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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].

Cousins

Primary Hierarchy

Parents
Combinatorial PI code
Children
The Pollatsek-Ruskai code is equivalent to the \(Q_{2,1,2,-}\) combinatorial PI code [1; Sec. 5.2].

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

Cite as:

“Combinatorial PI code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/combinatorial_permutation_invariant

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