高斯玻色子采样核心的对偶性
A duality at the heart of Gaussian boson sampling
摘要：
高斯玻色子采样（GBS）是一个近期的量子计算框架，被认为是经典的棘手的，但仍有丰富的潜在应用。在本文中，我们研究了从GBS设备与图形匹配多项式的样本类定义的分布之间的密切关系。为此，我们引入了一个新的图形多项式称为位移GBS多项式，其系数是粗粒度的光子数概率的任意无向图$G$编码在GBS设备。我们报告发现的位移GBS多项式的$G$和匹配多项式的$G\,\square\,P_2(x)$之间的对偶-笛卡尔图形产品的$G$与一个单一的加权边缘也被称为棱镜在$G$。除了对高斯玻色子采样的令人惊讶的洞察之外，它还为经典模拟高斯玻色子采样设备的新方式打开了大门。此外，它激发了最近用于构建特征图的新型粗粒度量子统计的成功[Schuld et al. 2019]。
英文摘要：
Gaussian boson sampling (GBS) is a near-term quantum computation framework that is believed to be classically intractable, but yet rich of potential applications. In this paper we study the intimate relation between distributions defined over classes of samples from a GBS device with graph matching polynomials. For this purpose, we introduce a new graph polynomial called the displaced GBS polynomial, whose coefficients are the coarse-grained photon-number probabilities of an arbitrary undirected graph $G$ encoded in a GBS device. We report a discovery of a duality between the displaced GBS polynomial of $G$ and the matching polynomial of $G\,\square\,P_2(x)$ - the Cartesian graph product of $G$ with a single weighted edge also known as the prism over $G$. Besides the surprising insight gained into Gaussian boson sampling, it opens the door for the new way of classically simulating the Gaussian boson sampling device. Furthermore, it motivates the recent success of a new type of coarse-grained quantum statistics used to construct feature maps in [Schuld et al. 2019].
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