Measurement-only quantum circuits offer a versatile platform for realizing intriguing quantum phases of matter. However, gapless symmetry-protected topological (gSPT) states remain insufficiently explored in these settings. Here, we generalize the notion of gSPT to the critical steady state in measurement-only circuits. Using large-scale Clifford circuit simulations, we investigate the steady-state phase diagram across several families of measurement-only circuits that exhibit topological nontrivial edge states at criticality. In the cluster Ising circuits, we uncover a symmetry-enriched nonunitary critical point, termed symmetry-enriched percolation, characterized by both topologically nontrivial edge states and string operators. Additionally, we demonstrate the realization of a steady-state gSPT phase in a ${\mathbb{Z}}_4$ circuit model. This phase features topological edge modes and remains robust under symmetry-preserving perturbations. Furthermore, we provide a unified theoretical framework by mapping the system to Majorana loop models, offering deeper insights into the underlying mechanisms.

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• Conformal field theory
• Dynamical phase transitions
• Nonequilibrium statistical mechanics
• Quantum circuits
• Quantum entanglement
• Symmetry protected topological states
• Nonequilibrium systems
• Topological materials
• Symmetries in condensed matter
• Thermal & statistical models