论文标题
任意$ 2^w \ times 2^w $统一矩阵的分解成签名的置换矩阵
The decomposition of an arbitrary $2^w\times 2^w$ unitary matrix into signed permutation matrices
论文作者
论文摘要
伯克霍夫（Birkhoff）的定理告诉任何双重随机矩阵，都可以分解为置换矩阵的加权总和。类似的定理表明，任何统一矩阵都可以分解为复杂置换矩阵的加权总和。尺寸的单一矩阵等于〜2（例如$ 2^w $）的功率，值得特别注意，因为它们代表量子量子电路。我们研究了哪些符号置换矩阵的子组足以分解任意的矩阵。事实证明，它是矩阵组的矩阵组{\ bf e} $ _ {2^{2W+1}}^+$的顺序$ 2^{2W {2W+1} $。相关的投影组订单$ 2^{2W} $同样就足够了。
Birkhoff's theorem tells that any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. A similar theorem reveals that any unitary matrix can be decomposed as a weighted sum of complex permutation matrices. Unitary matrices of dimension equal to a power of~2 (say $2^w$) deserve special attention, as they represent quantum qubit circuits. We investigate which subgroup of the signed permutation matrices suffices to decompose an arbitrary such matrix. It turns out to be a matrix group isomorphic to the extraspecial group {\bf E}$_{2^{2w+1}}^+$ of order $2^{2w+1}$. An associated projective group of order $2^{2w}$ equally suffices.
