\([[7,1,3]]\) Steane code[1] 

Description

A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [2]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors.

The code's stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both the parity-check matrix for the \([7,4,3]\) Hamming code, \begin{align} H_{X} = H_{Z} = \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}

The stabilizer group for the Steane code has six generators, three \(X\)-type and three \(Z\)-type, which can be thought of as lying on the three trapezoids of the following tiling of the triangle. Figure I.

Figure I: Stabilizer generators of the Steane code.

The Steane code can also be thought of as a code on all corners of a cube except one [3].

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*{(2)}\end{align}

The automorphism group of the code is \(PGL(3,2)\) [4]. It is one of sixteen distinct \([[7,1,3]]\) codes [5].

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 ([6], Fig. 10.14).Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity [7].

Transversal Gates

Gates

Fault-tolerant logical zero and magic state preparation [10]. Magic-state preparation converts unbiased noise into biased noise [11].Pieceable fault-tolerant CCZ gate [12].

Decoding

Shor error correction fidelity calculation [13,14].

Fault Tolerance

A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [1518].A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [19].Fault-tolerant logical zero and magic state preparation [10]. Magic-state preparation converts unbiased noise into biased noise [11].Fault-tolerant logical zero and logical plus state preparation on all-to-all and 2D grid qubit connectivity [7].Pieceable fault-tolerant CCZ gate [12].Syndrome measurement can be done with ancillary flag qubits [20,21] or with no extra qubits [22]. The depth of syndrome extraction circuits can be lowered by using past syndrome values [23].

Realizations

Trapped-ion devices: seven-qubit device in Blatt group [24]. Ten-qubit QCCD device by Quantinuum [25] 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 [26]). Fault-tolerant universal two-qubit gate set using T injection by Monz group [27]. 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 [28]; 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 [29]. Fault-tolerant universal gate set via code switching between the Steane code and the \([[10,1,2]]\) code [19]. Post-selected fault-tolerant logical Bell-state preparation with logical error rates at least 10 times lower than physical rate on a device by Quantinuum [30]. The quantum Fourier transform on three code blocks [31]. Fault-tolerant transversal and lattice-surgery state teleportation protocols as well as Knill error correction [32]. Rains shadow enumerators have been measured [33].Rydberg atom arrays: Lukin group [34]; ten logical qubits, transversal CNOT gate performed, logical ten-qubit GHZ state initialized with break-even fidelity, and fault-tolerant logical two-qubit GHZ state initialized [35].

Parents

Cousins

  • Cyclic quantum code — The Steane code is equivalent to a cyclic code via qubit permutations [41; Exam. 1].
  • Codeword stabilized (CWS) code — The Steane code is equivalent via a single-qubit Clifford unitary to a CWS code for a particular graph and classical code [41; Exam. 4].
  • Graph quantum code — Four non-isomorphic graphs yield graph quantum codes that are equivalent to the Steane code under a single-qubit-Clifford circuit [42].
  • EA qubit stabilizer code — The Steane code is globally equivalent to a \([[6,1,3;1]]\) code, which is the smallest EA CSS code with that distance [2].
  • \([7,4,3]\) Hamming code — The Steane code is constructed from the \([7,4,3]\) classical Hamming code via the CSS construction.
  • Concatenated cat code — Two-component cat codes concatenated with Steane and Golay codes are estimated to be fault tolerant against photon loss noise with rate \(\eta < 5\times 10^{-4}\) provided that \(\alpha > 1.2\) [43].
  • Tensor-network code — The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [44].
  • \(((7,2,3))\) Pollatsek-Ruskai code — The Pollatsek-Ruskai code can be continuously deformed to the Steane code [45].
  • \([[10,1,2]]\) CSS code — A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[10,1,2]]\) code [19].
  • \([[4,2,2]]\) Four-qubit code — The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [44].
  • \([[5,1,2]]\) rotated surface code — The \([[5,1,2]]\) rotated surface code can be obtained by morphing the Steane code [46].
  • Quantum divisible code — A fault-tolerant \(T\) gate on the Steane code can be obtained by concatenating with particular quantum divisible codes [47].
  • Raussendorf-Bravyi-Harrington (RBH) cluster-state code — Concatenation of the RBH code with small codes such as the \([[2,1,1]]\) repetition code, \([[4,1,1,2]]\) subsystem code, or Steane code can improve thresholds [48].
  • \([[15,1,3]]\) quantum Reed-Muller code — The \([[15,1,3]]\) code can be viewed as a (gauge-fixed) doubled color code obtained from the Steane code via the doubling transformation [49]. A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [1518,50]. An \([[105,1,3]]\) alternative concatenation of the \([[15,1,3]]\) and Steane codes allows for a universal gate set consisting of gates that are close to transversal [51,52].

