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.

中文翻译：

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量子元胞自动机的分类

存在一个索引理论来对一维中的严格局部量子细胞自动机进行分类（Fidkowski 等人在 Interacting invariants for Floquet phase of fermions in twodimensional, 2017. arXiv:1703.07360 ; Gross 等人在 Commun Math Phys 310(2) :419–454, 2012; Po 等人在 Phys Rev B 96: 245116, 2017)。我们考虑两个分类问题。首先，我们研究该指数理论在多大程度上可以通过降维应用于更高维度，通过流形模扭的第一同调群找到分类。其次，在二维中，我们表明该指数理论的扩展（包括扭转）可以完全分类量子细胞自动机，至少在没有费米子自由度的情况下是这样。由于最近有证据表明三维中存在非平凡自动机（Haah 等人在更高维度的非平凡量子细胞自动机，2018. arXiv:1812.01625 中），因此索引理论在一维和二维上的完整分类预计不会扩展到更高维度. 最后，我们讨论了量子细胞自动机分类的一些群理论方面，并在更高维的实射影空间上考虑这些自动机。