用多项式小均值近似随机矩阵的永久性：零点与普遍性
Approximating the Permanent of a Random Matrix with Polynomially Small Mean: Zeros and Universality
AI总结：
研究在矩阵条目略微偏离零时近似随机矩阵永久性的算法，通过零点区域分析揭示复杂性理论中的核心问题与贡献。
摘要：
我们研究在矩阵条目略微偏离零时近似随机矩阵永久性的算法。此问题受理解经典线性光学复杂性及玻色采样（Aaronson和Arkhipov '11；Eldar和Mehraban '17）的动机驱动。Barvinok的插值方法允许高效近似永久性，前提是能够证明多项式per(zJ + W)的足够大零点自由区域。我们证明当W的条目为标准复高斯分布时，随机多项式per(zJ + W)的所有零点均在半径为~O(n^{-1/3})的圆内，这在条目偏置为~Ω(n^{-1/3})时提供近似算法。此前，对于偏置小于1/polylog(n)的情况，没有高效的算法，且不清楚是否存在通常有零点z满足|z| ≥ 1。作为补充结果，我们证明大部分零点（即(1 - ε)n个）的模长为Θ(n^{-1/2})，这防止了插值方法与永久性近似的平均情况难度猜想相矛盾。我们还建立了一般图上硬核模型的零点自由区域，以及随机矩阵W具有i.i.d.次指数条目的普遍性结果，证明了零点自由区域。
英文摘要：
We study algorithms for approximating the permanent of a random matrix when the entries are slightly biased away from zero. This question is motivated by the goal of understanding the classical complexity of linear optics and \emph{boson sampling} (Aaronson and Arkhipov '11; Eldar and Mehraban '17). Barvinok's interpolation method enables efficient approximation of the permanent, provided one can establish a sufficiently large zero-free region for the polynomial $\mathrm{per}(zJ + W)$, where $J$ is the all-ones matrix and $W$ is a random matrix with independent mean-zero entries. We show that when the entries of $W$ are standard complex Gaussians, all zeros of the random polynomial $\mathrm{per}(zJ + W)$ lie within a disk of radius $\tilde{O}(n^{-1/3})$, which yields an approximation algorithm when the bias of the entries is $\tildeΩ(n^{-1/3})$. Previously, there were no efficient algorithms at biases smaller than $1/\mathrm{polylog}(n)$, and it was unknown whether there typically exist zeros $z$ with $|z| \ge 1$. As a complementary result, we show that the bulk of the zeros, namely $(1 - ε)n$ of them, have magnitude $Θ(n^{-1/2})$. This prevents our interpolation method from contradicting the conjectured average-case hardness of approximating the permanent. We also establish analogous zero-free regions for the hardcore model on general graphs with complex vertex fugacities. In addition, we prove universality results establishing zero-free regions for random matrices $W$ with i.i.d. subexponential entries.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
