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.