\([[7,1,3]]\) Steane code[1]
Description
A \([[7,1,3]]\) CSS code that is the smallest qubit CSS code to correct a single-qubit error. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors.
The parity-check matrix for the \([7,4,3]\) Hamming code is \begin{align} H = \left(\begin{matrix} 0&0&0&1&1&1&1\\ 0&1&1&0&0&1&1\\ 1&0&1&0&1&0&1 \end{matrix}\right), \tag*{(1)}\end{align} and the check matrix for the Steane code is therefore \begin{align} \left(\begin{matrix} 0&H\\ H&0 \end{matrix}\right). \tag*{(2)}\end{align} The stabilizer group for the Steane code has six generators. Logical codewords are \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{\sqrt{8}}\Big(|0000000\rangle+|1010101\rangle+|0110011\rangle+|1100110\rangle\\&\,\,\,\,\,\,\,\,+|0001111\rangle+|1011010\rangle+|0111100\rangle+|1101001\rangle\Big)\\|\overline{1}\rangle&=\frac{1}{\sqrt{8}}\Big(|1111111\rangle+|0101010\rangle+|1001100\rangle+|0011001\rangle\\&\,\,\,\,\,\,\,\,+|1110000\rangle+|0100101\rangle+|1000011\rangle+|0010110\rangle\Big)~. \end{split} \tag*{(3)}\end{align} The automorphism group of the code is \(PGL(3,2)\) [2].
Protection
Encoding
Transversal Gates
Gates
Fault Tolerance
Realizations
Parents
- Color code — Steane code is the smallest 2D color code.
- \([[2^r-1, 1, 3]]\) quantum Reed-Muller code
- \([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code
- \([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured Reed-Muller code
Cousins
- \([7,4,3]\) Hamming code — The Steane code is constructed from the \([7,4,3]\) classical Hamming code.
- Quantum divisible code — A fault-tolerant \(T\) gate on the Steane code can be obtained by concatenating with particular quantum divisible codes.
References
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- [15]
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Page edit log
- Victor V. Albert (2022-08-04) — most recent
- Victor V. Albert (2022-03-14)
- Joseph T. Iosue (2021-12-19)
Cite as:
“\([[7,1,3]]\) Steane code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/steane