  We prove that a classical computer can efficiently sample from the
photon-number probability distribution of a Gaussian state prepared by using an
optical circuit that is shallow and local. Our work generalizes previous known
results for qubits to the continuous-variable domain. The key to our proof is
the observation that the adjacency matrices characterizing the Gaussian states
generated by shallow and local circuits have small bandwidth. To exploit this
structure, we devise fast algorithms to calculate loop hafnians of banded
matrices. Since sampling from deep optical circuits with exponential-scaling
photon loss is classically simulable, our results pose a challenge to the
feasibility of demonstrating quantum supremacy on photonic platforms with local
interactions.
