  The goal of demonstrating a quantum advantage with currently available
experimental systems is of utmost importance in quantum information science.
While this remains elusive for quantum computation, the field of communication
complexity offers the possibility to already explore and showcase this
advantage for useful tasks. Here, we define such a task, the Sampling Matching
problem, which is inspired by the Hidden Matching problem and features an
exponential gap between quantum and classical protocols in the one-way
communication model. Our problem allows by its conception a photonic
implementation based on encoding in the phase of coherent states of light, the
use of a fixed size linear optic circuit, and single-photon detection. This
enables us to demonstrate experimentally an advantage in the transmitted
information resource beyond a threshold input size, which would have been
impossible to reach for the original Hidden Matching problem. Our demonstration
has implications in various communication and cryptographic settings, for
example for quantum retrieval games and quantum money.
