Classification of Quantum Cellular Automata

Understand

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.