Topology of quantum cellular automata: how to measure the difference between Hamiltonians and time evolutions

On lattice systems, a global unitary evolution that has a notion of lightcone is not necessarily generated by a local Hamiltonian. This statement (that is perhaps surprising) is best understood for those with abrupt lightcones and quantum circuits in 1+1d. These discrete objects form a group, which unfortunately does not include any continuous time evolutions by interacting Hamiltonians. Here, I will discuss ongoing efforts to treat continuous time evolutions on the same footing as the quantum circuits. The spirit is to endow the group G of all approximately locality-preserving automorphisms on operator algebras with an appropriate metric topology, by which we consider approximation of continuous time evolutions by discrete ones.