GNU permutation-invariant code[1]

Description

Can be expressed in terms of Dicke states where the logical states are \begin{align} |\overline{\pm}\rangle = \sum_{\ell=0}^{n} \frac{(\pm 1)^\ell}{\sqrt{2^n}} \sqrt{n \choose \ell} |D^m_{g \ell}\rangle~. \end{align} Here, \(m\) is the number of particles used for encoding \(1\) qubit, and \(g, n \leq m\) are arbitrary positive integers. The state \(|D^m_w\rangle\) is a Dicke state -- a normalized permutation-invariant state on \(m\) qubits with \(w\) excitations, i.e., a normalized sum over all basis elements with \(w\) ones and \(m - w\) zeroes.

A qudit extension of such codes, based on a correspondence with binomial codes, exists [2].

Protection

Depends on the family. One family which is completely symmetrized versions of Bacon-Shor codes (parameterized by \(t\)) protects against arbitrary weight-\(t\) qubit errors. Additionally, codes with large enough length \((t+1)(3t+1)+t\) can approximately correct \(t\) spontaneous decay errors.

Parent

Cousins

Zoo code information

Internal code ID: gnu_permutation_invariant

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: gnu_permutation_invariant

Cite as:
“GNU permutation-invariant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/gnu_permutation_invariant
BibTeX:
@incollection{eczoo_gnu_permutation_invariant, title={GNU permutation-invariant code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/gnu_permutation_invariant} }
Permanent link:
https://errorcorrectionzoo.org/c/gnu_permutation_invariant

References

[1]
Y. Ouyang, “Permutation-invariant quantum codes”, Physical Review A 90, (2014). DOI; 1302.3247
[2]
V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018). DOI; 1708.05010
[3]
Y. Ouyang, “Quantum storage in quantum ferromagnets”, Physical Review B 103, (2021). DOI; 1904.01458

Cite as:

“GNU permutation-invariant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/gnu_permutation_invariant

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/gnu_permutation_invariant.yml.