Higher-dimensional HGP code

Higher-dimensional HGP code[1]

Alternative names: Multi-sector HGP code, Higher-dimensional QHG code, Higher-dimensional Tillich-Zemor product code, Multi-sector QHG code, Multi-sector Tillich-Zemor product code.

Description

A generalization of hypergraph product codes constructed from a hypergraph product of two or more classical linear binary codes. The stabilizer generator matrices of an \(m\)-dimensional hypergraph product code are the boundary and co-boundary operators of a 2-dimensional chain complex contained within an \(m\)-complex that is recursively constructed by taking the tensor product of an \((m-1)\)-complex and a 1-complex, with each 1-complex corresponding to some classical linear binary code.

Transversal Gates

Transversal gates for hypergraph product codes of all product dimensions lie in the Clifford group [2].

Cousins

Linear binary code— Higher-dimensional HGP codes are constructed using a hypergraph product of two or more linear binary codes
Quantum expander code— Quantum expander codes have been generalized to hypergraph products of 3 or more expander codes [3].
XYZ product code— The XYZ product code is a non-CSS code constructed using a variant of the hypergraph product of three linear binary codes.
\(D\)-dimensional twisted toric code— The non-twisted \(D\)-dimensional planar (toric) code can be obtained from a hypergraph product of \(D\) repetition (cyclic) codes [1].
Subsystem homological product code— SP codes are projected higher-dimensional HGP codes [4].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Parents

Qubit stabilizer code

Generalized homological-product codeQLDPC Stabilizer Hamiltonian-based QECC Quantum

Higher-dimensional HGP code

Children

Hypergraph product (HGP) code

3D surface code

The 3D planar (3D toric) code can be obtained from a hypergraph product of three repetition (cyclic) codes [1][5; Exam. A.1].

Loop toric code

The 4D loop planar (toric) code can be obtained from a hypergraph product of four repetition (cyclic) codes [1].

References

[1]: W. Zeng and L. P. Pryadko, “Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates”, Physical Review Letters 122, (2019) arXiv:1810.01519 DOI
[2]: X. Fu, H. Zheng, Z. Li, and Z.-W. Liu, “No-go theorems for logical gates on product quantum codes”, (2025) arXiv:2507.16797
[3]: L. Golowich, K. Chang, and G. Zhu, “Constant-Overhead Addressable Gates via Single-Shot Code Switching”, (2025) arXiv:2510.06760
[4]: W. Zeng and L. P. Pryadko, “Minimal distances for certain quantum product codes and tensor products of chain complexes”, Physical Review A 102, (2020) arXiv:2007.12152 DOI
[5]: L. Berent, T. Hillmann, J. Eisert, R. Wille, and J. Roffe, “Analog Information Decoding of Bosonic Quantum Low-Density Parity-Check Codes”, PRX Quantum 5, (2024) arXiv:2311.01328 DOI

Page edit log

Victor V. Albert (2025-10-15) — most recent

Cite as:

“Higher-dimensional HGP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/multisector_hypergraph

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/homological/balanced_product/hypergraph/multisector_hypergraph.yml.