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We introduce a model of non-unitary quantum dynamics that exhibits infinitely long-lived discrete spatiotemporal order robust against any unitary or dissipative perturbation. Ergodicity is evaded by combining a sequence of projective measurements with a local feedback rule that is inspired by Toom's `North-East-Center' classical cellular automaton. The measurements in question only partially collapse the wavefunction of the system, allowing some quantum coherence to persist. We demonstrate our claims using numerical simulations of a Clifford circuit in two spatial dimensions which allows access to large system sizes, and also present results for more generic dynamics on modest system sizes. We also devise explicit experimental protocols realising this dynamics using one- and two-qubit gates that are available on present-day quantum computing platforms., 6+7 pages, 2+6 figures

We investigate how information spreads in three paradigmatic one-dimensional models with spatial disorder. The models we consider are unitarily related to a system of free fermions and are thus manifestly integrable. A slow logarithmic-in-time growth of the entanglement entropy is observed following a quench from an unentangled initial state. We attribute this growth to the presence of strong zero modes, which gives rise to an exponential hierarchy of time scales upon ensemble averaging. We also demonstrate that out-of-time-order correlators can spread slowly beyond the single-particle localization length, atypical for Anderson localized systems. This phenomenon is shown to be due to the nonlocal relationship between elementary excitations and the physical degrees of freedom. We argue that this non-locality becomes relevant when considering time-dependent correlation functions. Our work on disordered integrable systems complements the rich phenomenology of information spreading and we discuss broader implications for general systems with non-local correlations.