The Steane code is a tool in quantum error correction introduced by Andrew Steane in 1996. It is a perfect CSS code (Calderbank-Shor-Steane), using the classical binary [7,4,3] Hamming code to correct for qubit flip errors (X errors) and the dual of the Hamming code, the [7,3,4] code, to correct for phase flip errors (Z errors). The Steane code is able to correct arbitrary single qubit errors.
In the stabilizer formalism, the Steane code has 6 generators, and the check matrix in standard form is
\( {\displaystyle {\begin{bmatrix}H&0\\0&H\end{bmatrix}}} \)
where H is the parity-check matrix of the Hamming code and is given by
\( {\displaystyle H={\begin{bmatrix}1&0&0&1&0&1&1\\0&1&0&1&1&0&1\\0&0&1&0&1&1&1\end{bmatrix}}.} \)
The \( {\displaystyle [[7,1,3]]} \)Steane code is the first in the family of quantum Hamming codes, codes with parameters \( {\displaystyle [[2^{r}-1,2^{r}-1-2r,3]]} \) for integers \( {\displaystyle r\geq 3} \). It is also a quantum color code.
References
Steane, Andrew (1996). "Multiple-Particle Interference and Quantum Error Correction". Proc. Roy. Soc. Lond. A. 452 (1954): 2551–2577. arXiv:quant-ph/9601029. Bibcode:1996RSPSA.452.2551S. doi:10.1098/rspa.1996.0136.
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