Abstract: Quantum measurements are critical to virtually any aspect of quantum information processing--for example: quantum error correction, distillation protocols, or state preparation. We discuss the evolution of quantum information under Pauli measurement circuits. We define "local reversibility" in context of measurement circuits, which guarantees that quantum information is preserved and remain "localized" after measurement. We find that measurement circuits can exhibit a richer set of behaviour in comparison to their unitary counterparts. For example, a finite depth measurement circuit can implement translation in one dimension, exhibiting a net flow of information. We introduce "measurement quantum cellular automata" which unifies these ideas; and find a Z_2 bulk invariant for two-dimensional Floquet topological codes which indicates an obstruction to having a trivial boundary. We show that the Hastings-Haah honeycomb code belongs to a class with such obstruction, which means that any boundary must have either nonlocal dynamics, period doubled, or admits anomalous boundary flow of quantum information.
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