Permutation-invariant code[1]


Codes which are stabilized by the symmetric group \(S_n\) on \(n\) elements, in a generalization of stabilizer codes to binary codes utilizing (non-abelian) group actions (in particular, \(S_n\) is non-abelian).


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].




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

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Internal code ID: permutation_invariant

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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={} }
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Harriet Pollatsek and Mary Beth Ruskai, “Permutationally Invariant Codes for Quantum Error Correction”. quant-ph/0304153
Y. Ouyang, “Permutation-invariant quantum codes”, Physical Review A 90, (2014). DOI; 1302.3247
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; 1809.09801
Y. Ouyang, “Permutation-invariant quantum coding for quantum deletion channels”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021). DOI; 2102.02494
T. Shibayama and M. Hagiwara, “Permutation-Invariant Quantum Codes for Deletion Errors”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021). DOI; 2102.03015
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). DOI; 1904.07358
M. T. Johnsson et al., “Geometric Pathway to Scalable Quantum Sensing”, Physical Review Letters 125, (2020). DOI; 1908.01120
Y. Ouyang, “Permutation-invariant qudit codes from polynomials”, Linear Algebra and its Applications 532, 43 (2017). DOI

Cite as:

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