Overhead and noise threshold of fault-tolerant quantum error correction

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 $\gamma$ and memory $\epsilon$ 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 $\left[\right[23,1,7\left]\right]$ Golay code give higher thresholds than those based on the $\left[\right[7,1,3\left]\right]$ Hamming code under most conditions. The threshold gate noise ${\gamma }_0$ is a function of $\epsilon /\gamma$ and $t_m;$ example values are $\{\epsilon /\gamma {,t}_m,{\gamma }_0\}=\{{1,1,10}^{-3}\},$ $\{0.01,1,3\times {10}^{-3}\},$ $\{{1,100,10}^{-4}\},$ $\{0.01,100,2\times {10}^{-3}\},$ 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.