AI 中文总结
研究玻色采样中随机高斯矩阵的永久反集中性，通过对随机高斯永久式超指数小于其标准差概率上界界定建立弱反集中界，还建立了高斯永久式典型量级，结合相关框架表明超指数模拟玻色采样会使多项式层级坍塌。
AI 中文摘要
近期量子计算优势的证明很大程度上由采样问题推动。一个突出模型是玻色采样，它涉及从线性光学网络的输出分布中采样。但其经典难度取决于两个看似合理但研究较少的猜想：近似高斯永久式的平均情况难度和永久反集中猜想（PACC）。PACC是关于随机高斯矩阵分布特性的纯数学断言。离散随机矩阵的永久式典型量级已确定，而控制线性光学网络跃迁振幅的复高斯情况仍未解决。本文通过对随机高斯永久式超指数小于其标准差的概率进行上界界定，建立了一个弱反集中界。收紧此界到逆多项式分数将证明原始的PACC。作为推论，我们建立了高斯永久式的典型量级，与陶哲轩和武丁对伯努利矩阵的开创性结果相当。结合阿隆森 - 阿尔基波夫框架，我们的结果意味着假设其余猜想成立，在超指数小的总变差距离内经典模拟玻色采样将使多项式层级坍塌。
英文摘要
Recent demonstrations of quantum computational advantage have been driven largely by sampling problems. A prominent model, boson sampling, involves sampling from the output distribution of a linear optical network. However, its classical hardness hinges on two plausible yet less-studied conjectures: the average-case hardness of approximating Gaussian permanents, and the permanent anti-concentration conjecture (PACC). The PACC is a purely mathematical assertion regarding the distributional properties of random Gaussian matrices. While the typical magnitude of the permanent has been established for discrete random matrices, the complex Gaussian case, which governs transition amplitudes in linear optical networks, has remained open. Here, we establish a weak anti-concentration bound by upper-bounding the probability that a random Gaussian permanent is superexponentially smaller than its standard deviation. Tightening this bound to an inverse-polynomial fraction would prove the original PACC. As a corollary, we establish the typical magnitude of Gaussian permanents, on par with Tao and Vu's seminal result for Bernoulli matrices. Combined with the Aaronson-Arkhipov framework, our result implies that classically simulating boson sampling to within a superexponentially small total variation distance would collapse the polynomial hierarchy, assuming the remaining conjectures hold.
Comments12 pages. Accepted in 2026 IEEE International Conference on Quantum Computing and Engineering (QCE)