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]
B. Shaw et al., “Encoding one logical qubit into six physical qubits”, Physical Review A 78, (2008) arXiv:0803.1495 DOI
[3]
R. Raussendorf, “Key ideas in quantum error correction”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, 4541 (2012) DOI
[4]
H. Hao, “Investigations on Automorphism Groups of Quantum Stabilizer Codes”, (2021) arXiv:2109.12735
[5]
S. Yu, Q. Chen, and C. H. Oh, “Graphical Quantum Error-Correcting Codes”, (2007) arXiv:0709.1780
[6]
M. Nakahara, “Quantum Computing”, (2008) DOI
[7]
R. Zen et al., “Quantum Circuit Discovery for Fault-Tolerant Logical State Preparation with Reinforcement Learning”, (2024) arXiv:2402.17761
[8]
P. W. Shor, “Fault-tolerant quantum computation”, (1997) arXiv:quant-ph/9605011
[9]
B. Zeng, A. Cross, and I. L. Chuang, “Transversality versus Universality for Additive Quantum Codes”, (2007) arXiv:0706.1382
[10]
H. Goto, “Minimizing resource overheads for fault-tolerant preparation of encoded states of the Steane code”, Scientific Reports 6, (2016) DOI
[11]
N. Fazio, R. Harper, and S. Bartlett, “Logical Noise Bias in Magic State Injection”, (2024) arXiv:2401.10982
[12]
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
[13]
Y. S. Weinstein, “Fidelity of an encoded [7,1,3] logical zero”, Physical Review A 84, (2011) arXiv:1101.1950 DOI
[14]
S. D. Buchbinder, C. L. Huang, and Y. S. Weinstein, “Encoding an Arbitrary State in a [7,1,3] Quantum Error Correction Code”, (2011) arXiv:1109.1714
[15]
A. Paetznick and B. W. Reichardt, “Universal Fault-Tolerant Quantum Computation with Only Transversal Gates and Error Correction”, Physical Review Letters 111, (2013) arXiv:1304.3709 DOI
[16]
J. T. Anderson, G. Duclos-Cianci, and D. Poulin, “Fault-Tolerant Conversion between the Steane and Reed-Muller Quantum Codes”, Physical Review Letters 113, (2014) arXiv:1403.2734 DOI
[17]
D.-X. Quan et al., “Fault-tolerant conversion between adjacent Reed–Muller quantum codes based on gauge fixing”, Journal of Physics A: Mathematical and Theoretical 51, 115305 (2018) arXiv:1703.03860 DOI
[18]
D. Banfield and A. Kay, “Implementing Logical Operators using Code Rewiring”, (2023) arXiv:2210.14074
[19]
I. Pogorelov et al., “Experimental fault-tolerant code switching”, (2024) arXiv:2403.13732
[20]
T. J. Yoder and I. H. Kim, “The surface code with a twist”, Quantum 1, 2 (2017) arXiv:1612.04795 DOI
[21]
R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
[22]
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
[23]
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
[24]
D. Nigg et al., “Quantum computations on a topologically encoded qubit”, Science 345, 302 (2014) arXiv:1403.5426 DOI
[25]
C. Ryan-Anderson et al., “Realization of real-time fault-tolerant quantum error correction”, (2021) arXiv:2107.07505
[26]
P. Ball, “Real-Time Error Correction for Quantum Computing”, Physics 14, (2021) DOI
[27]
L. Postler et al., “Demonstration of fault-tolerant universal quantum gate operations”, Nature 605, 675 (2022) arXiv:2111.12654 DOI
[28]
C. Ryan-Anderson et al., “Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code”, (2022) arXiv:2208.01863
[29]
L. Postler et al., “Demonstration of fault-tolerant Steane quantum error correction”, (2023) arXiv:2312.09745
[30]
M. P. da Silva et al., “Demonstration of logical qubits and repeated error correction with better-than-physical error rates”, (2024) arXiv:2404.02280
[31]
