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.

翻译：高斯玻色采样被普遍认为是一种能够实现量子霸权计算模型，并已在实验中得到验证，据称其性能已超越当今最强大超级计算机的经典模拟能力。然而，当前实验中受限于光子损耗和噪声的方法是否能提供可扩展的量子优势路径仍是一个悬而未决的问题。为理解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度相关的渐近算子纠缠熵标度。结果表明，在光子存活数$N_\text{out}$与输入光子数$N$满足$N_\text{out}\propto\sqrt{N}$标度律时，高效的张量网络模拟很可能成为可能。我们利用具有$U(1)$对称性的张量网络算法数值验证了这一结论，并通过硬件加速克服了高斯玻色采样中因局部希尔伯特空间维度过大带来的先前挑战。此外，我们发现通过增大压缩参数增加光子数不会显著提升纠缠熵。最后，我们数值求解了固定精度模拟所需的关键维数，为张量网络的复杂度提供了更直接的证据。