Concatenated Steane code[1,2] 

Description

A member of the family of \([[7^m,1,3^m]]\) CSS codes, each of which is a recursive level-\(m\) concatenatenation of the Steane code. This family is one of the first to admit a concatenated threshold [15].

Protection

Code performance against general Pauli channels has been worked out [6,7].

Decoding

There exist fault-tolerant syndrome extraction protocols for the concatenated Steane code [8].Randomized compiling helps reduce logical error rate for some noise models [9].

Fault Tolerance

There exist fault-tolerant syndrome extraction protocols for the concatenated Steane code [8].

Code Capacity Threshold

This family is one of the first to admit a concatenated threshold [15].

Threshold

Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [10]; see also [11].A measurement threshold of one [12].

Parents

Child

Cousin

References

[1]
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
[2]
A. M. Steane, “Efficient fault-tolerant quantum computing”, Nature 399, 124 (1999) arXiv:quant-ph/9809054 DOI
[3]
A. M. Steane, “Overhead and noise threshold of fault-tolerant quantum error correction”, Physical Review A 68, (2003) arXiv:quant-ph/0207119 DOI
[4]
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
[5]
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
[6]
B. Rahn, A. C. Doherty, and H. Mabuchi, “Exact and Approximate Performance of Concatenated Quantum Codes”, (2001) arXiv:quant-ph/0111003
[7]
B. Rahn, A. C. Doherty, and H. Mabuchi, “Exact performance of concatenated quantum codes”, Physical Review A 66, (2002) arXiv:quant-ph/0206061 DOI
[8]
B. Pato, T. Tansuwannont, and K. R. Brown, “Concatenated Steane code with single-flag syndrome checks”, (2024) arXiv:2403.09978
[9]
A. Jain et al., “Improved quantum error correction with randomized compiling”, Physical Review Research 5, (2023) arXiv:2303.06846 DOI
[10]
A. W. Cross, D. P. DiVincenzo, and B. M. Terhal, “A comparative code study for quantum fault-tolerance”, (2009) arXiv:0711.1556
[11]
B. W. Reichardt, “Improved ancilla preparation scheme increases fault-tolerant threshold”, (2004) arXiv:quant-ph/0406025
[12]
D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
[13]
Z. W. E. Evans et al., “Error correction optimisation in the presence of X/Z asymmetry”, (2007) arXiv:0709.3875
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Zoo Code ID: concatenated_steane

Cite as:
“Concatenated Steane code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/concatenated_steane
BibTeX:
@incollection{eczoo_concatenated_steane, title={Concatenated Steane code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/concatenated_steane} }
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Cite as:

“Concatenated Steane code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/concatenated_steane

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/concatenated/concatenated_steane.yml.