Description
A CE code designed to detect and correct AD 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
- Qubit code
- Amplitude-damping (AD) code — Jump codes are designed to protect against qubit AD noise.
- Constant-excitation (CE) code
Cousins
- \([[4,2,2]]\) Four-qubit code — The subcode \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [1] is a \(((4,2,1))_2\) jump code correcting a single AD 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 — 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, Th. Beth, Ch. Charnes, A. Delgado, M. Grassl, and M. Mussinger, “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, Th. Beth, Ch. Charnes, A. Delgado, M. Grassl, and M. Mussinger, “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, C. Charnes, M. Grassl, G. Alber, A. Delgado, and M. Mussinger, 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
Page edit log
- Victor V. Albert (2024-05-11) — most recent
Cite as:
“Jump code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/jump