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Gaussian boson sampling (GBS) is a near-term quantum computation frameworkthat is believed to be classically intractable, but yet rich of potentialapplications. In this paper we study the intimate relation betweendistributions defined over classes of samples from a GBS device with graphmatching polynomials. For this purpose, we introduce a new graph polynomialcalled the displaced GBS polynomial, whose coefficients are the coarse-grainedphoton-number probabilities of an arbitrary undirected graph $G$ encoded in aGBS device. We report a discovery of a duality between the displaced GBSpolynomial of $G$ and the matching polynomial of $G\,\square\,P_2(x)$ - theCartesian graph product of $G$ with a single weighted edge also known as theprism over $G$. Besides the surprising insight gained into Gaussian bosonsampling, it opens the door for the new way of classically simulating theGaussian boson sampling device. Furthermore, it motivates the recent success ofa new type of coarse-grained quantum statistics used to construct feature mapsin [Schuld et al. 2019].

A duality at the heart of Gaussian boson sampling