Concatenated Steane code

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 [1–5].

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 [1–5,10]; see the book [11].

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”, Physical Review A 110, (2024) arXiv:2403.09978 DOI
[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

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