Measurement-induced criticality in extended and long-range unitary circuits

We explore the dynamical phases of unitary Clifford circuits with variable-range interactions, coupled to a monitoring environment. investigate two classes models, distinguished by action gates, which either are organized in clusters finite-range two-body or pair-wise interactions randomly distributed throughout system power-law distribution. find range plays key role characterizing both and their measurement-induced transitions. For cluster gates we transition between phase volume-law scaling entanglement entropy area-law entropy. Our results indicate that universality class is compatible short hybrid circuits. Oppositely, case changes continuously parameter controlling interactions. In particular, for intermediate values control parameter, non-conformal critical line separates from one sub-extensive scaling. Within this region, logarithmic negativity present cross-over algebraic growth size, an phase.