Combinatorial PI code[1]
Description
A member of a family of PI quantum codes whose correction properties are derived from solving a family of combinatorial identities. The code encodes one logical qubit in superpositions of Dicke states whose coefficients are square roots of ratios of binomial coefficients.
Protection
A code \(Q_{g,m,\delta,\epsilon}\) is defined for nonnegative integers \(g\), \(m\), and \(\delta\) as well as a sign \(\epsilon\) [1; Construction 5.1]. The number of qubits is \(n = 2g+m+\delta+1\). The code corrects errors on up to \(t\) qubits for \(m\geq t\), \(\delta\geq 2t\), and either \((g\geq 2t,\epsilon=-)\) or \((g\geq 2t+1,\epsilon=+)\).
Notes
See Quantum News and Views article [2].
Parent
Child
- \(((7,2,3))\) Pollatsek-Ruskai code — The Pollatsek-Ruskai code is equivalent to the \(Q_{2,1,2,-}\) combinatorial PI code [1; Sec. 5.2].
Cousins
- Binary dihedral PI code — The \(Q_{3,1,2m-4,+}\) and \(Q_{3,1,2^m-4,+}\) combinatorial PI codes reduce to the \(((2m+3,2,3))\) and \(((2^{m-1}+3,2,3))\) binary dihedral PI codes, respectively [1; Prop. 5.6].
- Four-qubit single-deletion code — The combinatorial PI code \(Q_{1,1,1,-}\) is another example of a four-qubit code correcting a single deletion error [1; Sec. 5.1].
- GNU PI code — Combinatorial PI codes \(Q_{g,(m-1)/2,g-1,+}\) are GNU codes for odd \(m\) [1; Prop. 5.4].
References
- [1]
- A. Aydin, M. A. Alekseyev, and A. Barg, “A family of permutationally invariant quantum codes”, Quantum 8, 1321 (2024) arXiv:2310.05358 DOI
- [2]
- Y. Ouyang, “A new take on permutation-invariant quantum codes”, Quantum Views 8, 80 (2024) DOI
Page edit log
- Victor V. Albert (2024-04-18) — most recent
Cite as:
“Combinatorial PI code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/combinatorial_permutation_invariant