=3
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x = gens(GA)[1]
+        y = gens(GA)[2]
+        z = x*y
+        A = reshape([x^4 + z^3], (1, 1))
+        B = reshape([x^4 + x + z^4 + y], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 30 &&  code_k(c) == 6
+
+        # [[48, 6, 6]]
+        l=4; m=6
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x = gens(GA)[1]
+        y = gens(GA)[2]
+        z = x*y
+        A = reshape([x^2 + y^4], (1, 1))
+        B = reshape([x^3 + z^3 + y^2 + y], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 48 &&  code_k(c) == 6
+
+        # [[40, 4, 6]]
+        l=4; m=5
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x = gens(GA)[1]
+        y = gens(GA)[2]
+        z = x*y
+        A = reshape([x^2 + y], (1, 1))
+        B = reshape([y^4 + y^2 + x^3 + x], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 40 &&  code_k(c) == 4
+
+        # [[48, 4, 6]]
+        l=4; m=6
+        GA = group_algebra(GF(2), abelian_group([l, m]))
+        x = gens(GA)[1]
+        y = gens(GA)[2]
+        z = x*y
+        A = reshape([x^3 + y^5], (1, 1))
+        B = reshape([x + z^5 + y^5 + y^2], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 48 &&  code_k(c) == 4
+    end
+
+    @testset "Weight-7 QLDPC codes" begin
+        # [[30, 4, 5]]
+        l=5; m=3;
+        GA = group_algebra(GF(2), abelian_group([l, m]));
+        x = gens(GA)[1]
+        y = gens(GA)[2]
+        z = x*y
+        A = reshape([x^4 + x^2], (1, 1))
+        B = reshape([x + x^2 + y + z^2 + z^3], (1, 1))
+        c = LPCode(A, B)
+        @test code_n(c) == 30 &&  code_k(c) == 4
+    end
+end