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 . Other related codes protect against amplitude damping  while admitting a constant number of excitations, and against deletion errors .
With quantum harmonic oscillators (superconducting charge qubits in a ultrastrong coupling regime) in \(O(N)\) as in . Can be done in \(O(N^2)\) steps using quantum circuits , or using geometric phase gates in \(O(N)\) .
For a family of codes, using projection, probability amplitude rebalancing, and gate teleportation can be done in \(O(N^2)\) .
Can be constructed using real polynomials for high-dimensional qudit spaces .
- Quantum cyclic code — The cyclic group of these codes is a subgroup of the \(S_n\) symmetric group used in permutation invariant codes.
Zoo code information
- 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
“Permutation-invariant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/permutation_invariant