Overhead and noise threshold of fault-tolerant quantum error correction

By Andrew M. Steane. 2002

333 citations recorded by the source corpus.

Fault-tolerant quantum error correction (QEC) networks are studied by a combination of numerical and approximate analytical treatments. The probability of failure of the recovery operation is calculated for a variety of Calderbank-Shor-Steane codes, including large block codes and concatenated codes. Recent insights into the syndrome extraction process, which render the whole process more efficient and more noise tolerant, are incorporated. The average number of recoveries that can be completed without failure is thus estimated as a function of various parameters. The main parameters are the gate $\ensuremath{\gamma}$ and memory $\ensuremath{\epsilon}$ failure rates, the physical scale-up of the computer size, and the time ${t}_{m}$ required for measurements and classical processing. The achievable computation size is given as a surface in parameter space. This indicates the noise threshold as well as other information. It is found that concatenated codes based on the $[[23,1,7]]$ Golay code give higher thresholds than those based on the $[[7,1,3]]$ Hamming code under most conditions. The threshold gate noise ${\ensuremath{\gamma}}_{0}$ is a function of $\ensuremath{\epsilon}/\ensuremath{\gamma}$ and ${t}_{m};$ example values are ${\ensuremath{\epsilon}/\ensuremath{\gamma}{,t}_{m},{\ensuremath{\gamma}}_{0}}={{1,1,10}^{\ensuremath{-}3}},$ ${0.01,1,3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}},$ ${{1,100,10}^{\ensuremath{-}4}},$ ${0.01,100,2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}},$ assuming zero cost for information transport. This represents an order of magnitude increase in tolerated memory noise, compared with previous calculations, which is made possible by recent insights into the fault-tolerant QEC process.

Subfields: Artificial Intelligence, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/c85d6e6f0dbc001d02c3eac60de3f4dc6cea27d1

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Citations in the corpus

Linear optical quantum computing with photonic qubits (2005, methodology): “General fault-tolerant thresholds for quantum computing have been derived by Steane (2003) and Knill (2005), and here we address LOQC specific error correction and fault-tolerant thresholds.”
Fault-tolerant measurement-based quantum computing with continuous-variable cluster states. (2013, background): “, concatenated codes) vary, with typical thresholds being 10−6 [25–28], 10−4 [31], and 3 × 10−3 [40]—and up to a few percent with postselection [29].”
NMR techniques for quantum control and computation (2004, background): “Many assumptions are made in obtaining pth, such as the availability of local, fast, parallel classical control resources, but the generally accepted theoretical optimal value of pth is about 10 −4 (126; 132), with optimistic estimates ranging as high as 10 with additional restrictions (133).”
Quantum computing with realistically noisy devices (2004, methodology): “This effect can be quantified by comparing theC4/C6 architecture to that of Steane using the illustrative examp le atγ ≈ 10 worked out in [9].”
Materials challenges and opportunities for quantum computing hardware (2021, background): “Threshold, error scaling, and overhead requirements depend strongly on choice of code ( 31 ) and on the physical nature of the error.”
A coherent all-electrical interface between polar molecules and mesoscopic superconducting resonators (2006, methodology): “By analogy with detailed analysis carried out for ion trap approache”
Quantum accuracy threshold for concatenated distance-3 codes (2005, background): “It would be interesting to analyze other codes that might conceivably yield a higher threshold, such as the [[23,1,7]] Golay code [12], or polynomial codes [42].”
Superconducting nanocircuits for topologically protected qubits (2008, background): “For implementation of quantum correction codes, the decoherence time of a qubit, τd, should be at least 10(4) times longer than the time of a single operation, τ0 [1,2].”
Quantum error correction for quantum memories (2013, neutral): “Other studies of coding overhead for several families of codes were undertaken by Steane (2003) and Cross, DiVincenzo, and Terhal (2009).”