A periodic classification of local unitary operators

We present a complete topological classification of single particle local unitary operators, which are usually represented by local unitary matrices. The classification is performed in the ten symmetry classes and leads to a “periodic table”. Local unitary operators arise naturally in systems with an evolution generated by a quantum possibly time-varying (Hermitian) Hamiltonian. They also describe unitary operations in more general contexts, such as in quantum walks or models for information flow. The classification is complete in the sense that two local unitary operators have the same topological invariants if and only if they can be continuously deformed into each other. It allows one to distinguish “locally generated” unitary operators in a given symmetry class (generated by local Hamiltonians) from those that are not locally generated and provides a complete table of local operators which are not locally generated.