High Threshold Error Correction for the Surface Code

An algorithm is presented for error correction in the surface code quantum memory. This is shown to correct depolarizing noise up to a threshold error rate of 18.5%, exceeding previous results and coming close to the upper bound of 18.9%. The time complexity of the algorithm is found to be polynomial with error suppression, allowing efficient error correction for codes of realistic sizes.

Figure 1

The spin lattice on which an $L\times L$ planar code is defined, with $s$ plaquettes shown in blue and $p$ plaquettes in white. A spin-$1/2$ particle resides on each vertex. The linear size $L$ is characterized by the number of $s$ plaquettes along each side, with $L=4$ in this case.

Figure 2

Results for the first variant of the algorithm are presented in (a), (b), (c), and (d). Those for the second are in (e) and (f). Both were run using the nearest odd integer to $L$ for $N_c$, ten metropolis iterations per step, $ϵ=0.1$ and $\mathrm{TOPS}=10$. The first used $\mathrm{SEQ}=2$ and the second used $\mathrm{SEQ}=10$. (a) and (e) show plots of the logical bit flip error rate after correction, $P$, against linear system size, $L$. Each point was averaged over ${10}^4$ samples. Fit lines for the threshold values are shown as a guide to the eye. (b) and (f) show plots of $T$, the number of steps required by the algorithm before convergence, against $L$. Since only the order of magnitude of the run time is important, the number of samples averaged over was reduced to between 10 and 100, allowing higher system sizes to be probed. The results in (c) and (d) are both for the first variant, and show what $T$ and $L$ are required to achieve a logical error rate $P$.