The topological classification of one-dimensional symmetric quantum walks

Understand

We give a topological classification of quantum walks on an infinite 1D lattice, which obey one of the discrete symmetry groups of the tenfold way, have a gap around some eigenvalues at symmetry protected points, and satisfy a mild locality condition.

• No translation invariance is assumed.
• The classification is parameterized by three indices, taking values in a group, which is either trivial, the group of integers, or the group of integers modulo 2, depending on the type of symmetry.
• The classification is complete in the sense that two walks have the same indices if and only if they can be connected by a norm continuous path along which all the mentioned properties remain valid.