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\([[2(M+1),M]]\) single-loss AD code[1]

Description

A member of a class of \([[2(M+1),M]]\) CSS codes for \(M\geq 1\) that generalizes the \([[4,1]]\) approximate amplitude-damping code of Ref. [2]. The stabilizer group is generated by an all-\(X\) Pauli string together with \(M+1\) pairwise \(Z\)-checks acting on disjoint qubit pairs [1].

Protection

The \([[2(M+1),M]]\) family is designed to correct a single AD error [1].

Cousin

  • \([[4,2,2]]\) Four-qubit code— The \([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) (approximately) corrects a single AD error [2] and is the smallest member of the amplitude-damping stabilizer family of Ref. [1].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Qubit CSS code
Self-complementary qubit codeAmplitude-damping (AD) AQECC Qubit QECC Quantum
The \([[2(M+1),M]]\) single-loss AD code is self-complementary, with the all-\(X\) stabilizer enforcing the complement-pair structure of their codewords [1].
\([[2(M+1),M]]\) single-loss AD code

References

[1]
A. S. Fletcher, P. W. Shor, and M. Z. Win, “Channel-Adapted Quantum Error Correction for the Amplitude Damping Channel”, (2007) arXiv:0710.1052
[2]
D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y. Yamamoto, “Approximate quantum error correction can lead to better codes”, Physical Review A 56, 2567 (1997) arXiv:quant-ph/9704002 DOI
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Zoo Code ID: ampdamp_stabilizer

Cite as:
“\([[2(M+1),M]]\) single-loss AD code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/ampdamp_stabilizer
BibTeX:
@incollection{eczoo_ampdamp_stabilizer, title={\([[2(M+1),M]]\) single-loss AD code}, booktitle={The Error Correction Zoo}, year={2026}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/ampdamp_stabilizer} }
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Permanent link:
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Cite as:

“\([[2(M+1),M]]\) single-loss AD code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/ampdamp_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/ampdamp_stabilizer.yml.