\([[7,1,3]]\) Steane code[1]
Description
A \([[7,1,3]]\) CSS code that uses 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} 1&0&0&1&0&1&1\\ 0&1&0&1&1&0&1\\ 0&0&1&0&1&1&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
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].
Realizations
Trapped-ion qubits: seven-qubit device in Blatt group [10], ten-qubit QCCD device by Quantinuum [11] (see APS Physics Synopsys [12]). Fault-tolerant universal two-qubit gate set by Monz group [13]. 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 [14]; logical fidelity interval of the combined preparation-CNOT-measurement procedure was higher than that of the unencoded physical qubits.Rydberg atom arrays: Lukin group [15].
Parents
- \([[2^r-1, 1, 3]]\) 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
- Color code — Steane code is the smallest 2D color 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. Nigg et al., “Quantum computations on a topologically encoded qubit”, Science 345, 302 (2014) arXiv:1403.5426 DOI
- [11]
- C. Ryan-Anderson et al., “Realization of real-time fault-tolerant quantum error correction”, (2021) arXiv:2107.07505
- [12]
- P. Ball, “Real-Time Error Correction for Quantum Computing”, Physics 14, (2021) DOI
- [13]
- L. Postler et al., “Demonstration of fault-tolerant universal quantum gate operations”, Nature 605, 675 (2022) arXiv:2111.12654 DOI
- [14]
- C. Ryan-Anderson et al., “Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code”, (2022) arXiv:2208.01863
- [15]
- 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