AI 中文总结
本文提出了一种显式构造的矩阵族，其CNOT复杂度达到$4n - o(n)$，超越了传统循环置换的复杂度。
AI 中文摘要
一个可逆矩阵在$\mathbb{F}_2$上的CNOT复杂度是指合成对应线性可逆算子所需的最少CNOT门数量。虽然所有$n \times n$矩阵的最大CNOT复杂度已知为$\Theta(n^2 / \log n)$，但目前尚无明确的矩阵族需要超线性数量的CNOT门，直到现在已知的最困难的显式族是循环置换，其CNOT复杂度为$3(n-1)$。我们证明了非可逆线性算子的加法复杂度的下界可以仅损失少量的情况下提升到可逆设置中。作为应用，我们使用此方法描述了一个由纠错码的奇偶校验矩阵构造的显式矩阵族，其CNOT复杂度至少为$4n - o(n)$，渐进上超过了循环置换。此外，该构造产生了一个显式矩阵$A \in \mathrm{GL}_{n}(\mathbb{F}_2)$，$n = 17167$，其CNOT复杂度超过了$n$符号的循环置换。
英文摘要
The CNOT-complexity of an invertible matrix over $\mathbb{F}_2$ is the minimum number of CNOT gates needed to synthesize the corresponding linear reversible operator. While the maximum CNOT-complexity over all $n \times n$ matrices is known to be $Θ(n^2 / \log n)$, no explicit family of matrices requiring a superlinear number of CNOT gates is known, and until now the hardest explicitly known family has been the cyclic permutations, with CNOT-complexity $3(n-1)$. We show that lower bounds for the additive complexity of not-necessarily-reversible linear operators can be lifted to the reversible setting with only a small loss. As an application, we use this to describe an explicit family of matrices, constructed from parity-check matrices of error-correcting codes, with CNOT-complexity at least $4n - o(n)$, asymptotically surpassing the cyclic permutations. Moreover, this construction yields an explicit matrix $A \in \mathrm{GL}_{n}(\mathbb{F}_2)$, $n = 17167$, whose CNOT-complexity exceeds that of the cyclic permutation on $n$ symbols.
