超立方体的近不变性
Near invariance of the hypercube
摘要：
我们给出了一个几乎完整的描述正交矩阵$M$的顺序$n$，“旋转一个不可忽略的分数的布尔超立方体$C_n=\{-1,1\}^n$到自己”，在这个意义上，$$P_{x\in C_n}(Mx\in C_n) \ge n^{-C},\mbox{ for some positive constant } C,$$，其中$x$是均匀采样$C_n$。特别是，我们表明，这样的矩阵$M$必须非常接近的产品置换和反射矩阵。这一结果是表征具有大积算式的正交和酉矩阵的一步，这是一个应用于线性光学量子计算的问题。
英文摘要：
We give an almost-complete description of orthogonal matrices $M$ of order $n$ that "rotate a non-negligible fraction of the Boolean hypercube $C_n=\{-1,1\}^n$ onto itself," in the sense that $$P_{x\in C_n}(Mx\in C_n) \ge n^{-C},\mbox{ for some positive constant } C,$$ where $x$ is sampled uniformly over $C_n$. In particular, we show that such matrices $M$ must be very close to products of permutation and reflection matrices. This result is a step toward characterizing those orthogonal and unitary matrices with large permanents, a question with applications to linear-optical quantum computing.
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