Complexity of tensor network simulation for noisy quantum circuits

We aim to rigorously address how local noise affects classical simulability of quantum dynamics benchmarked by tensor-network methods. Using operator entanglement entropy (OEE), we prove the following: (1) For single-qubit depolarizing noise on arbitrary circuits, tensor networks with poly(n) bond dimension suffice for fixed absolute Hilbert-Schmidt error after order{1} depth, while relative error demands order{log n} depth; and this bound is optimal. (2) For single-qubit depolarizing noise on 1D local circuits, the existence of whole-trajectory error-bounded matrix product operator (MPO) of poly(n) bond dimension at all depths. (3) For general single-qubit noise in 1D brickwall circuits, random two-design gates with contraction coefficient c<1/3 yield an order{1} OEE plateau with probability 1-Te^-Ω(n), while arbitrary gates with c<1/48 give order{log n} OEE in the worst case. (4) In higher dimensions, these bounds yield uniform-in-depth poly(n) average boundary-bond dimensions for projected entangled pair operators (PEPO) across every cut - under depolarizing noise at either absolute or relative accuracy, and under general noise with strong contraction at absolute accuracy. Our results establish a rigorous connection between certain noise models, circuit types, and their classical simulability.