Steane code

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

{\begin{bmatrix}H&0\\0&H\end{bmatrix}}

where H is the parity-check matrix of the Hamming code and is given by

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 $[[7,1,3]]$ Steane code is the first in the family of quantum Hamming codes, codes with parameters $[[2^{r}-1,2^{r}-1-2r,3]]$ for integers $r\geq 3$ . It is also a quantum color code.