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Entanglement transitions in quantum dynamics present a novel class of phasetransitions in non-equilibrium systems. When a many-body quantum systemundergoes unitary evolution interspersed with monitored random measurements,the steady-state can exhibit a phase transition between volume and area-lawentanglement. There is a correspondence between measurement-induced transitionsin non-unitary quantum circuits in $d$ spatial dimensions and classicalstatistical mechanical models in $d+1$ dimensions. In certain limits thesemodels map to percolation, but there is analytical and numerical evidence tosuggest that away from these limits the universality class should genericallybe distinct from percolation. Intriguingly, despite these arguments, numericson 1D qubit circuits give bulk exponents which are nonetheless close to thoseof 2D percolation, with possible differences in surface behavior. In the firstpart of this work we study the critical properties of 2D Clifford circuits. Inthe bulk, we find many properties suggested by the percolation picture,including matching bulk exponents, and an inverse power-law for the criticalentanglement growth, $S(t,L) \sim L(1 - a/t)$, which saturates to an area-law.We then utilize a graph-state based algorithm to analyze in 1D and 2D thecritical properties of entanglement clusters in the steady state. We show thatin a model with a simple geometric map to percolation, the projectivetransverse field Ising model, the entanglement clusters are governed bypercolation surface exponents. However, in the Clifford models we find largedeviations in the cluster exponents from those of surface percolation,highlighting the breakdown of any possible geometric map to percolation. Giventhe evidence for deviations from the percolation universality class, ourresults raise the question of why nonetheless many bulk properties behavesimilarly to percolation.

Measurement-induced criticality and entanglement clusters: a study of 1D and 2D Clifford circuits