Description
A non-CSS QLDPC code obtained from a three-fold variant of the hypergraph product applied to three classical binary codes with parity-check matrices \(H_1,H_2,H_3\). Unlike CSS three-fold hypergraph product codes, the third input code acts through Pauli-\(Y\) checks [2]. Under mild assumptions, the code dimension is determined by a tensor Sylvester equation over \(\mathbb{F}_2\), and the minimum-distance problem reduces up to constant factors to how closely a related inhomogeneous tensor Sylvester equation can be satisfied [2]. When the underlying classical codes are repetition codes, the construction yields the Chamon model code.Protection
The natural logical operators are membrane-like two-dimensional objects rather than strings [2]. For cyclic XYZ product codes, fractal operators yield large-weight errors with constant-weight syndromes, ruling out local testability [2].Rate
The logical dimension depends on properties of the input classical codes, specifically similarity invariants of the matrices \(H_i H_i^T\). 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 a non-CSS three-fold variant of the hypergraph product built from three classical linear binary codes [2].
- Higher-dimensional homological product code— The XYZ product code is a non-CSS three-fold variant of the hypergraph product built from three classical linear binary codes [2].
- Asymmetric quantum code (AQC)— 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
Quantum spatially coupled (SC-QLDPC) codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
Parents
XYZ product stabilizer generator matrices can be used as sub-matrices to define a 2D SC-QLDPC code [4].
The XYZ product code is a non-CSS three-fold variant of the hypergraph product built from three classical linear binary codes [2].
XYZ product code
Children
The Chamon model code can be obtained from an XYZ product of three repetition codes [1], in a construction different from the 3D surface code; see [2; Sec. 3.4].
References
- [1]
- D. Maurice. Codes correcteurs quantiques pouvant se décoder itérativement. PhD thesis, 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”, (2025) arXiv:2408.03123
- [4]
- S. Yang and R. Calderbank, “Spatially-Coupled QLDPC Codes”, Quantum 9, 1693 (2025) arXiv:2305.00137 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