论文标题
统一动态的拓扑阶段：克利福德类别的分类
Topological phases of unitary dynamics: Classification in Clifford category
论文作者
论文摘要
根据定义，量子蜂窝自动机（QCA）或因果统一是本地操作员代数的自动形态，将本地操作员映射到本地运营商。量子量很小的量子回路，局部的哈密顿量变化，而翻译（换档）就是例子。 Clifford QCA是将任何保利操作员映射到Pauli操作员的有限张量产品的QCA。 Here, we obtain a complete table of groups $\mathfrak C(\mathsf d,p)$ of translation invariant Clifford QCA in any spatial dimension $\mathsf d \ge 0$ modulo Clifford quantum circuits and shifts over prime $p$-dimensional qudits, where the circuits and shifts are allowed to obey only coarser translation invariance.组$ \ Mathfrak C（\ Mathsf d，p）$仅对$ \ Mathsf d = 2k+3 $如果$ p = 2 $ = 2 $和$ \ MATHSF d = 4k+3 $如果$ p $是奇数〜$ k \ ge 0 $是任何整数，在这种情况下，在这种情况下w}（\ Mathbb f_p）$，有限字段$ \ mathbb f_p $的经典二次二次形式的经典witt组。众所周知，$ \ widetilde {\ Mathfrak w}（\ Mathbb f_2）\ cong \ Mathbb z/2 \ Mathbb z $，$ \ widetilde {\ Mathfrak W}（\ Mathbb f_p） $ \ widetilde {\ Mathfrak w}（\ Mathbb f_p）\ cong \ Mathbb Z/2 \ Mathbb Z \ oplus \ oplus \ Mathbb z/2 \ Mathbb z $如果$ p = 1 \ bmod 4 $。分类是通过维下降来实现的，这是拓扑中代数$ l $ - $ l $ groups of laurent延长定理的降低。
A quantum cellular automaton (QCA) or a causal unitary is by definition an automorphism of local operator algebra, by which local operators are mapped to local operators. Quantum circuits of small depth, local Hamiltonian evolutions for short time, and translations (shifts) are examples. A Clifford QCA is one that maps any Pauli operator to a finite tensor product of Pauli operators. Here, we obtain a complete table of groups $\mathfrak C(\mathsf d,p)$ of translation invariant Clifford QCA in any spatial dimension $\mathsf d \ge 0$ modulo Clifford quantum circuits and shifts over prime $p$-dimensional qudits, where the circuits and shifts are allowed to obey only coarser translation invariance. The group $\mathfrak C(\mathsf d,p)$ is nonzero only for $\mathsf d = 2k+3$ if $p=2$ and $\mathsf d = 4k+3$ if $p$ is odd where~$k \ge 0$ is any integer, in which case $\mathfrak C(\mathsf d,p) \cong \widetilde{\mathfrak W}(\mathbb F_p)$, the classical Witt group of nonsingular quadratic forms over the finite field $\mathbb F_p$. It is well known that $\widetilde{\mathfrak W}(\mathbb F_2) \cong \mathbb Z/2\mathbb Z$, $\widetilde{\mathfrak W}(\mathbb F_p) \cong \mathbb Z/4\mathbb Z$ if $p = 3 \bmod 4$, and $\widetilde{\mathfrak W}(\mathbb F_p)\cong \mathbb Z/2\mathbb Z \oplus \mathbb Z/2\mathbb Z$ if $p = 1 \bmod 4$. The classification is achieved by a dimensional descent, which is a reduction of Laurent extension theorems for algebraic $L$-groups of surgery theory in topology.
