Measurement-Induced Entanglement in Noisy 2D Random Circuits

We study measurement-induced entanglement (MIE) generated by column-by-column sampling of noisy 2D random circuits of size N and depth T . Focusing primarily on Clifford circuits and using the operator entanglement S op of the sampling-induced boundary state as a proxy for computational complexity, first, we reproduce in the noiseless limit a finite-depth transition from area- to volume-law scaling at a threshold depth T c = 6 . In contrast, in the presence of single-qubit depolarizing noise at any constant rate p > 0 , we find that the operator entanglement S op obeys an area law, with its maximum value scaling approximately linearly with T / p in the regime T > T c . By analyzing the spatial distribution of stabilizer generators, we observe exponential localization of stabilizer generators; these both account for the scaling of the maximal S op and imply an exponential decay of conditional mutual information across buffered tripartitions, which we also confirm numerically. Together, these results indicate that constant local noise destroys long-range MIE in 2D random Clifford circuits, and that a tensor-network-based algorithm can efficiently sample from noisy 2D random Clifford circuits (i) at sub-logarithmic depths T = o ( log N ) for any constant noise rate p = Ω ( 1 ) , and (ii) at constant depths T = O ( 1 ) for noise rates p = Ω ( log − 1 N ) . Finally, we turn to depth T = 4 Haar-random and measurement-based quantum computing-type circuits, providing evidence that MIE in noisy 2D Haar-random circuits exhibits the same qualitative behavior as in random Clifford circuits, and that noise destroys the volume-law scaling of MIE in non-Clifford circuits.