## Complete homotopy invariants for translation invariant symmetric quantum walks on a chain. (arXiv:1804.04520v1 [quant-ph])

We provide a classification of translation invariant one-dimensional quantum
walks with respect to continuous deformations preserving unitarity, locality,
translation invariance, a gap condition, and some symmetry of the tenfold way.
The classification largely matches the one recently obtained (arXiv:1611.04439)
for a similar setting leaving out translation invariance. However, the
translation invariant case has some finer distinctions, because some walks may
be connected only by breaking translation invariance along the way, retaining
only invariance by an even number of sites. Similarly, if walks are considered
equivalent when they differ only by adding a trivial walk, i.e., one that
allows no jumps between cells, then the classification collapses also to the
general one. The indices of the general classification can be computed in
practice only for walks closely related to some translation invariant ones. We
prove a completed collection of simple formulas in terms of winding numbers查看全文