Measurement-induced entanglement in noisy 2D random Clifford circuits

We study measurement-induced entanglement generated by column-by-column sampling of noisy 2D random Clifford circuits of size N and depth T. Focusing on the operator entanglement S_op of the sampling-induced boundary state, first, we reproduce in the noiseless limit a finite-depth transition from area- to volume-law scaling. With on-site probablistic trace noise at any constant rate p>0, the maximal S_op attained along the sampling trajectory obeys an area law in the boundary length and scales approximately linearly with T/p. 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_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, volume-law measurement-induced entanglement in 2D random Clifford circuits. Finally, based on the observed scaling, we conjecture 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(1/ log N).