Threshold error rates for the toric and planar codes (pp0456-0469)
          David S. Wang, Austin G. Fowler, Ashley M. Stephens, and Lloyd C.L. Hollenberg
          doi: https://doi.org/10.26421/QIC10.5-6-6
Abstracts: The planar code scheme for quantum computation features a 2d array of nearest-neighbor coupled qubits yet claims a threshold error rate approaching 1% [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 pth = 15.5% under ideal syndrome extraction, and pth = 7.8 ื 10−3 for the non-ideal case, in agreement with [1]. Simulations of the planar code indicate that the threshold is close to that of the toric code.
Key words: toric, planar codes