Gapless Symmetry-Protected Topological States in Measurement-Only Circuits

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. In this work, 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 non-unitary critical point, termed symmetry-enriched percolation, characterized by both topologically nontrivial edge states and string operator. 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.