Topological classification of Local Unitary Operators*

Local unitary operators arise naturally as time evolution operators generated by Hamiltonians and also in the context of quantum walks. In one dimension, the flow index is a topological invariant which distinguishes classes of local unitary operators up to equivalence by smooth deformation. We propose a higher-dimensional generalization of this index and obtain a topological classification of non-interacting local unitary operators. We extend this classification to local unitary operators with symmetry and show that the result can be represented in a periodic table with a period of eight, similar to the periodic table of topological insulators. Our classification naturally distinguishes unitary operators that are locally generated from those that cannot be locally generated.