Generalized Toric code on twisted tori via Oscar's Laurent polynomials (#574)

* Generalized Toric Code

* more tests

* improve documentation

* test invariant properties (row and column weights) of the [[144,12,12]] Generalized Toric Code

* add changelog

Author: Fe-r-oz

CHANGELOG.md

@@ -7,6 +7,7 @@
 
 ## v0.10.1-dev
 
+- The `GeneralizedToricCode` on twisted tori via `Oscar`'s Laurent polynomials is now implemented in the ECC submodule.
 - The `HomologicalProductCode` and `DoubleHomologicalProductCode` are now implemented via `Oscar`'s homological algebra in the ECC submodule.
 - Adapt.jl can now be used to convert various types to GPU-backed storage.
 - The `GeneralizedBicycleCode` and `ExtendedGeneralizedBicycleCode` are now implemented via `Hecke`'s polynomial ring in the ECC submodule.

docs/src/references.bib

@@ -1056,3 +1056,27 @@ @article{Campbell_2019
   year={2019},
   month=feb, pages={025006}
 }
+
+@misc{liang2025generalizedtoriccodestwisted,
+  title={Generalized toric codes on twisted tori for quantum error correction}, 
+  author={Zijian Liang and Ke Liu and Hao Song and Yu-An Chen},
+  year={2025},
+  eprint={2503.03827},
+  archivePrefix={arXiv},
+  primaryClass={quant-ph},
+  url={https://arxiv.org/abs/2503.03827}, 
+}
+
+@article{Liang_2024,
+  title={Extracting Topological Orders of Generalized Pauli Stabilizer Codes in Two Dimensions},
+  volume={5},
+  ISSN={2691-3399},
+  url={http://dx.doi.org/10.1103/PRXQuantum.5.030328},
+  DOI={10.1103/prxquantum.5.030328},
+  number={3},
+  journal={PRX Quantum},
+  publisher={American Physical Society (APS)},
+  author={Liang, Zijian and Xu, Yijia and Iosue, Joseph T. and Chen, Yu-An},
+  year={2024},
+  month=aug
+}

docs/src/references.md

@@ -69,6 +69,7 @@ For quantum code construction routines:
 - [Quintavalle_2021](@cite)
 - [xu2024fastparallelizablelogicalcomputation](@cite)
 - [bravyi2013homologicalproductcodes](@cite)
+- [liang2025generalizedtoriccodestwisted](@cite)
 
 For classical code construction routines:
 - [muller1954application](@cite)

ext/QuantumCliffordOscarExt/QuantumCliffordOscarExt.jl

@@ -9,28 +9,35 @@ import QuantumClifford: Stabilizer
 import Nemo
 import Nemo: FqFieldElem
 import Hecke: group_algebra, GF, abelian_group, gens, quo, one, GroupAlgebra,
-   GroupAlgebraElem, direct_product, sub, ZZ, lift
+   GroupAlgebraElem, direct_product, sub, ZZ, lift, polynomial_ring
 import Oscar
 import Oscar: free_group, small_group_identification, describe, order, FPGroupElem, FPGroup,
     BasicGAPGroupElem, DirectProductGroup, cyclic_group, free_module, hom, transpose, tensor_product,
     chain_complex, total_complex, map, summands, MatElem, matrix, nrows, ncols, kernel, dim, range, image,
     base_ring, ComplexOfMorphisms, coefficients, zero_matrix, hcat, circshift, size, zeros, enumerate,
-    kronecker_product, FqMatrix, identity_matrix, iszero, FqPolyRingElem
+    kronecker_product, FqMatrix, identity_matrix, iszero, FqPolyRingElem, laurent_polynomial_ring,
+    hnf_with_transform, ideal, intersect, ==, is_coprime, quo, groebner_basis, length, FqMPolyRingElem,
+    first, length
 import Oscar.Generic.MatSpaceElem
+import Oscar.Generic.DirectSumModule
+import Oscar.Generic.LaurentMPolyWrap
+import Oscar.Generic.exponent_vectors
+import Oscar.IdealGens
 
 import QuantumClifford.ECC: two_block_group_algebra_codes, twobga_from_direct_product, twobga_from_fp_group,
-    boundary_maps
+    boundary_maps, max_xy_exponents
 
 import QECCore: AbstractECC, CSS, RepCode, AbstractCSSCode,
     hgp, code_k, code_n, code_s, distance, parity_matrix_x, parity_matrix_z, parity_matrix_xz, parity_matrix,
     metacheck_matrix_x, metacheck_matrix_z, metacheck_matrix
 
 # exported from extension so that Documenter.jl sees them when autogenerating API lists
 export twobga_from_direct_product, twobga_from_fp_group, DDimensionalSurfaceCode, DDimensionalToricCode, boundary_maps,
-    HomologicalProductCode, DoubleHomologicalProductCode
+    HomologicalProductCode, DoubleHomologicalProductCode, GeneralizedToricCode
 
 include("types.jl")
 include("direct_product.jl")
+include("generalized_toric.jl")
 include("group_presentation.jl")
 include("d_dimensional_codes.jl")
 include("homological_product_codes.jl")