  Boson sampling, a computational task believed to be classically hard to
simulate, is expected to hold promise for demonstrating quantum computational
advantage using near-term quantum devices. However, noise in experimental
implementations poses a significant challenge, potentially rendering boson
sampling classically simulable and compromising its classical intractability.
Numerous studies have proposed classical algorithms under various noise models
that can efficiently simulate boson sampling as noise rates increase with
circuit depth. To address this issue particularly related to circuit depth, we
explore the viability of achieving quantum computational advantage through
boson sampling with shallow-depth linear optical circuits. Specifically, as the
average-case hardness of estimating output probabilities of boson sampling is a
crucial ingredient in demonstrating its classical intractability, we make
progress on establishing the average-case hardness confined to
logarithmic-depth regimes. We also obtain the average-case hardness for
logarithmic-depth Fock-state boson sampling subject to lossy environments and
for the logarithmic-depth Gaussian boson sampling. By providing
complexity-theoretical backgrounds for the classical simulation hardness of
logarithmic-depth boson sampling, we expect that our findings will mark a
crucial step towards a more noise-tolerant demonstration of quantum advantage
with shallow-depth boson sampling.