K. Mayer et al., “Benchmarking logical three-qubit quantum Fourier transform encoded in the Steane code on a trapped-ion quantum computer”, (2024) arXiv:2404.08616
[32]
C. Ryan-Anderson et al., “High-fidelity and Fault-tolerant Teleportation of a Logical Qubit using Transversal Gates and Lattice Surgery on a Trapped-ion Quantum Computer”, (2024) arXiv:2404.16728
[33]
D. Miller et al., “Experimental measurement and a physical interpretation of quantum shadow enumerators”, (2024) arXiv:2408.16914
[34]
D. Bluvstein et al., “A quantum processor based on coherent transport of entangled atom arrays”, Nature 604, 451 (2022) arXiv:2112.03923 DOI
[35]
D. Bluvstein et al., “Logical quantum processor based on reconfigurable atom arrays”, Nature (2023) arXiv:2312.03982 DOI
[36]
C.-Y. Lai and C.-C. Lu, “A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices”, IEEE Transactions on Information Theory 57, 7163 (2011) arXiv:0712.0103 DOI
[37]
A. Ashikhmin, C.-Y. Lai, and T. A. Brun, “Quantum Data-Syndrome Codes”, IEEE Journal on Selected Areas in Communications 38, 449 (2020) arXiv:1907.01393 DOI
[38]
Y. Fujiwara, “Ability of stabilizer quantum error correction to protect itself from its own imperfection”, Physical Review A 90, (2014) arXiv:1409.2559 DOI
[39]
B. Audoux and A. Couvreur, “On tensor products of CSS Codes”, (2018) arXiv:1512.07081
[40]
A. Denys and A. Leverrier, “Quantum error-correcting codes with a covariant encoding”, (2024) arXiv:2306.11621
[41]
A. A. Kovalev, I. Dumer, and L. P. Pryadko, “Design of additive quantum codes via the code-word-stabilized framework”, Physical Review A 84, (2011) arXiv:1108.5490 DOI
[42]
M. Grassl, A. Klappenecker, and M. Rotteler, “Graphs, quadratic forms, and quantum codes”, Proceedings IEEE International Symposium on Information Theory, arXiv:quant-ph/0703112 DOI
[43]
A. P. Lund, T. C. Ralph, and H. L. Haselgrove, “Fault-Tolerant Linear Optical Quantum Computing with Small-Amplitude Coherent States”, Physical Review Letters 100, (2008) arXiv:0707.0327 DOI
[44]
C. Cao and B. Lackey, “Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks”, PRX Quantum 3, (2022) arXiv:2109.08158 DOI
[45]
M. Du et al., “Characterizing Quantum Codes via the Coefficients in Knill-Laflamme Conditions”, (2024) arXiv:2410.07983
[46]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[47]
J. Hu, Q. Liang, and R. Calderbank, “Divisible Codes for Quantum Computation”, (2022) arXiv:2204.13176
[48]
Z. Li, I. Kim, and P. Hayden, “Concatenation Schemes for Topological Fault-tolerant Quantum Error Correction”, Quantum 7, 1089 (2023) arXiv:2209.09390 DOI
[49]
S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
[50]
F. Butt et al., “Fault-Tolerant Code-Switching Protocols for Near-Term Quantum Processors”, PRX Quantum 5, (2024) arXiv:2306.17686 DOI
[51]
T. Jochym-O’Connor and R. Laflamme, “Using Concatenated Quantum Codes for Universal Fault-Tolerant Quantum Gates”, Physical Review Letters 112, (2014) arXiv:1309.3310 DOI
[52]
T. Jochym-O’Connor, A. Kubica, and T. J. Yoder, “Disjointness of Stabilizer Codes and Limitations on Fault-Tolerant Logical Gates”, Physical Review X 8, (2018) arXiv:1710.07256 DOI
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.), 2024. https://errorcorrectionzoo.org/c/steane
BibTeX:
@incollection{eczoo_steane, title={\([[7,1,3]]\) Steane code}, booktitle={The Error Correction Zoo}, year={2024}, 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.), 2024. https://errorcorrectionzoo.org/c/steane

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