Classical-product code[1,2] 

Description

A CSS code constructed by separately constructing the \(X\) and \(Z\) check matrices using product constructions from classical codes. A particular \([[512,174,8]]\) code performed well [2] against erasure and depolarizing noise when compared to other notable CSS codes, such as the asymptotically good quantum Tanner codes.

For example, for \(H_i^x\), \(H_i^z\) being the \(X\) and \(Z\) check matrices of CSS codes \(C_i\) with \(i\in\{1,2,3,4\}\), the 2-fold symmetric classical product code is given by \begin{align} H_{\otimes}^x &:=\left(\begin{array}{c} H_1^x \otimes H_2^x \otimes I \otimes I \\ I \otimes I \otimes H_3^x \otimes H_4^x \end{array}\right) \tag*{(1)}\\ H_{\otimes}^z &:=\left(\begin{array}{c} H_1^z \otimes I \otimes H_3^z \otimes I \\ I \otimes H_2^z \otimes I \otimes H_4^z \end{array}\right)~. \tag*{(2)}\end{align}

Parents

Cousins

  • Quantum Tanner code — A \([[512,174,8]]\) classical-product code performed well [2] against erasure and depolarizing noise when compared to a member of an asymptotically good quantum Tanner code family.
  • Single parity-check (SPC) code — SPC codes are used as component codes in classical-product code constructions.
  • Tensor-product code — Tensor-product codes are utilized in classical-product code constructions.

References

[1]
M. Hivadi, “On quantum SPC product codes”, Quantum Information Processing 17, (2018) DOI
[2]
D. Ostrev et al., “Classical product code constructions for quantum Calderbank-Shor-Steane codes”, (2022) arXiv:2209.13474
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Zoo Code ID: classical_product

Cite as:
“Classical-product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/classical_product
BibTeX:
@incollection{eczoo_classical_product, title={Classical-product code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/classical_product} }
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“Classical-product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/classical_product

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