Classification of Quantum Cellular Automata

There exists an index theory to classify strictly local quantum cellular automata in one dimension (Fidkowski et al. in Interacting invariants for Floquet phases of fermions in two dimensions, 2017. arXiv:1703.07360 ; Gross et al. in Commun Math Phys 310(2):419–454, 2012; Po et al. in Phys Rev B 96: 245116, 2017). 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 (Haah et al. in Nontrivial quantum cellular automata in higher dimensions, 2018. arXiv:1812.01625 ). Finally, we discuss some group theoretical aspects of the classification of quantum cellular automata and consider these automata on higher dimensional real projective spaces.