BosonSampling is a problem of sampling events according to the transition probabilities of indistinguishable photons in a linear optical network. Computational hardness of BosonSampling depends on photon-number statistics of the input light. BosonSampling with multi-photon Fock states at the input is believed to be classically intractable but there exists an efficient classical algorithm for classical input states. In this paper, we present a mathematical connection between BosonSampling with quantum and classical light inputs. Specifically, we show that the generating function of a transition probability for Fock-state BosonSampling (FBS) can be expressed as a transition probability of thermal-light inputs. The closed-form expression of a thermal-light transition probability allows all possible transition probabilities of FBS to be obtained by calculating a single matrix permanent. Moreover, the transition probability of FBS is shown to be expressed as an integral involving a Gaussian function multiplied by a Laguerre polynomial, resulting in a fast oscillating integrand. Our work sheds new light on computational hardness of FBS by identifying the mathematical connection between BosonSampling with quantum and classical light.

中文翻译：

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具有量子和经典输入状态的BosonSampling之间的连接

BosonSampling是根据线性光网络中不可区分的光子的跃迁概率对事件进行采样的问题。BosonSampling的计算硬度取决于输入光的光子数统计。在输入端具有多光子Fock状态的BosonSampion被认为是经典的，但是对于经典的输入状态，存在一种有效的经典算法。在本文中，我们介绍了具有量子和经典光输入的BosonSampling之间的数学联系。具体而言，我们表明，Fock状态玻色子采样（FBS）的跃迁概率的生成函数可以表示为热光输入的跃迁概率。热光跃迁几率的闭式表达式允许通过计算单个矩阵永久性来获得FBS的所有可能跃迁几率。而且，FBS的跃迁概率被表示为包含高斯函数乘以Laguerre多项式的积分，从而得到快速振荡的积分。我们的工作通过确定BosonSampling与量子光和经典光之间的数学联系，为FBS的计算难度提供了新的思路。