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

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

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

Encoding

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].

Gates

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].

Realizations

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].

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

[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
[2]: H. Hao, “Investigations on Automorphism Groups of Quantum Stabilizer Codes”, (2021) arXiv:2109.12735
[3]: M. Nakahara, “Quantum Computing”, (2008) DOI
[4]: P. W. Shor, “Fault-tolerant quantum computation”, (1997) arXiv:quant-ph/9605011
[5]: B. Zeng, A. Cross, and I. L. Chuang, “Transversality versus Universality for Additive Quantum Codes”, (2007) arXiv:0706.1382
[6]: T. J. Yoder, R. Takagi, and I. L. Chuang, “Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes”, Physical Review X 6, (2016) arXiv:1603.03948 DOI
[7]: T. J. Yoder and I. H. Kim, “The surface code with a twist”, Quantum 1, 2 (2017) arXiv:1612.04795 DOI
[8]: R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
[9]: 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
[10]: D. Bhatnagar et al., “Low-Depth Flag-Style Syndrome Extraction for Small Quantum Error-Correction Codes”, (2023) arXiv:2305.00784
[11]: D. Nigg et al., “Quantum computations on a topologically encoded qubit”, Science 345, 302 (2014) arXiv:1403.5426 DOI
[12]: C. Ryan-Anderson et al., “Realization of real-time fault-tolerant quantum error correction”, (2021) arXiv:2107.07505
[13]: P. Ball, “Real-Time Error Correction for Quantum Computing”, Physics 14, (2021) DOI
[14]: L. Postler et al., “Demonstration of fault-tolerant universal quantum gate operations”, Nature 605, 675 (2022) arXiv:2111.12654 DOI
[15]: C. Ryan-Anderson et al., “Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code”, (2022) arXiv:2208.01863
[16]: D. Bluvstein et al., “A quantum processor based on coherent transport of entangled atom arrays”, Nature 604, 451 (2022) arXiv:2112.03923 DOI

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

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