AI 中文总结
该研究构造了$\boldsymbol{\text{Z}_2}$上具有$\boldsymbol{4n-\text{o}(n)}$行复杂度的显式可逆矩阵，证明其置换群同构于可逆仿射变换群，将量子复杂度归约为行约简复杂度，得到其量子复杂度下界。
AI 中文摘要
本文提出了一类显式的$\boldsymbol{n\times n}$可逆$\boldsymbol{\text{Z}_2}$上的矩阵，其CNOT和行复杂度至少为$\boldsymbol{4n-\text{o}(n)}$；等价地，将这些矩阵约简为单位矩阵至少需要$\boldsymbol{4n-\text{o}(n)}$次初等行操作。此外，在更强的计算模型中，即CNOT门被任意局部线性逻辑门（作用于坐标对的任意可逆线性变换）取代时，相同的复杂度下界仍然成立。令$\boldsymbol{G_n}$表示作用于长度为$\boldsymbol{n}$的二进制串集合的局部逻辑门生成的置换群，我们证明$\boldsymbol{G_n}$自然同构于向量空间$\boldsymbol{\text{Z}_2^n}$上所有可逆仿射变换组成的群，从而将估计$\boldsymbol{G_n}$中置换的量子复杂度问题归约为$\boldsymbol{\text{Z}_2}$上可逆矩阵的行约简复杂度问题。作为应用，我们表明与我们的显式矩阵相关联的置换具有至少$\boldsymbol{4n-\text{o}(n)}$的量子复杂度。
英文摘要
In this article, we present an explicit family of invertible $n\times n$ matrices over $\mathbb Z_2$ whose CNOT and row complexity is at least $4n-\text{o}(n)$; equivalently, reducing these matrices to the identity requires at least $4n-\text{o}(n)$ elementary row operations. Moreover, the same complexity lower bound holds in the stronger computational model where the CNOT gates are replaced by arbitrary local linear logic gates, namely arbitrary invertible linear transformations acting on pairs of coordinates. Let $G_n$ denote the permutation group generated by local logic gates acting on the set of binary strings of length $n$. We prove that $G_n$ is naturally isomorphic to the group of all invertible affine transformations of the vector space $\mathbb Z_2^n$, thus reducing the problem of estimating the quantum complexity of permutations in $G_n$ to the row reduction complexity of invertible matrices over $\mathbb Z_2$. As an application, we show that the permutations associated with our explicit matrices have quantum complexity at least $4n-\text{o}(n)$.
