全连接下的CNOT距离是NP完全问题
CNOT-Distance is NP-complete under all-to-all connectivity
AI总结：
该研究证明全连接下的CNOT距离问题是NP完全的，构造特定矩阵建立其与最小顶点覆盖的关联，还得出相关CNOT电路优化问题是APX难的结论。
AI中文摘要：
给定 $A\in\operatorname{GL}(N,2)$ 和整数 $K$，我们询问在全连接的固定带标签线路上，是否可以用至多 $K$ 个 CNOT 门实现矩阵 $A$。我们证明该问题是 NP 完全的。从有限简单图 $G=(V,E)$ 出发，我们构造一个上三角幺幂矩阵 $A_G\in\operatorname{GL}(2|V|+|E|+1,2)$，满足 $\ell_{\mathrm{CNOT}}(A_G)=2|V|+2|E|+\tau(G)$，其中 $\tau(G)$ 是最小顶点覆盖的大小。每个目标矩阵具有 $O(N)$ 个非零元，且行汉明重量至多为 4。下界将任意 CNOT 电路展开为异或有向无环图，并应用投影 - 收缩操作，允许中间奇偶性的抵消和无限制复用。对于该族，任意有限数量的必须被恢复的干净或借用辅助线路不会改变最优值。多项式时间解码器进一步得出：在每个固定加性常数内的近似是 NP 难的，并且通过从立方图上的最小顶点覆盖进行 L 归约，相关的 CNOT 电路优化问题是 APX 难的。
英文摘要：
Given $A\in\operatorname{GL}(N,2)$ and an integer $K$, we ask whether $A$ can be implemented by at most $K$ CNOT gates on fixed labelled wires with all-to-all connectivity. We prove that this problem is NP-complete. From a finite simple graph $G=(V,E)$, we construct an upper-unitriangular matrix $A_G\in\operatorname{GL}(2|V|+|E|+1,2)$ satisfying $\ell_{\mathrm{CNOT}}(A_G)=2|V|+2|E|+τ(G)$, where $τ(G)$ is the minimum vertex-cover size. Each target matrix has $O(N)$ nonzero entries and row Hamming weight at most four. The lower bound unfolds an arbitrary CNOT circuit into an XOR directed acyclic graph and applies projection--contraction operations, allowing cancellation and unrestricted reuse of intermediate parities. For this family, the optimum is unchanged by any finite number of clean or borrowed ancillary wires that must be restored. A polynomial-time decoder further yields NP-hardness of approximation within every fixed additive constant and, through an L-reduction from Minimum Vertex Cover on cubic graphs, APX-hardness of the associated CNOT-circuit optimisation problem.
