六个双量子比特门的每种架构在三量子比特上都是局部通用的

中文摘要 AI 辅助

我们首次对三量子比特上任意双量子比特门的每种固定架构的可达维度进行了精确解析确定。支撑词表示每个双量子比特门作用于哪对量子比特，其约化长度通过合并同一对上的连续门获得。若约化长度为 $r$，则可实现的三量子比特酉算子集合具有可达维度 $d(w)=\min\{63,9r+9\}$。因此，六个任意双量子比特门对于局部通用性是必要且充分的：当约化长度至少为六时，每种固定架构都达到 $\mathrm{SU}(8)$ 的一个非空开子集。该结果强于存在一个有利架构。每个约化的六槽支撑词都是局部通用的，包括交替最近邻线路 $AB,BC,AB,BC,AB,BC$。上界由标准参数计数论证得出。匹配的下界通过泡利基中的雅可比证书证明。在量子比特重标号和反转的意义下，长度为二到六的约化架构共有 $22$ 种。对于每一种，有理泡利旋转的乘积在素数 $1{,}000{,}003$ 模下产生一个非零最大子式。因此，用六个双量子比特门实现全局通用性的任何障碍都必须超越参数计数、连通性和微分秩。

英文摘要

We present the \emph{first} analytical determination of the exact accessible dimension for every fixed architecture of arbitrary two-qubit gates on three qubits. A support word represents which pair of qubits each two-qubit gate acts on, and its reduced length is obtained by merging consecutive gates on the same pair. If the reduced length is $r$, then the set of implementable three-qubit unitaries has accessible dimension $d(w)=\min\{63,9r+9\}$. Consequently, six arbitrary two-qubit gates are \emph{necessary and sufficient for local universality}: every fixed architecture reaches a nonempty open subset of $\mathrm{SU}(8)$ when its reduced length is at least six. The result is stronger than existence of one favorable architecture. Every reduced six-slot support word is locally universal, including the alternating nearest-neighbor line $AB,BC,AB,BC,AB,BC$. The upper bound follows from a standard parameter-counting argument. The matching lower bounds are proved by Jacobian certificates in the Pauli basis. Up to qubit relabeling and reversal, there are $22$ reduced architectures of lengths two through six. For each one, a product of rational Pauli rotations yields a nonzero maximal minor modulo the prime $1{,}000{,}003$. Thus, any obstruction to global universality with six two-qubit gates must go beyond parameter counting, connectivity, and differential rank.

发表机构

• NextQuantum Innovation Research Center（NextQuantum创新研究中心）
• Seoul National University（首尔大学）

机构由 AI 辅助整理，请以论文原文为准。

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80 pages