Permutation-invariant (PI) 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 \(S_n\).


PI codes of distance \(d\) can protect against \(d-1\) deletion errors [25], i.e., erasures of subsystems at unknown locations.

Other protection depends on the code family. The GNU PI family (parameterized by \(t\)) protects against arbitrary weight \(t\) qubit errors and approximately corrects spontaneous decay errors [6,7]. Other related codes protect against amplitude damping [8] while admitting a constant number of excitations.


With quantum harmonic oscillators (superconducting charge qubits in a ultrastrong coupling regime) in \(O(N)\) as in [9]. Can be done in \(O(N^2)\) steps using quantum circuits [10], or using geometric phase gates in \(O(N)\) [11].


PI codes can be constructed using real polynomials for high-dimensional qudit spaces [12].Qubit and qudit PI codes obtained from numerical optimization routines are useful for entanglement distillation [13; Appx. B.1].


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



  • Editing code — PI codes of distance \(d\) can protect against \(d-1\) (quantum) deletion errors.


H. Pollatsek and M. B. Ruskai, “Permutationally Invariant Codes for Quantum Error Correction”, (2004) arXiv:quant-ph/0304153
M. Hagiwara and A. Nakayama, “A Four-Qubits Code that is a Quantum Deletion Error-Correcting Code with the Optimal Length”, (2020) arXiv:2001.08405
A. Nakayama and M. Hagiwara, “Single Quantum Deletion Error-Correcting Codes”, (2020) arXiv:2004.00814
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
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
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
B. Desef and M. B. Plenio, “Optimizing quantum codes with an application to the loss channel with partial erasure information”, Quantum 6, 667 (2022) arXiv:2105.13233 DOI
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Zoo Code ID: permutation_invariant

Cite as:
“Permutation-invariant (PI) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_permutation_invariant, title={Permutation-invariant (PI) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Permutation-invariant (PI) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.