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We study measurement-induced symmetry-protected topological (SPT) order in awide class of quantum random circuit models by combining calculations withinthe stabilizer formalism with tensor network simulations. We construct a familyof quantum random circuits, generating the out-of-equilibrium version of allgeneralized cluster models, and derive a set of non-local string orderparameters to distinguish different SPT phases. We apply this framework toinvestigate a random circuit realization of the XZX cluster model, and use thestring order parameter to demonstrate that the phase diagram is stable againstextending the class of unitary gates in the circuit, from Clifford gates toHaar unitaries. We then turn to the XZZX generalized cluster model, anddemonstrate the coexistence of SPT order and spontaneous symmetry breaking, byrelying on string order parameters and a connected correlation function.

Detecting and stabilizing measurement-induced symmetry-protected topological phases in generalized cluster models