XYZ product code[1,2] 

Description

A non-CSS QLDPC code constructed from three classical codes. The construction of an XYZ product code is similar to that of a hypergraph product code and related 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.

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].

Parents

Children

  • Chamon model code — 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].
  • 3D surface code — The 3D planar (3D toric) code can be obtained from a hypergraph product of three repetition (cyclic) codes [4; Exam. A.1], but done in a different way than the Chamon code; see [2; Sec. 3.4].

Cousin

  • Hypergraph product (HGP) code — Hypergraph (XYZ) product codes are constructed out of hypergraph products of two (three) classical linear codes.

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]
S. Yang and R. Calderbank, “Spatially-Coupled QDLPC Codes”, (2023) arXiv:2305.00137
[4]
L. Berent et al., “Analog information decoding of bosonic quantum LDPC codes”, (2023) arXiv:2311.01328
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Zoo Code ID: xyz_product

Cite as:
“XYZ product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/xyz_product
BibTeX:
@incollection{eczoo_xyz_product, title={XYZ product code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/xyz_product} }
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Cite as:

“XYZ product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/xyz_product

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