Classification of quantum cellular automata (Q2184742)

The reviewed paper concerns a classification of the so-called local Quantum Cellular Automata (QCA), understood here as (local) automorphisms of certain tensor products of matrix algebras, indexed by a control space, whose metric structure makes it possible to quantify the locality condition. The key tool is the notion of the GNVW index introduced for one-dimensional systems in the article [\textit{D. Gross} et al., Commun. Math. Phys. 310, No. 2, 419--454 (2012; Zbl 1238.81154)]. Note that the dimension refers here to the dimension of the control space, and the matrix dimension of individual nodes does not play a key role, as often one considers stable equivalence, allowing local tensoring by additional matrix algebras. In the first part of the paper the authors exploit methods of algebraic topology to define higher-dimensional version of the GNVW index, called also the flux of the QCA in question, and to develop methods of computing it from the one-dimensional reductions. The second part is more algebraic in nature: here the notion of a boundary algebra is introduced and applied to provide a new definition of the GNVW index and to obtain certain criteria for blending of two QCAs, understood as building a new QCA which locally coincides with two given ones. A special role is played by the `finite depth quantum circuits', given by adjoint action of finitely many local unitaries. The article is best read together with the paper of Gross, Nesme, Vogts and Werner mentioned above, as significant parts of the discussion remain on a semi-formal level and certain relevant initial definitions are only stated in detail in the earlier work. On the other hand the authors include several examples, which illustrate well the theory being developed.