An Index for Quantum Cellular Automata on Fusion Spin Chains

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 \(\textbf{Fib}\) $\mathbf{Fib}$ , the index is a complete invariant for the group of QCA modulo finite depth circuits.