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 in the number of input photons. 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.

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计算模型，已在实验中得到验证，并声称其超越了当前最强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与噪声影响的方法能否为量子优势提供可扩展路径仍是一个未解问题。为探究光子损耗对高斯玻色采样可扩展性的影响，我们解析推导了与模拟复杂性相关的渐近算符纠缠熵标度律。结果表明，当幸存光子数与输入光子数满足标度关系$N_{\text{out}}\propto\sqrt{N}$时，高效的张量网络模拟可能实现。我们采用具有$U(1)$对称性的张量网络算法对该结果进行数值验证，并借助硬件加速克服了高斯玻色采样中局部希尔伯特空间维度大带来的先前挑战。此外，我们观察到通过增大压缩参数来增加光子数并不会显著提升纠缠熵。最后，通过数值方法确定了固定精度模拟所需的关键维数，为张量网络复杂性提供了更直接的证据。