标题： 对称子代数上的量子细胞自动机 显示英文标题

标题： Quantum Cellular Automata on Symmetric Subalgebras

摘要： 我们研究在有限阿贝尔群对称性下的全局部算子代数的子代数——对称子代数上的二维自旋系统上的量子细胞自动机（QCA）$G$。 对于每个位置都携带$G$的正则表示的系统，我们基于两个拓扑不变量对这种子代数QCA进行了完整的分类：(1) 从子代数QCA群到$(2+1)d$ $G$ 规范理论中的任意子排列对称群的满同态；以及 (2) 对Gross-Nesme-Vogts-Werner（GNVW）指标的一种推广，该指标描述了对称子代数的流动。 具体来说，如果两个子代数QCA具有相同的任意子排列并具有相同的指标，则它们仅通过由$G$对称局部门组成的有限深度单元电路有所不同。 我们还识别出一组操作，这些操作通过有限次组合可以生成所有子代数QCA。 作为示例，我们研究在$\mathbb{Z}_2$对称子代数上的Kramers-Wannier对偶性，证明它在二维拓扑码中映射到$e$-$m$置换，并具有无理指标$\sqrt{2}$。因此，它不能扩展为整个局部算子代数上的QCA，并且与晶格平移非平凡地混合。 显示英文摘要

摘要： We investigate quantum cellular automata (QCA) on one-dimensional spin systems defined over a subalgebra of the full local operator algebra - the symmetric subalgebra under a finite Abelian group symmetry $G$. For systems where each site carries a regular representation of $G$, we establish a complete classification of such subalgebra QCAs based on two topological invariants: (1) a surjective homomorphism from the group of subalgebra QCAs to the group of anyon permutation symmetries in a $(2+1)d$ $G$ gauge theory; and (2) a generalization of the Gross-Nesme-Vogts-Werner (GNVW) index that characterizes the flow of the symmetric subalgebra. Specifically, two subalgebra QCAs correspond to the same anyon permutation and share the same index if and only if they differ by a finite-depth unitary circuit composed of $G$-symmetric local gates. We also identify a set of operations that generate all subalgebra QCAs through finite compositions. As an example, we examine the Kramers-Wannier duality on a $\mathbb{Z}_2$ symmetric subalgebra, demonstrating that it maps to the $e$-$m$ permutation in the two-dimensional toric code and has an irrational index of $\sqrt{2}$. Therefore, it cannot be extended to a QCA over the full local operator algebra and mixes nontrivially with lattice translations.