\([[7,1,3]]\) 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]. It is one of sixteen distinct \([[7,1,3]]\) codes [3].

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 ([4], Fig. 10.14).

Transversal Gates

All single-qubit Clifford gates, which realize the \(2O\) binary octahedral subgroup of \(SU(2)\) [5,6].

Gates

Fault-tolerant logical zero and magic state preparation [7]. Magic-state preparation converts unbiased noise into biased noise [8].Pieceable fault-tolerant CCZ gate [9].

Fault Tolerance

Fault-tolerant logical zero and magic state preparation [7]. Magic-state preparation converts unbiased noise into biased noise [8].Pieceable fault-tolerant CCZ gate [9].Syndrome measurement can be done with ancillary flag qubits [10,11] or with no extra qubits [12]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [13].

Realizations

Trapped-ion qubits: seven-qubit device in Blatt group [14]. Ten-qubit QCCD device by Quantinuum [15] realizing repeated syndrome extraction, real-time look-up-table decoding (yielding lower logical SPAM error rate than physical SPAM), and non-fault-tolerant magic-state distillation (see APS Physics Synopsis [16]). Fault-tolerant universal two-qubit gate set using T injection by Monz group [17]. Logical CNOT gate and Bell-pair creation between two logical qubits (yielding a logical fidelity higher than physical), including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 20-qubit device by Quantinuum [18]; logical fidelity interval of the combined preparation-CNOT-measurement procedure was higher than that of the unencoded physical qubits. Multiple rounds of Steane error correction [19].Rydberg atom arrays: Lukin group [20]; transversal CNOT gate performed on distance \(3\), \(5\), and \(7\) codes, logical ten-qubit GHZ state initialized with break-even fidelity, fault-tolerant logical two-qubit GHZ state initialized [21].

Parents

Cousins

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]
S. Yu, Q. Chen, and C. H. Oh, “Graphical Quantum Error-Correcting Codes”, (2007) arXiv:0709.1780
[4]
M. Nakahara, “Quantum Computing”, (2008) DOI
[5]
P. W. Shor, “Fault-tolerant quantum computation”, (1997) arXiv:quant-ph/9605011
[6]
B. Zeng, A. Cross, and I. L. Chuang, “Transversality versus Universality for Additive Quantum Codes”, (2007) arXiv:0706.1382
[7]
H. Goto, “Minimizing resource overheads for fault-tolerant preparation of encoded states of the Steane code”, Scientific Reports 6, (2016) DOI
[8]
N. Fazio, R. Harper, and S. Bartlett, “Logical Noise Bias in Magic State Injection”, (2024) arXiv:2401.10982
[9]
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
[10]
T. J. Yoder and I. H. Kim, “The surface code with a twist”, Quantum 1, 2 (2017) arXiv:1612.04795 DOI
[11]
R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
[12]
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
[13]
D. Bhatnagar et al., “Low-Depth Flag-Style Syndrome Extraction for Small Quantum Error-Correction Codes”, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) (2023) arXiv:2305.00784 DOI
[14]
D. Nigg et al., “Quantum computations on a topologically encoded qubit”, Science 345, 302 (2014) arXiv:1403.5426 DOI
[15]
C. Ryan-Anderson et al., “Realization of real-time fault-tolerant quantum error correction”, (2021) arXiv:2107.07505
[16]
P. Ball, “Real-Time Error Correction for Quantum Computing”, Physics 14, (2021) DOI
[17]
L. Postler et al., “Demonstration of fault-tolerant universal quantum gate operations”, Nature 605, 675 (2022) arXiv:2111.12654 DOI
[18]
C. Ryan-Anderson et al., “Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code”, (2022) arXiv:2208.01863
[19]
L. Postler et al., “Demonstration of fault-tolerant Steane quantum error correction”, (2023) arXiv:2312.09745
[20]
D. Bluvstein et al., “A quantum processor based on coherent transport of entangled atom arrays”, Nature 604, 451 (2022) arXiv:2112.03923 DOI
[21]
D. Bluvstein et al., “Logical quantum processor based on reconfigurable atom arrays”, Nature (2023) arXiv:2312.03982 DOI
[22]
A. Denys and A. Leverrier, “Multimode bosonic cat codes with an easily implementable universal gate set”, (2023) arXiv:2306.11621
[23]
E. Knill, R. Laflamme, and W. H. Zurek, “Resilient quantum computation: error models and thresholds”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 365 (1998) arXiv:quant-ph/9702058 DOI
[24]
A. M. Steane, “Efficient fault-tolerant quantum computing”, Nature 399, 124 (1999) arXiv:quant-ph/9809054 DOI
[25]
A. M. Steane, “Overhead and noise threshold of fault-tolerant quantum error correction”, Physical Review A 68, (2003) arXiv:quant-ph/0207119 DOI
[26]
K. M. Svore, B. M. Terhal, and D. P. DiVincenzo, “Local fault-tolerant quantum computation”, Physical Review A 72, (2005) arXiv:quant-ph/0410047 DOI
[27]
K. M. Svore, D. P. DiVincenzo, and B. M. Terhal, “Noise Threshold for a Fault-Tolerant Two-Dimensional Lattice Architecture”, (2006) arXiv:quant-ph/0604090
[28]
A. Jain et al., “Improved quantum error correction with randomized compiling”, Physical Review Research 5, (2023) arXiv:2303.06846 DOI
[29]
D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
[30]
S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
[31]
J. Hu, Q. Liang, and R. Calderbank, “Divisible Codes for Quantum Computation”, (2022) arXiv:2204.13176
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

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
BibTeX:
@incollection{eczoo_steane, title={\([[7,1,3]]\) Steane code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/steane} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/steane

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/edit/main/codes/quantum/qubits/small_distance/small/steane.yml.