Permutation-invariant code[1] 


Block quantum code 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
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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.
  title={Permutation-invariant code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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Cite as:

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