Permutation-invariant code[1]


A code \(C\) constructed in a physical space consisting of a tensor product of \(n\) identical subsystems (e.g., qubits, modular qudits, or Galois qudits) such that any permutation of the subsystems leaves any codeword invariant. In other words, the automorphism group of the code contains the symmetric group \(S_n\).


Depends on the family. The GNU permutation-invariant family (parameterized by \(t\)) protects against arbitrary weight \(t\) qubit errors and approximately corrects spontaneous decay errors [2][3]. Other related codes protect against amplitude damping [4] while admitting a constant number of excitations, and against deletion errors [5][6].


With quantum harmonic oscillators (superconducting charge qubits in a ultrastrong coupling regime) in \(O(N)\) as in [7]. Can be done in \(O(N^2)\) steps using quantum circuits [8], or using geometric phase gates in \(O(N)\) [9].


For a family of codes, using projection, probability amplitude rebalancing, and gate teleportation can be done in \(O(N^2)\) [5].


Can be constructed using real polynomials for high-dimensional qudit spaces [10].


  • Cyclic quantum code — The cyclic group of these codes is a subgroup of the \(S_n\) symmetric group used in permutation invariant codes.




H. Pollatsek and M. B. Ruskai, “Permutationally Invariant Codes for Quantum Error Correction”, (2004) arXiv:quant-ph/0304153
Y. Ouyang, “Permutation-invariant quantum codes”, Physical Review A 90, (2014) arXiv:1302.3247 DOI
Y. Ouyang and J. Fitzsimons, “Permutation-invariant codes encoding more than one qubit”, Physical Review A 93, (2016) DOI
Y. Ouyang and R. Chao, “Permutation-Invariant Constant-Excitation Quantum Codes for Amplitude Damping”, IEEE Transactions on Information Theory 66, 2921 (2020) DOI
Y. Ouyang, “Permutation-invariant quantum coding for quantum deletion channels”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021) arXiv:2102.02494 DOI
T. Shibayama and M. Hagiwara, “Permutation-Invariant Quantum Codes for Deletion Errors”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021) arXiv:2102.03015 DOI
C. Wu et al., “Initializing a permutation-invariant quantum error-correction code”, Physical Review A 99, (2019) DOI
A. Bärtschi and S. Eidenbenz, “Deterministic Preparation of Dicke States”, Fundamentals of Computation Theory 126 (2019) arXiv:1904.07358 DOI
M. T. Johnsson et al., “Geometric Pathway to Scalable Quantum Sensing”, Physical Review Letters 125, (2020) arXiv:1908.01120 DOI
Y. Ouyang, “Permutation-invariant qudit codes from polynomials”, Linear Algebra and its Applications 532, 43 (2017) DOI
Page edit log

Your contribution is welcome!

on (edit & pull request)

edit on this site

Zoo Code ID: permutation_invariant

Cite as:
“Permutation-invariant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_permutation_invariant, title={Permutation-invariant code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:

Cite as:

“Permutation-invariant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.