Steady-state phases in long-range measurement-only quantum circuits

Measurements can drive quantum many-body systems into nontrivial steady states and induce interesting dynamical phase transitions, rendering measurement-only quantum circuits a useful platform for exploring quantum many-body phases beyond those of equilibrium Hamiltonian systems. Here we study a class of long-range measurement-only quantum circuits with competing two-qubit and three-qubit measurements. We demonstrate that these circuits exhibit rich steady-state structure and uncover a strong influence of the measurement range on the resulting phases. In particular, states with symmetry-protected topological (SPT) order can emerge with sufficiently short-range measurements beyond the nearest-neighbor limit. These states feature robust topological edge modes, which can also be detected from circuit dynamics. With longer-range measurements, an extended parameter regime emerges in which conventional order parameters are suppressed while spatial correlations remain nontrivial. Moreover, we show that in this circuit model sufficiently long-range measurements can produce significant entanglement with scaling beyond an area law despite the absence of any unitary evolution.