  Boson sampling is a sampling task proven to be hard to simulate efficiently
using classical computers under plausible assumptions, which makes it an
appealing candidate for quantum supremacy. However, due to a large noise rate
for near-term quantum devices, it is still unclear whether those noisy devices
maintain the quantum advantage for much larger quantum systems. Since the noise
rate typically grows with the circuit depth, an alternative is to find evidence
of simulation hardness at the shallow-depth quantum circuit. To find the
evidence, one way is to identify the minimum depth required for the
average-case hardness of approximating output probabilities, which is
considered a necessary condition for the state-of-the-art technique to prove
the simulation hardness of boson sampling. In this work, we analyze the output
probability distribution of shallow-depth boson sampling for Fock-states and
Gaussian states, and examine the limitation of the average-case hardness
argument at this shallow-depth regime for geometrically local architectures. We
propose a shallow-depth linear optical circuit architecture that can overcome
the problems associated with geometrically local architectures. Our numerical
results suggest that this architecture demonstrates possibilities of
average-case hardness properties in a shallow-depth regime, through its
resemblance to the global Haar-random boson sampling circuit. This result
implies that the corresponding architecture has the potential to be utilized
for scalable quantum supremacy with its shallow-depth boson sampling.
