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 those of the underlying chain complexes. Hypergraph products of multiple classical codes were considered first [1]. Homological products of CSS-code and classical chain complexes are used as a building block in Hastings’ weight-reduction construction [2], while later work studied explicit product-code families built from length-two chain complexes [3] (which correspond to qubit CSS codes per the qubit CSS-to-homology correspondence).
Ref. [4] gives explicit parameter formulas for the tensor product \(\mathcal{A}\times\mathcal{K}(P)\) of an \(m\)-complex \(\mathcal{A}\) with the 1-complex \(\mathcal{K}(P)\) built from an \(r\times c\) binary matrix \(P\). Let \(u=\mathrm{rank}(P)\) and \(\delta\) be the minimum distance of the binary code with parity-check matrix \(P\). Writing \(n_j\), \(k_j\), \(d_j\) for the block length, logical dimension, and homological distance of the \(j\)-th sector of \(\mathcal{A}\), and primes for the corresponding parameters of the product, the Künneth theorem gives \(n_j'=n_{j-1}c+n_j r\) and \(k_j'=k_{j-1}(c-u)+k_j(r-u)\). The main result of [4; Thm. 1] establishes tight bounds on the homological distance: \(d_j'=\min(d_j,d_{j-1}\delta)\) when \(r>u\) (P does not have full row rank), and \(d_j'=d_{j-1}\delta\) when \(r=u\) (P has full row rank).
Rate
A notable special case uses a single full-row-rank \(r\times c\) seed matrix \(P\) (so \(r=u\), \(\kappa=c-r\) logical bits per block, classical distance \(\delta\), row/column weights bounded by \((\omega,\upsilon)\)). Taking \(a\) copies of \(\mathcal{K}(P)\) and \(b\) copies of the dual 1-complex \(\tilde{\mathcal{K}}=\mathcal{K}(P^T)\) yields the \((a+b)\)-complex \(\mathcal{K}^{(a,b)}=\mathcal{K}^{\times a}\times \tilde{\mathcal{K}}^{\times b}\), which has a single non-trivial homology sector \(H_a\) with \begin{align} n_a&=\sum_{i=0}^{a} c^{2i} r^{a+b-2i} {a\choose i}{b\choose i} < (r+c)^{a+b}~,\tag*{(1)}\\ k_a&=\kappa^{a+b}~, \tag*{(2)}\end{align} yielding a quantum CSS code with \(d_X=\delta^a\), \(d_Z=\delta^b\), and stabilizer-generator weights bounded by \((a+b)\max(\omega,\upsilon)\) [4]. The parameters \(a\) and \(b\) can be chosen independently to trade off \(X\)- and \(Z\)-distance, enabling asymmetric codes optimized for channels with unequal bit-flip and phase-flip rates. For asymptotically good LDPC seed-code families, these higher-dimensional HGP families have finite rate and \(d_X d_Z\) scaling linearly with block length.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].Fault Tolerance
For \((a,b)\) constructions with \(a,b>1\), the check matrices \(G_X=K_a\) and \(G_Z=K_{a+1}^T\) satisfy many linear relations coming from neighboring boundary maps; these redundant checks can be used to handle syndrome-measurement errors in repeated-measurement schemes [4].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].
- Asymmetric quantum code (AQC)— The \((a,b)\)-complex construction yields asymmetric CSS codes with \(d_X=\delta^a\) and \(d_Z=\delta^b\), allowing independent tuning of the two distances for channels with unequal bit-flip and phase-flip rates [4].
- 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 three-fold variant of the hypergraph product built from three classical linear binary codes [8].
- 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][9; 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 [10].
Primary Hierarchy
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]
- A. Leverrier, S. Apers, and C. Vuillot, “Quantum XYZ Product Codes”, Quantum 6, 766 (2022) arXiv:2011.09746 DOI
- [9]
- 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
- [10]
- 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
- [11]
- 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