  Interpreting the GNVW index for 1D quantum cellular automata (QCA) in terms
of the Jones index for subfactors leads to a generalization of the index
defined for QCA on more general abstract spin chains. These include fusion spin
chains, which arise as the local operators invariant under a global
(categorical/MPO) symmetry, and as the boundary operators of 2D topological
codes. We show that for the fusion spin chains built from the fusion category
$\mathbf{Fib}$, the index is a complete invariant for the group of QCA modulo
finite depth circuits.
