  Gaussian boson sampling, a computational model that is widely believed to
admit quantum supremacy, has already been experimentally demonstrated and is
claimed to surpass the classical simulation capabilities of even the most
powerful supercomputers today. However, whether the current approach limited by
photon loss and noise in such experiments prescribes a scalable path to quantum
advantage is an open question. To understand the effect of photon loss on the
scalability of Gaussian boson sampling, we analytically derive the asymptotic
operator entanglement entropy scaling, which relates to the simulation
complexity. As a result, we observe that efficient tensor network simulations
are likely possible under the $N_\text{out}\propto\sqrt{N}$ scaling of the
number of surviving photons orange$N_\text{out}$ in the number of input photons
$N$. We numerically verify this result using a tensor network algorithm with
$U(1)$ symmetry, and overcome previous challenges due to the large local
Hilbert space dimensions in Gaussian boson sampling with hardware acceleration.
Additionally, we observe that increasing the photon number through larger
squeezing does not increase the entanglement entropy significantly. Finally, we
numerically find the bond dimension necessary for fixed accuracy simulations,
providing more direct evidence for the complexity of tensor networks.
