Classification of locality preserving unitaries with symmetry

Locality preserving unitaries (LPUs) are unitary operators that map local operators to nearby local operators. In a given dimension $d$, some LPUs can be written as finite time evolution of a local $d$-dimensional Hamiltonian. Such LPUs are called finite depth local unitaries (FDLUs). Other LPUs are nontrivial in the sense that they cannot be written in this way, and can only be realized at the boundary of a $(d+1)$ dimensional FDLU. These LPUs have been studied in the context of quantum cellular automata and Floquet topological phases. The study of LPUs without symmetry is already a rich subject, and the classification of nontrivial LPUs in higher dimensions is not well understood. In this talk, I will discuss an easier problem: classifying LPUs that are nontrivial only in the presence of a global symmetry. Parts of the classification are related to symmetry protected topological phases, and other parts of the classification are entirely new.