Gaussian boson sampling, a computational model that is widely believed to admit quantum supremacy, has already been experimentally demonstrated to surpasses the classical simulation capabilities of even with 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. For example, random circuit sampling with constant noise per gate was recently shown not to be a scalable approach to achieve quantum supremacy, although simulating intermediate scale systems is still difficult. To understand the effect of photon loss on the scability of Gaussian boson sampling, we use a tensor network algorithm with $U(1)$ symmetry to examine the asymptotic operator entanglement entropy scaling, which relates to the simulation complexity. We develop a custom-built algorithm that significantly reduces the computational time with state-of-the-art hardware accelerators, enabling simulations of much larger systems. With this capability, we observe, for Gaussian boson sampling, the crucial $N_\text{out}\propto\sqrt{N}$ scaling of the number of surviving photons in the number of input photons that marks the boundary between efficient and inefficient classical simulation. We further theoretically show that this should be general for other input states.

翻译：高斯玻色采样作为一种被广泛认为能够实现量子霸权的计算模型，已在实验上证明其超越当前最强大超级计算机的经典模拟能力。然而，这些实验中受限于光子损耗和噪声的现有方法，是否能为量子优势提供可扩展路径仍是一个开放问题。例如，最近研究表明，虽然中等规模系统的模拟仍然困难，但恒定门噪声下的随机电路采样并非实现量子霸权的可扩展方法。为理解光子损耗对高斯玻色采样可扩展性的影响，我们利用具有$U(1)$对称性的张量网络算法，考察与模拟复杂度相关的渐近算符纠缠熵标度。通过定制算法结合最先进的硬件加速器，我们显著降低了计算时间，从而能模拟更大规模系统。基于这一能力，我们观测到高斯玻色采样中决定经典模拟效率界限的关键标度律——存活光子数$N_\text{out}\propto\sqrt{N}$（其中$N$为输入光子数）。此外，我们通过理论论证表明该标度律对其他输入态具有普适性。