Steane code

The Steane Code is a perfect CSS code. It uses the classical binary [7,4,3] Hamming code to correct for qubit flip errors (X errors) and the dual of the Hamming code, the [7,3,3] 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

In particular,

$H={\begin{bmatrix}1&0&0&1&0&1&1\\0&1&0&1&1&0&1\\0&0&1&0&1&1&1\end{bmatrix}}$