Steane code
≈ 3 min read
Names in other languages
No names in other languages are held for this entry yet.
Article bodies are English and are not machine-translated. A name in another language is not a translation of the article.
The Steane code is a tool in quantum error correction introduced by Andrew Steane in 1996. It is a CSS code (Calderbank-Shor-Steane), using the classical binary [7,4,3] Hamming code to correct for both qubit flip errors (X errors) and phase flip errors (Z errors). The Steane code encodes one logical qubit in 7 physical qubits and is able to correct arbitrary single qubit errors. Its 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.
The Steane code is a tool in quantum error correction introduced by Andrew Steane in 1996. It is a CSS code (Calderbank-Shor-Steane), using the classical binary [7,4,3] Hamming code to correct for both qubit flip errors (X errors) and phase flip errors (Z errors). The Steane code encodes one logical qubit in 7 physical qubits and is able to correct arbitrary single qubit errors. Its 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. X_L & = XXXXXXX \\ Z_L & = ZZZZZZZ. \end{align} The logical | 0 \rangle and | 1 \rangle states of the Steane code are \begin{align} | 0 \rangle_L = & \frac{1}{\sqrt{8}} [ | 0000000 \rangle + | 1010101 \rangle + | 0110011 \rangle + | 1100110 \rangle \\ & + | 0001111 \rangle + | 1011010 \rangle + | 0111100 \rangle + | 1101001 \rangle ] \\ | 1 \rangle_L = & X_L | 0 \rangle_L. \end{align} Arbitrary codestates are of the form | \psi \rangle = \alpha | 0 \rangle_L + \beta | 1 \rangle_L . ==References==
Text derived from https://en.wikipedia.org/w/index.php?oldid=1221445652 under CC BY-SA 4.0; modifications have been made.