  Sampling from a quantum distribution can be exponentially hard for classical
computers and yet could be performed efficiently by a noisy intermediate-scale
quantum device. A prime example of a distribution that is hard to sample is
given by the output states of a linear interferometer traversed by $N$
identical boson particles. Here, we propose a scheme to implement such a boson
sampling machine with ultracold atoms in a polarization-synthesized optical
lattice. We experimentally demonstrate the basic building block of such a
machine by revealing the Hong-Ou-Mandel interference of two bosonic atoms in a
four-mode interferometer. To estimate the sampling rate for large $N$, we
develop a theoretical model based on a master equation that accounts for
particle losses, but not include technical errors. Our results show that atomic
samplers have the potential to achieve quantum advantage over today's best
supercomputers with $N \gtrsim 40$.
