In the Heisenberg picture of quantum mechanics, time evolution is a one-parameter family of automorphisms of operator algebra. Restricted to short times the equivalence between time evolution and quantum circuits, especially the property that it maps a local operator to another, has been implanted in theoretical studies of topological phases of matter. In this talk, I will explain recent findings that not all locality-preserving automorphisms, also called quantum cellular automata, can be written as quantum circuits --- there exists a "discrete time dynamics" that cannot have a "Hamiltonian." These are tightly related to static, topological many-body states. I will give results on the classification of these automorphisms, and connect them to locally generated subalgebras in one lower dimension.
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Vasily Pestun & Slava Rychkov