Error-correction and noise-decoherence thresholds for coherent errors in planar-graph surface codes

Abstract

We numerically study coherent errors in surface codes on planar graphs, focusing on noise of the form of Z or X rotations of individual qubits. We find that, similar to the case of incoherent bit and phase flips, a trade-off between resilience against coherent X and Z rotations can be made via the connectivity of the graph. However, our results indicate that, unlike in the incoherent case, the error-correction thresholds for the various graphs do not approach a universal bound. We also study the distribution of final states after error correction. We show that graphs fall into three distinct classes, each resulting in qualitatively distinct final-state distributions. In particular, we show that a graph class exists where the logical-level noise exhibits a decoherence threshold slightly above the error-correction threshold. In these classes, therefore, the logical level noise above the error-correction threshold can retain a significant amount of coherence even for large-distance codes. To perform our analysis, we develop a Majorana-fermion representation of planar-graph surface codes and describe the characterization of logical-state storage using fermion-linear-optics-based simulations. We thereby generalize the approach introduced for the square lattice by Bravyi et al. [npj Quantum Inf. 4, 55 (2018)10.1038/s41534-018-0106-y] to surface codes on general planar graphs.