Four-qubit single-deletion code[1,2] 

Description

Four-qubit PI code that is the smallest qubit code to correct one deletion error.

In terms of Dicke states, a basis of logical codewords is \begin{align} \begin{split} |0_{L}\rangle&=\frac{1}{\sqrt{2}}\left(|D_{0}^{4}\rangle+|D_{4}^{4}\rangle\right)\\ |1_{L}\rangle&=|D_{2}^{4}\rangle~. \end{split} \tag*{(1)}\end{align}

Protection

The smallest qubit code to correct one deletion error.

Parents

Cousins

  • Binomial code — The four-qubit single-deletion code can be obtained from the "0-2-4" single-mode binomial code by substituting Fock states with Dicke states.
  • \([[4,2,2]]\) CSS code — A basis of codewords for the four-qubit single-deletion code consists of the \(|\overline{00}\rangle\) and \(|\overline{01}\rangle+|\overline{10}\rangle+|\overline{11}\rangle\)states of the four-qubit code.
  • Combinatorial PI code — The combinatorial PI code \(Q_{1,1,1,-}\) is another example of a four-qubit code correcting a single deletion error [4; Sec. 5.1].

References

[1]
M. Hagiwara and A. Nakayama, “A Four-Qubits Code that is a Quantum Deletion Error-Correcting Code with the Optimal Length”, (2020) arXiv:2001.08405
[2]
A. Nakayama and M. Hagiwara, “Single Quantum Deletion Error-Correcting Codes”, (2020) arXiv:2004.00814
[3]
Y. Ouyang, “Permutation-invariant quantum coding for quantum deletion channels”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021) arXiv:2102.02494 DOI
[4]
A. Aydin, M. A. Alekseyev, and A. Barg, “A family of permutationally invariant quantum codes”, Quantum 8, 1321 (2024) arXiv:2310.05358 DOI
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Zoo Code ID: four_qubit_permutation_invariant

Cite as:
“Four-qubit single-deletion code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/four_qubit_permutation_invariant
BibTeX:
@incollection{eczoo_four_qubit_permutation_invariant, title={Four-qubit single-deletion code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/four_qubit_permutation_invariant} }
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Cite as:

“Four-qubit single-deletion code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/four_qubit_permutation_invariant

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/permutation_invariant/gnu/four_qubit_permutation_invariant.yml.