Error threshold of toric code state under the simultaneous occurrence of incoherent and coherent errors

Quantum error correction codes are robust against various errors within a certain threshold of error strength. However, obtaining the actual error threshold is usually tricky when different types of errors occur simultaneously. Recent machine learning estimation of error-threshold under both types of errors in the paradigmatic toric code state [1] invites more rigorous and systematic studies. In this work, we study such decodability transitions through a rigorous mapping to a classical statistical mechanics model. Specifically, we map the probability distribution of the stabilizer measurement outcome from a noisy toric code to that associated with a classical random bond Ashkin-Teller model with correlated bond disorders. We then conduct Monte Carlo simulations of the classical model to observe various ordering transitions as a function of error rates. This enables us to rigorously obtain error thresholds under the co-occurrence of both types of noise through the phase boundary between the paramagnetic phase and various ordered phases. We determine the nature of the phase transitions across the error threshold through finite-size scaling.



[1] Kim, H., Zhou, Y., Xu, Y., Varma, K., Karamlou, A.H., Rosen, I.T., Hoke, J.C., Wan, C., Zhou, J.P., Oliver, W.D. and Lensky, Y.D., 2024. Attention to Quantum Complexity. arXiv preprint arXiv:2405.11632.