We prove that on any two-dimensional lattice of qudits of a prime dimension, every translation invariant Pauli stabilizer group with local generators and with code distance being the linear system size, is decomposed by a Clifford circuit of constant depth into $\mathcal T^{\oplus n} \oplus \mathcal Z$ for some integer $n \ge 0$, where $\mathcal T$ is the stabilizer group of the toric code (abelian discrete gauge theory) on the square lattice and $\mathcal Z$ is a stabilizer group whose code space encodes zero logical qudit in any finite periodic lattice. The direct summand $\mathcal Z$ is mapped to the trivial stabilizer group for a product state under a locality-preserving automorphism of the complex operator algebra on the lattice which maps every Pauli matrix to a product of Pauli matrices (Clifford QCA). In other words, up to Clifford QCA the integer $n$ is the complete invariant of such a stabilizer group. Previously, the same conclusion was obtained by assuming nonchirality for qubit codes or the Calderbank-Shor-Steane structure for prime qudit codes; we do not assume any of these.

中文翻译：

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二维格子上素维量子数的平移不变拓扑泡利稳定码的分类

我们证明，在质数维度的量子点的任何二维格子上，每个具有本地生成器且代码距离为线性系统大小的平移不变泡利稳定器群，都被恒定深度的 Clifford 电路分解为 $\mathcal T^ {\oplus n} \oplus \mathcal Z$ 表示某个整数 $n \ge 0$，其中 $\mathcal T$ 是方格上的复曲面码（阿贝尔离散规范理论）和 $\mathcal Z 的稳定群$ 是一个稳定器群，其代码空间在任何有限周期晶格中编码零逻辑量子。在将每个泡利矩阵映射到泡利矩阵的乘积 (Clifford QCA) 的点阵上复算子代数的局部保持自同构下，直接被加数 $\mathcal Z$ 被映射到产品状态的平凡稳定器群。换句话说，直到 Clifford QCA，整数 $n$ 是这种稳定器群的完全不变量。以前，通过假设 qubit 代码的非手性或质数 qudit 代码的 Calderbank-Shor-Steane 结构，得到了相同的结论；我们不承担任何这些。