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 $S_{\rm op}$ 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 $S_{\rm op}$ 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; this both accounts for the scaling of the maximal $S_{\rm op}$ and implies 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 $p = \Omega(1)$, and (ii) at constant depths $T = O(1)$ for noise rates $p = \Omega(\log^{-1}N)$. 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.