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 self-dual [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

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

References

• Steane, Andrew (1996). "Multiple-Particle Interference and Quantum Error Correction". Proc. Roy. Soc. Lond. A 452 (1954): 2551–2577. doi:10.1098/rspa.1996.0136.