Error threshold estimates for surface code with loss of qubits

Abstract: We estimate optimal thresholds for surface code in the presence of loss via an analytical method developed in statistical physics. The optimal threshold for the surface code is closely related to a special critical point in a finite-dimensional spin glass, which is disordered magnetic material. We compare our estimations to the heuristic numerical results reported in earlier studies. Further application of our method to the depolarizing channel, a natural generalization of the noise model, unveils its wider robustness even with loss of qubits.