Point processes with Gaussian boson sampling

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 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.

Figure 1

Illustration of a random point pattern in a honeycomb. The sealed yellow cells contain developing bees and the dark holes are empty cells that have been randomly vacated. The appearance of empty cells can be modeled as a point process generating clusters of points in the hexagonal lattice.

Figure 2

Schematic illustration of a point process implemented with GBS. A symmetric kernel matrix $K$ can be encoded by appropriately selecting the squeezing levels $r_1,r_2,...,r_m$ and the interferometer unitary $U$. Each point in the state space is associated with an output mode such that the modes where photons are observed determine the specific point pattern that has been sampled.

Figure 3

Samples generated with (a) determinantal, (b) Poisson, and (c) Torontonian spatial point processes in a one-dimensional space containing 20 points. DPP samples have points that are scattered and spread out in space. PPPs treat all patterns uniformly, so both clustering and repulsion are typically present. For TPPs, sample points are likely to occur in clusters.

Figure 4

Samples generated with (a) determinantal, (b) Poisson, and (c) Torontonian spatial point processes in a two-dimensional space containing 100 points. DPP samples have points that are scattered and spread out in space. PPPs treat all patterns uniformly, so both clustering and repulsion are typically present. For TPPs, sample points are likely to occur in clusters.

Figure 5

Normalized distributions of the nearest-neighbor distance for (a) determinantal, (b) Poisson, and (c) Torontonian spatial point processes in a two-dimensional space containing 100 points. We set a value of $\sigma =1$ for the kernel matrix parameter in Eq. (46). The statistics were taken over ten independent samples and the bars correspond to one standard deviation. For DPPs, the most common nearest-neighbor distance is noticeably larger than those for PPPs and TPPs, showcasing the scattering property of DPPs. Conversely, the significant majority of TPP points have their nearest neighbor at the closest possible distances, a signal of the clustering property of this point process.

Figure 6

Examples of Voronoi cell diagrams for points obtained from (a) determinantal, (b) Poisson, and (c) Torontonian point processes in a two-dimensional space containing 100 points. We set a value of $\sigma =1$ for the kernel matrix parameter in Eq. (46). DPPs lead to Voronoi cells that have comparable areas, whereas TPPs lead to cells that have either small areas (around clusters) or very large areas (in between clusters). The PPP cells are also inhomogeneous but the level of size variation is smaller than that in the TPP sample.

Figure 7

Normalized distributions of the Voronoi cell areas for (a) determinantal, (b) Poisson, and (c) Torontonian point processes in a two-dimensional space containing 100 points. We set a value of $\sigma =1$ for the kernel matrix parameter in Eq. (46). The statistics were taken over ten independent samples and the bars correspond to one standard deviation. DPPs have a high peak in the distribution, indicating that the Voronoi cells have roughly the same area since the points are scattered evenly. For TPPs, there is a significantly large probability of small cell areas, which is to be expected when the patterns form several clusters of nearby points. The PPP distribution reflects the intermediate level of inhomogeneity in the PPP cell sizes, compared to the DPP and TPP samples.

Figure 8

Samples generated with (a) determinantal, (b) Poisson, and (c) GBS-inspired permanental point processes in a two-dimensional space containing 2500 points. DPP samples have points that are scattered and spread out in space. PPPs treat all patterns uniformly, so both clustering and repulsion are typically present. For the GBS-inspired PerPP, sample points are likely to occur in clusters of nearby points.

Figure 9

Samples generated with (a) determinantal, (b) Poisson, and (c) Torontonian point processes in a two-dimensional inhomogeneous space containing two dense clusters and a sparse random background. The parameter $\sigma$ in the kernel matrix in Eq. (46) was set to 1. The DPP points are spread out in space and the PPP points are located in both dense and sparse regions, while the TPP points are all located in the dense regions.

Figure 10

TPP point pattern sampled with a kernel matrix that has been rescaled according to a Gaussian density background. The darker regions correspond to higher density.

Figure 11

Typical samples of correlation matrices of stocks selected by applying (a) determinantal, (b) Poisson, and (c) Torontonian point processes. The kernel matrix used to implement the point processes is the correlation matrix constructed from the market data of the stocks of the S&P 500 index [95]. The point process samples are subsets of stocks, shown here by their stock symbols. For each subset, we display the corresponding correlation matrix. Lighter points correspond to large entries of the correlation matrix. The TPP sample contains stocks that are highly correlated, while the DPP and PPP samples contain stocks with lower levels of correlation.

Figure 12

Samples generated with determinantal, Poisson, and Torontonian spatial point processes in a two-dimensional space containing 100 points. DPP was set to generate 10 points for each sample. The patterns generated with PPP and TPP usually contain different numbers of points but only those samples that contained 10 points are shown here for consistency.