Threshold error rates for the toric and planar codes

Abstract

The planar code scheme for quantum computation features a 2d array of nearest-neighbor coupled qubits yet claims a threshold error rate approaching 1\%. This result was obtained for the toric code, from which the planar code is derived, and surpasses all other known codes restricted to 2d nearest-neighbor architectures by several orders of magnitude. We describe in detail an error correction procedure for the toric and planar codes, which is based on polynomial-time graph matching techniques and is efficiently implementable as the classical feed-forward processing step in a real quantum computer. By applying one and two qubit depolarizing errors of equal probability $p$, we determine the threshold error rates for the two codes (differing only in their boundary conditions) for both ideal and non-ideal syndrome extraction scenarios. We verify that the toric code has an asymptotic threshold of $p_{\textrm{th}} = 15.5\%$ under ideal syndrome extraction, and $p_{\textrm{th}} = 7.8 \times 10^{-3}$ for the non-ideal case, in agreement with \cite{Raus07d}. Simulations of the planar code indicate that the threshold is close to that of the toric code.