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Periodically driven (Floquet) systems can exhibit new possibilities beyondwhat can be obtained in equilibrium. Both in Floquet systems and in the relatedproblems of discrete-time quantum walks and quantum cellular automata, a basicdistinction arises among unitary time evolution operators: while all physicaloperators are local, not all are locally generated (i.e., generated by somelocal Hamiltonian). In this paper, we define the notion of equivalence up to alocally generated unitary in all Altland-Zirnbauer symmetry classes. We thenclassify non-interacting unitaries in all dimensions on this basis by showingthat equivalence up to a locally generated unitary is identical to homotopyequivalence.

Classification of Unitary Operators by Local Generatability