Error Tolerance in Constructing Cluster States

We consider the problem often encountered in constructing large cluster states, that is, the δ_x measurements in the depolarizing noise, and present a regulation to calculate the error propagation and the accumulation. Rohde et al. first pointed out the question A quantitative analysis is given on the disadvantage influence of the probabilistic gate and noisy environment on the construction of two-dimensional (2D) scalable cluster states. Positivity of partial transpose criteria is used to predict whether the final state is distillable or not. A critical value on the error tolerance in constructing cluster states is determined, which could be used as a test-bed for successful construction of scalable 2D cluster states. Furthermore, our quantitative analysis would also be helpful to find optimal approach to suppress the noise accumulation.