A sampling-based quasi-probability simulation for fault-tolerant quantum error correction on the surface codes under coherent noise.

We propose a sampling-based simulation for fault-tolerant quantum error correction under coherent noise. A mixture of incoherent and coherent noise, possibly due to over-rotation, is decomposed into Clifford channels with a quasi-probability distribution. Then, an unbiased estimator of the logical error probability is constructed by sampling Clifford channels with an appropriate post-processing. We characterize the sampling cost via the channel robustness and find that the proposed sampling-based method is feasible even for planar surface codes with relatively large code distances intractable for full state-vector simulations. As a demonstration, we simulate repetitive faulty syndrome measurements on the planar surface code of distance 5 with 81 qubits. We find that the coherent error increases the logical error rate. This is a practical application of the quasi-probability simulation for a meaningful task and would be useful to explore experimental quantum error correction on the near-term quantum devices.