Combinatorial PI code

Combinatorial PI code[1]

Alternative names: AAB code.

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 \((g\geq 2t+1,\epsilon=+)\).

Transversal Gates

A class of combinatorial PI codes called \((b,g,1)\)-codes has been identified that admits logical gates in the diagonal Clifford hierarchy from transversal \(Z\)-axis rotations [2].

Notes

See Quantum News and Views article [3].

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] (see also [2]).
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].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

PI qubit codePI Cyclic quantum QECC Quantum

Parents

PI qubit code

Combinatorial PI code

Children

\(((7,2,3))\) Pollatsek-Ruskai code

The Pollatsek-Ruskai code is equivalent to the \(Q_{2,1,2,-}\) combinatorial PI code [1; Sec. 5.2]. It is a seven-qubit PI code that realizes gates from the binary icosahedral group transversally.

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, Y. Jing, and G. K. Brennen, “Measurement-free code-switching for low overhead quantum computation using permutation invariant codes”, (2025) arXiv:2411.13142
[3]: 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

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