Description
A non-CSS QLDPC code constructed from a particular hypergraph product of three linear binary codes. The idea is that rather than taking a product of only two classical codes to produce a CSS code, a third classical code is considered, acting with Pauli-\(Y\) operators. When the underlying classical codes are 1D (e.g., repetition codes), the XYZ product yields a 3D code. Higher dimensional versions have been constructed [3].Rate
Not much has been proven about the relationship between XYZ product codes and other codes. The logical dimension depends on properties of the input classical codes, specifically similarity invariants from abstract algebra. It is conjectured that specific instances of XYZ product codes have a constant encoding rate and a minimum distance of \(d \in \Theta(n^{2/3})\) [2].Cousins
- Linear binary code— The XYZ product code is constructed using a particular hypergraph product of three linear binary codes
- Asymmetric quantum code— XYZ product codes can be used to protect against biased noise [3].
- Hypergraph product (HGP) code— Hypergraph (XYZ) product codes are constructed out of hypergraph products of two (three) classical linear codes.
Primary Hierarchy
Parents
XYZ product codes result from a tensor product of three classical-code chain complexes.
Quantum spatially coupled (SC-QLDPC) codeLattice stabilizer QLDPC Stabilizer Hamiltonian-based Qubit QECC Quantum
XYZ product stabilizer generator matrices can be used as sub-matrices to define a 2D SC-QLDPC code [4].
XYZ product code
Children
The Chamon model code can be obtained from a hypergraph product of three repetition codes [1], but done in a different way than the 3D surface code; see [2; Sec. 3.4].
The 3D planar (3D toric) code can be obtained from a hypergraph product of three repetition (cyclic) codes [5; Exam. A.1], but done in a different way than the Chamon code; see [2; Sec. 3.4].
References
- [1]
- Maurice, Denise. Codes correcteurs quantiques pouvant se décoder itérativement. Diss. Université Pierre et Marie Curie-Paris VI, 2014.
- [2]
- A. Leverrier, S. Apers, and C. Vuillot, “Quantum XYZ Product Codes”, Quantum 6, 766 (2022) arXiv:2011.09746 DOI
- [3]
- Z. Liang, Z. Yi, F. Yang, J. Chen, Z. Wang, and X. Wang, “High-dimensional quantum XYZ product codes for biased noise”, (2024) arXiv:2408.03123
- [4]
- S. Yang and R. Calderbank, “Spatially-Coupled QLDPC Codes”, (2025) arXiv:2305.00137
- [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
- Finnegan Voichick (2021-12-01) — most recent
Cite as:
“XYZ product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/xyz_product