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 PI code to correct one deletion error.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.
- \([[4,2,2]]\) Four-qubit code— Projecting the four-qubit code into the PI subspace yields the four-qubit single-deletion 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 [3; Sec. 5.1].
- \(((3,3,2))_{3}\) Three-qutrit single-deletion code— The four-qubit (three-qutrit) single-deletion code is the smallest PI qubit (qutrit) code to correct one deletion error.
Primary Hierarchy
Parents
The four-qubit single-deletion code is a GNU code for \(g=m=2\) [4].
\(((n,2,2))\) Bravyi-Lee-Li-Yoshida PI codeQubit PI Cyclic quantum Small-distance block quantum Hamiltonian-based QECC Quantum
The Bravyi-Lee-Li-Yoshida code reduces to the four-qubit single-deletion code for \(n=4\).
\(((4,2,2))\) Four-qubit single-deletion code
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]
- A. Aydin, M. A. Alekseyev, and A. Barg, “A family of permutationally invariant quantum codes”, Quantum 8, 1321 (2024) arXiv:2310.05358 DOI
- [4]
- Y. Ouyang, “Permutation-invariant quantum coding for quantum deletion channels”, 2021 IEEE International Symposium on Information Theory (ISIT) 1499 (2021) arXiv:2102.02494 DOI
Page edit log
- Victor V. Albert (2023-04-18) — most recent
Cite as:
“\(((4,2,2))\) Four-qubit single-deletion code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/four_qubit_permutation_invariant