{
  "id": "enwiki-001221445652",
  "slug": "steane-code",
  "title": "Steane code",
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  "lead": "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.",
  "body": "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==",
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    "title": "Steane code",
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        "family": "Anticloud FZ LLE",
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    "container-title": "encyclopegia.org",
    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/steane-code",
    "issued": {
      "date-parts": [
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          2026,
          11,
          13
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