Abstract

We provide a classification of one-dimensional quantum walks which satisfy a symmetry condition with respect to some sitewise unitary or antiunitary reflections, and also have a spectral gap up to eigenvalues of finite multiplicity. No translation invariance whatsoever is assumed. The classification is stable under arbitrary local perturbations and is in terms of two indices whose range (either the integers or the integers mod 2) depend on the symmetry type. The indices can be computed from the behavior of the walk far to the right and far to the left, respectively. Their sum is a lower bound to the combined multiplicities of the eigenspaces at 1 and -1 respectively. For translation invariant walks this classification is the same as the K-theoretic classification of band projections over the quasi-momentum torus. Therefore we confirm the predictions in the heuristic literature that when joining two walks in different topological phases (here: with different index) a bound state at +-1 will necessarily appear.