Measurement Quantum Cellular Automata and Anomalies in Floquet Codes

We investigate the evolution of quantum information under Pauli measurement circuits. We focus on the case of one- and two-dimensional systems, which are relevant to the recently introduced Floquet topological codes. We define local reversibility in context of measurement circuits, which allows us to treat finite depth measurement circuits on a similar footing to finite depth unitary circuits. In contrast to the unitary case, a finite depth locally reversible measurement circuit can implement a translation in one dimension. A locally reversible measurement circuit in two dimensions may also induce a flow of logical information along the boundary. We introduce "measurement quantum cellular automata" which unifies these ideas and define an index in one dimension to characterize the flow of logical operators. We find a $\mathbb{Z}_2$ bulk invariant for two-dimensional Floquet topological codes which indicates an obstruction to having a trivial boundary. We prove 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.