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 $\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}(n)$ bond dimension suffice for fixed absolute Hilbert-Schmidt error after $\text{order}1$ depth, while relative error demands $\text{order}\log n$ 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 $\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}(n)$ bond dimension at all depths. (3) For general single-qubit noise in 1D brickwall circuits, random two-design gates with contraction coefficient $c<1\mathrm{/}3$ yield an $\text{order}1$ OEE plateau with probability $1-T{e}^{-\mathrm{\Omega }(n)}$, while arbitrary gates with $c<1\mathrm{/}48$ give $\text{order}\log n$ OEE in the worst case. (4) In higher dimensions, these bounds yield uniform-in-depth $\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}(n)$ 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.