Random point patterns are ubiquitous in nature, and statistical models such as point processes, i.e., algorithms that generate stochastic collections of points, are commonly used to simulate and interpret them. We propose an application of quantum computing to statistical modeling by establishing a connection between point processes and Gaussian Boson Sampling, an algorithm for photonic quantum computers. We show that Gaussian Boson Sampling can be used to implement a class of point processes based on hard-to-compute matrix functions which, in general, are intractable to simulate classically. We also discuss situations where polynomial-time classical methods exist. This leads to a family of efficient quantum-inspired point processes, including a new fast classical algorithm for permanental point processes. We investigate the statistical properties of point processes based on Gaussian Boson Sampling and reveal their defining property: like bosons that bunch together, they generate collections of points that form clusters. Finally, we analyze properties of these point processes for homogeneous and inhomogeneous state spaces, describe methods to control cluster location, and illustrate how to encode correlation matrices.

中文翻译：

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高斯玻色子采样的点过程

随机点模式本质上是普遍存在的，并且统计模型（例如点过程）（即生成点的随机集合的算法）通常用于模拟和解释它们。通过建立点过程与高斯玻色子采样（一种用于光子量子计算机的算法）之间的联系，我们建议将量子计算应用于统计建模。我们表明，高斯玻色子采样可用于基于难以计算的矩阵函数来实现一类点过程，这些矩阵函数通常很难经典地模拟。我们还将讨论存在多项式时间经典方法的情况。这导致了一系列有效的量子启发式点过程，包括用于永久点过程的新的快速经典算法。我们研究基于高斯玻色子采样的点过程的统计特性，并揭示它们的定义特性：就像玻色子聚在一起，它们生成形成簇的点的集合。最后，我们分析了均匀和不均匀状态空间的这些点过程的性质，描述了控制簇位置的方法，并说明了如何对相关矩阵进行编码。