Jump code[13] 

Description

A CE code designed to detect and correct amplitude damping errors. An \(((n,K))\) jump code is denoted as \(((n,K,t))_w\) (which conflicts with modular-qudit notation), where \(t\) is the maximum number of qubits that can be corrected after each one has undergone a jump error \(|0\rangle\langle 1|\), and where each codeword is a uniform superposition of qubit basis states with Hamming weight \(w\).

Protection

Various code bounds, including an upper bound on \(K\) given the other parameters, are provided in Ref. [3]. For example, one can apply bit flips to all qubits of an \(((n,K,t))_w\) jump code to obtain an \(((n,K,t))_{n-w}\) jump code.

Rate

An infinite family of jump codes asymptotically attains an upper bound on \(K\) [3; Thm. 27].

Notes

Survey of jump codes [4].

Parents

Cousins

  • \([[4,2,2]]\) CSS code — The subcode \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [1] is a \(((4,2,1))_2\) jump code correcting a single amplitude damping error. This code can be extended to a \(((4,3,1))_2\) jump code \(\{|\overline{01}\rangle,|\overline{10}\rangle,|\overline{11}\rangle\}\) [2].
  • Chuang-Leung-Yamamoto (CLY) code — Jump codes can be thought of as qubit analogues of uniform CLY codes.
  • \([[2m,2m-2,2]]\) error-detecting code — The subcode of the \([[2m,2m-2,2]]\) error-detecting code consisting of codewords labeled by weight-\(m\) bitstrings is a \(((2m,\frac{1}{2}{2m \choose m},1))_{m}\) optimal jump code [2][3; Corr. 9].
  • Combinatorial design code — Certain types of combinatorial designs can be used to obtain jump codes [2,3,5].
  • Self-dual linear code — Iso-dual codes can be used to construct jump codes [3].

References

[1]
G. Alber et al., “Stabilizing Distinguishable Qubits against Spontaneous Decay by Detected-Jump Correcting Quantum Codes”, Physical Review Letters 86, 4402 (2001) arXiv:quant-ph/0103042 DOI
[2]
G. Alber et al., “Detected-jump-error-correcting quantum codes, quantum error designs, and quantum computation”, Physical Review A 68, (2003) arXiv:quant-ph/0208140 DOI
[3]
T. Beth et al., Designs, Codes and Cryptography 29, 51 (2003) DOI
[4]
Jimbo, Masakazu, and Keisuke Shiromoto. "Quantum jump codes and related combinatorial designs." Information Security, Coding Theory and Related Combinatorics. IOS Press, 2011. 285-311.
[5]
Y. Lin and M. Jimbo, “Extremal properties of t-SEEDs and recursive constructions”, Designs, Codes and Cryptography 73, 805 (2013) DOI
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Zoo Code ID: jump

Cite as:
“Jump code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/jump
BibTeX:
@incollection{eczoo_jump, title={Jump code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/jump} }
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Cite as:

“Jump code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/jump

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/nonstabilizer/jump.yml.