\([[7,1,3]]\) Steane code[1] 


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].


The Steane code is a distance 3 code. It detects errors on 2 qubits, corrects errors on 1 qubit.


Nine CNOT and four Hadamard gates ([3], Fig. 10.14).

Transversal Gates

All single-qubit Clifford gates, which realize the \(2O\) binary octahedral subgroup of \(SU(2)\) [4,5].


Pieceable fault-tolerant CCZ gate [6].

Fault Tolerance

Pieceable fault-tolerant CCZ gate [6].Syndrome measurement can be done with ancillary flag qubits [7,8] or with no extra qubits [9]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [10].


Trapped-ion qubits: seven-qubit device in Blatt group [11], ten-qubit QCCD device by Quantinuum [12] (see APS Physics Synopsys [13]). Fault-tolerant universal two-qubit gate set by Monz group [14]. Logical CNOT gate between two logical qubits, including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 20-qubit device by Quantinuum [15]; logical fidelity interval of the combined preparation-CNOT-measurement procedure was higher than that of the unencoded physical qubits.Rydberg atom arrays: Lukin group [16].



  • \([7,4,3]\) 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.


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Zoo Code ID: steane

Cite as:
\([[7,1,3]]\) Steane code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/steane
  title={\([[7,1,3]]\) Steane code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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\([[7,1,3]]\) Steane code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/steane

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/small_distance/small/steane.yml.