We develop closed-form expressions for the moments and cumulants of Gaussian states when measured in the photon-number basis. We express the photon-number moments of a Gaussian state in terms of the loop Hafnian, a function that when applied to a $(0,1)$-matrix representing the adjacency of a graph, counts the number of its perfect matchings. Similarly, we express the photon-number cumulants in terms of the Montrealer, a newly introduced matrix function that when applied to a $(0,1)$-matrix counts the number of Hamiltonian cycles of that graph. Based on these graph-theoretic connections, we show that the calculation of photon-number moments and cumulants are #P-hard. Moreover, we provide an exponential time algorithm to calculate Montrealers (and thus cumulants), matching well-known results for Hafnians. We then demonstrate that when a uniformly lossy interferometer is fed in every input with identical single-mode Gaussian states with zero displacement, all the odd-order cumulants but the first one are zero. Finally, we employ the expressions we derive to study the distribution of cumulants up to the fourth order for different input states in a Gaussian boson sampling setup where $K$ identical states are fed into an $\ell$-mode interferometer. We analyze the dependence of the cumulants as a function of the type of input state, squeezed, lossy squeezed, squashed, or thermal, and as a function of the number of non-vacuum inputs. We find that thermal states perform much worse than other classical states, such as squashed states, at mimicking the photon-number cumulants of lossy or lossless squeezed states.

中文翻译：

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高斯态的光子数矩和累积量


当以光子数为基础测量时，我们为高斯态的矩和累积量开发了闭式表达式。我们用循环哈夫尼安来表示高斯态的光子数矩，当该函数应用于表示图邻接性的 $（0,1）$ 矩阵时，会计算其完美匹配的数量。同样，我们用蒙特利尔来表示光子数累积量，这是一个新引入的矩阵函数，当应用于 $（0,1）$ 矩阵时，会计算该图的哈密顿循环数。基于这些图论联系，我们表明光子数矩和累积量的计算是 #P 难的。此外，我们提供了一种指数时间算法来计算蒙特利尔人（以及累积量），与哈夫尼人的众所周知的结果相匹配。然后，我们证明，当均匀有损干涉仪以相同的单模高斯态和零位移在每个输入中馈入时，除第一个累积量外，所有奇数阶累积量都为零。最后，我们利用推导的表达式来研究高斯玻色子采样设置中不同输入状态的四阶累积量分布，其中 $K$ 个相同的状态被馈送到 $\ell$ 模式干涉仪中。我们分析了累积量的依赖性，它是输入状态类型（挤压、有损挤压、挤压或热）的函数，以及非真空输入数量的函数。我们发现，热态在模拟有损或无损压缩态的光子数累积量方面比其他经典态（例如压缩态）差得多。