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 [1–5].
Protection
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
Threshold
Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [12]; see also [13].A measurement threshold of one [14].
Parents
- Qubit CSS code
- Concatenated qubit code
- Heptagon holographic code — A recursively concatenated Steane code is a heptagon holographic code on a tree tensor network.
Child
- \([[7,1,3]]\) Steane code — The concatenated Steane code at level \(m=1\) is the Steane code.
Cousins
- Asymmetric quantum code — Concatenating while taking into account noise bias can reduce resource overhead [15].
- \([[15,1,3]]\) quantum Reed-Muller code — The \([[105,1]]\) concatenation of the \([[15,1,3]]\) and Steane codes allows for a universal gate set consisting of gates that are transversal w.r.t. to two different partitions [16,17].
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]
- P. Aliferis, D. Gottesman, and J. Preskill, “Quantum accuracy threshold for concatenated distance-3 codes”, (2005) arXiv:quant-ph/0504218
- [11]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [12]
- A. W. Cross, D. P. DiVincenzo, and B. M. Terhal, “A comparative code study for quantum fault-tolerance”, (2009) arXiv:0711.1556
- [13]
- B. W. Reichardt, “Improved ancilla preparation scheme increases fault-tolerant threshold”, (2004) arXiv:quant-ph/0406025
- [14]
- D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
- [15]
- Z. W. E. Evans et al., “Error correction optimisation in the presence of X/Z asymmetry”, (2007) arXiv:0709.3875
- [16]
- 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
- [17]
- 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
- Victor V. Albert (2024-03-26) — most recent
Cite as:
“Concatenated Steane code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/concatenated_steane