Permutation-invariant code[1] 

Description

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 \(S_n\).

Dicke states: For \(n\)-qubit block codes, an often used basis for the \(n/2\)-dimensional permutation-invariant subspace consists of the Dicke states \(|D^n_w\rangle\) -- normalized permutation-invariant states \(w\) excitations, i.e., a normalized sum over all basis elements with \(w\) ones and \(n - w\) zeroes. Each Dicke state in the code can be shifted by adding a shift \(s\) to both \(n\) and \(w\).

Protection

Permutation invariant codes of distance \(d\) can protect against \(d-1\) deletion errors [2,3], i.e., erasures of qubits at unknown locations.

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

Encoding

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].

Decoding

Schur-Weyl-transform based decoder for qubit permutation-invariant codes [10]. Here, one first measures the total angular momentum of consecutive pairs of qubits, and then its projection modulo some spacing. Recovery can be performed by applying geometric phase gates [11] and the quantum Schur transform.

Notes

Can be constructed using real polynomials for high-dimensional qudit spaces [12].

Parent

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

Children

Cousin

References

[1]
H. Pollatsek and M. B. Ruskai, “Permutationally Invariant Codes for Quantum Error Correction”, (2004) arXiv:quant-ph/0304153
[2]
Y. Ouyang, “Permutation-invariant quantum coding for quantum deletion channels”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021) arXiv:2102.02494 DOI
[3]
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
[4]
Y. Ouyang, “Permutation-invariant quantum codes”, Physical Review A 90, (2014) arXiv:1302.3247 DOI
[5]
Y. Ouyang and J. Fitzsimons, “Permutation-invariant codes encoding more than one qubit”, Physical Review A 93, (2016) DOI
[6]
Y. Ouyang and R. Chao, “Permutation-Invariant Constant-Excitation Quantum Codes for Amplitude Damping”, IEEE Transactions on Information Theory 66, 2921 (2020) DOI
[7]
C. Wu et al., “Initializing a permutation-invariant quantum error-correction code”, Physical Review A 99, (2019) DOI
[8]
A. Bärtschi and S. Eidenbenz, “Deterministic Preparation of Dicke States”, Fundamentals of Computation Theory 126 (2019) arXiv:1904.07358 DOI
[9]
M. T. Johnsson et al., “Geometric Pathway to Scalable Quantum Sensing”, Physical Review Letters 125, (2020) arXiv:1908.01120 DOI
[10]
Y. Ouyang and G. K. Brennen, “Quantum error correction on symmetric quantum sensors”, (2023) arXiv:2212.06285
[11]
X. Wang and P. Zanardi, “Simulation of many-body interactions by conditional geometric phases”, Physical Review A 65, (2002) arXiv:quant-ph/0111017 DOI
[12]
Y. Ouyang, “Permutation-invariant qudit codes from polynomials”, Linear Algebra and its Applications 532, 43 (2017) DOI
[13]
E. Kubischta and I. Teixeira, “The Not-So-Secret Fourth Parameter of Quantum Codes”, (2023) arXiv:2310.17652
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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. https://errorcorrectionzoo.org/c/permutation_invariant
BibTeX:
@incollection{eczoo_permutation_invariant, title={Permutation-invariant code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/permutation_invariant} }
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“Permutation-invariant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/permutation_invariant

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/block/symmetric/permutation_invariant.yml.