  There exists an index theory to classify strictly local quantum cellular
automata in one dimension. We consider two classification questions. First, we
study to what extent this index theory can be applied in higher dimensions via
dimensional reduction, finding a classification by the first homology group of
the manifold modulo torsion. Second, in two dimensions, we show that an
extension of this index theory (including torsion) fully classifies quantum
cellular automata, at least in the absence of fermionic degrees of freedom.
This complete classification in one and two dimensions by index theory is not
expected to extend to higher dimensions due to recent evidence of a nontrivial
automaton in three dimensions. Finally, we discuss some group theoretical
aspects of the classification of quantum cellular automata and consider these
automata on higher dimensional real projective spaces.
