Lifted Product construction of bivariate bicycle using LPCode

QuantumSavory · Oct 3, 2024 · 656d5e8 · 656d5e8
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diff --git a/docs/Project.toml b/docs/Project.toml
@@ -9,6 +9,7 @@ Hecke = "3e1990a7-5d81-5526-99ce-9ba3ff248f21"
 LDPCDecoders = "3c486d74-64b9-4c60-8b1a-13a564e77efb"
 Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
 Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
+Oscar = "f1435218-dba5-11e9-1e4d-f1a5fab5fc13"
 PyQDecoders = "17f5de1a-9b79-4409-a58d-4d45812840f7"
 Quantikz = "b0d11df0-eea3-4d79-b4a5-421488cbf74b"
 QuantumClifford = "0525e862-1e90-11e9-3e4d-1b39d7109de1"

diff --git a/docs/src/references.bib b/docs/src/references.bib
@@ -487,3 +487,14 @@ @article{anderson2014fault
   year={2014},
   publisher={APS}
 }
+
+@article{bravyi2024high,
+  title={High-threshold and low-overhead fault-tolerant quantum memory},
+  author={Bravyi, Sergey and Cross, Andrew W and Gambetta, Jay M and Maslov, Dmitri and Rall, Patrick and Yoder, Theodore J},
+  journal={Nature},
+  volume={627},
+  number={8005},
+  pages={778--782},
+  year={2024},
+  publisher={Nature Publishing Group UK London}
+}
diff --git a/docs/src/references.md b/docs/src/references.md
@@ -40,6 +40,7 @@ For quantum code construction routines:
 - [steane1999quantum](@cite)
 - [campbell2012magic](@cite)
 - [anderson2014fault](@cite)
+- [bravyi2024high](@cite)
 
 For classical code construction routines:
 - [muller1954application](@cite)

diff --git a/ext/QuantumCliffordHeckeExt/lifted_product.jl b/ext/QuantumCliffordHeckeExt/lifted_product.jl
@@ -70,6 +70,33 @@ 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
+
+Bivariate Bicycle codes belong to a wider class of generalized bicycle (GB) codes, which are further generalized into of two block group algebra (2GBA) codes. They can be viewed as a special case of Lifted Product construction based on abelian group `ℤₗ x ℤₘ` where `ℤⱼ` cyclic group of order `j`.
+
+A [[756, 16, ≤ 34]] code from Table 3 of [bravyi2024high](@cite).
+
+```jldoctest
+julia> import Hecke: group_algebra, GF, abelian_group, gens; import Oscar: PcGroup;
+
+julia> l=21; m=18;
+
+julia> GA = group_algebra(GF(2), abelian_group(PcGroup, [l, m]));
+
+julia> x = gens(GA)[1];
+
+julia> y = gens(GA)[2];
+
+julia> A = reshape([x^3 + y^10 + y^17], (1, 1));
+
+julia> B = reshape([y^5 + x^3 + x^19], (1, 1));
+
+julia> c1 = LPCode(A, B);
+
+julia> code_n(c1), code_k(c1)
+(756, 16)
+```
+
 ## The representation function
 
 We use the default representation function `Hecke.representation_matrix` to convert a `GF(2)`-group algebra element to a binary matrix.