Steane \([[7,1,3]]\) code[1]

Description

A \([[7,1,3]]\) CSS code that uses the classical binary \([7,4,3]\) Hamming code for protecting against \(X\) errors and its dual \([7,3,4]\) for \(Z\) errors. The parity-check matrix for the \([7,4,3]\) Hamming code is \begin{align} H = \left(\begin{matrix} 1&0&0&1&0&1&1\\ 0&1&0&1&1&0&1\\ 0&0&1&0&1&1&1 \end{matrix}\right), \end{align} and the check matrix for the Steane code is therefore \begin{align} \left(\begin{matrix} 0&H\\ H&0 \end{matrix}\right). \end{align} The stabilizer group for the Steane code has six generators.

Protection

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

Gates

Pieceable fault-tolerant CCZ gate [2].

Fault Tolerance

Pieceable fault-tolerant CCZ gate [2].Syndrome measurement can be done with ancillary flag qubits [3][4] or with no extra qubits [5].

Realizations

Trapped-ion qubits: seven-qubit device in Blatt group [6], ten-qubit QCCD device by Quantinuum [7], fault-tolerant universal two-qubit gate realized by Monz group [8].Rydberg atom arrays: Lukin group [9].

Parents

Cousins

  • Hamming code — The Steane code is constructed from a 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.

Zoo code information

Internal code ID: steane

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: steane

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

References

[1]
“Multiple-particle interference and quantum error correction”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452, 2551 (1996). DOI; quant-ph/9601029
[2]
T. J. Yoder, R. Takagi, and I. L. Chuang, “Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes”, Physical Review X 6, (2016). DOI; 1603.03948
[3]
T. J. Yoder and I. H. Kim, “The surface code with a twist”, Quantum 1, 2 (2017). DOI; 1612.04795
[4]
R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018). DOI; 1705.02329
[5]
B. W. Reichardt, “Fault-tolerant quantum error correction for Steane’s seven-qubit color code with few or no extra qubits”, Quantum Science and Technology 6, 015007 (2020). DOI
[6]
D. Nigg et al., “Quantum computations on a topologically encoded qubit”, Science 345, 302 (2014). DOI; 1403.5426
[7]
C. Ryan-Anderson et al., “Realization of real-time fault-tolerant quantum error correction”. 2107.07505
[8]
Lukas Postler et al., “Demonstration of fault-tolerant universal quantum gate operations”. 2111.12654
[9]
Dolev Bluvstein et al., “A quantum processor based on coherent transport of entangled atom arrays”. 2112.03923

Cite as:

“Steane \([[7,1,3]]\) 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/steane.yml.