Alternative names: Higher-dimensional tensor product code, Multi-sector homological product code, Multi-sector tensor product code, Iterative homological product code, Iterative tensor product code.
Description
A qubit CSS code formulated using a tensor product of two or more chain complexes, each of length one or greater. The number of chain complexes participating in the product is the dimension of the code. When all chain complexes are length-one, meaning that they correspond to classical codes, the code is called a higher-dimensional HGP code (a.k.a. multi-sector HGP code or iterative HGP code).
The stabilizer generator matrices of an \(m\)-dimensional homological 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. There is freedom in choosing which 2-dimensional chain complex to pick for the code.
Protection
The Künneth formula gives code properties in terms of the properties of the underlying chain complexes. Hypergraph products of multiple classical codes were considered first [1], followed by products of length-two chain complexes [2,3] (which correspond to qubit CSS codes per the qubit CSS-to-homology correspondence). A general theory and distance bounds are formulated in Ref. [4].Transversal Gates
Transversal gates for multi-dimensional HGP codes of all dimensions lie in the Clifford group [5].Gates
Gates in the Clifford hierarchy can be implemented via constant-depth circuits [5].Parallel Pauli product measurements via homomorphic CNOT gates for 3- and 4-dimensional HGP codes [6].Cousins
- Linear binary code— \(D\)-dimensional HGP codes are constructed using a hypergraph product of \(D\) linear binary codes.
- Cyclic linear binary code— Higher-dimensional homological-product codes can be constructed out of CSS codes that in turn stem from cyclic codes [1; Sec. 4.2].
- Quantum Reed-Muller (RM) code— Higher-dimensional homological-product codes can be constructed out of quantum RM codes [1; Sec. 4.3].
- Quasi-cyclic LDPC (QC-LDPC) code— Higher-dimensional hypergraph-product codes can be constructed out of QC-LDPC codes [6; Table III].
- Quantum expander code— Quantum expander codes have been generalized to hypergraph products of 3 or more expander codes [7].
- 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.
- Finite-geometry (FG) qubit QLDPC code— Multi-dimensional homological products of PG-QLDPC codes yield families whose stabilizer-generator weights scale logarithmically with \(n\) [1; Cor. 2.23][1; Sec. 4.1].
- 3D surface code— The 3D planar and toric code on a cubic lattice can be obtained from a hypergraph product of three repetition codes [4][8; Exam. A.1].
- \((1,3)\) 4D toric code— The 4D \((1,3)\) planar (toric) code on a hypercubic lattice can be obtained from a particular choice of chain complex from a hypergraph product of four repetition codes [4].
- \((2,2)\) Loop toric code— The 4D loop planar (toric) code on a hypercubic lattice can be obtained from a particular choice of chain complex from a hypergraph product of four repetition codes [4].
- \(D\)-dimensional twisted toric code— The non-twisted \(D\)-dimensional planar and toric codes on a hypercubic lattice can be obtained from a hypergraph product of \(D\) repetition codes [4].
- Subsystem homological product code— SP codes are projected higher-dimensional HGP codes [9].
Primary Hierarchy
Generalized homological-product qubit CSS codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Higher-dimensional homological product code
Children
Tensor-product HDX codes result from iterated homological products of length-two chain complexes (i.e., quantum codes) based on Ramanujan complexes [10].
Multi-dimensional homological products of two length-two chain complexes reduce to homological product codes.
Multi-dimensional homological products of two length-one chain complexes reduce to HGP codes.
References
- [1]
- B. Audoux and A. Couvreur, “On tensor products of CSS Codes”, (2018) arXiv:1512.07081
- [2]
- M. B. Hastings, “Weight Reduction for Quantum Codes”, (2016) arXiv:1611.03790
- [3]
- E. T. Campbell, “A theory of single-shot error correction for adversarial noise”, Quantum Science and Technology 4, 025006 (2019) arXiv:1805.09271 DOI
- [4]
- W. Zeng and L. P. Pryadko, “Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates”, Physical Review Letters 122, (2019) arXiv:1810.01519 DOI
- [5]
- E. X. Fu, H. Zheng, Z. Li, and Z.-W. Liu, “No-go theorems for logical gates on product quantum codes”, (2025) arXiv:2507.16797
- [6]
- Q. Xu, H. Zhou, G. Zheng, D. Bluvstein, J. P. B. Ataides, M. D. Lukin, and L. Jiang, “Fast and Parallelizable Logical Computation with Homological Product Codes”, (2024) arXiv:2407.18490
- [7]
- L. Golowich, K. Chang, and G. Zhu, “Constant-Overhead Addressable Gates via Single-Shot Code Switching”, (2025) arXiv:2510.06760
- [8]
- 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
- [9]
- 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
- [10]
- N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021) arXiv:2103.06309 DOI
Page edit log
- Victor V. Albert (2025-10-15) — most recent
- Nathanan Tantivasadakarn (2025-10-15)
- Feroz Ahmed Mian (2025-10-15)
Cite as:
“Higher-dimensional homological product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/multisector_hypergraph