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可以实现量子霸权的计算模型，已在实验上证明其超越了当今最强大超级计算机的经典模拟能力。然而，当前受限于此类实验中光子损耗和噪声的方法，是否能够提供一条可扩展的量子优势路径仍是一个开放性问题。例如，最近研究表明，具有恒定门噪声的随机电路采样并非实现量子霸权的可扩展方法，尽管模拟中等规模系统仍然困难。为了理解光子损耗对高斯玻色采样可扩展性的影响，我们采用具有$U(1)$对称性的张量网络算法，考察与模拟复杂性相关的渐近算子纠缠熵标度。我们开发了一种定制算法，结合最先进的硬件加速器显著缩短了计算时间，从而能够模拟更大规模的系统。借助这一能力，我们观察到在高斯玻色采样中，幸存光子数随输入光子数呈关键的$N_\text{out}\propto\sqrt{N}$标度关系，这一标度标志着经典模拟有效与无效的分界。我们进一步从理论上证明，这一标度对于其他输入态也应具有普适性。