BosonSampling is a popular candidate for near-term quantum advantage, which has now been experimentally implemented several times. The original proposal of Aaronson and Arkhipov from 2011 showed that classical hardness of BosonSampling is implied by a proof of the "Gaussian Permanent Estimation" conjecture. This conjecture states that $e^{-n\log{n}-n-O(\log n)}$ additive error estimates to the output probability of most random BosonSampling experiments are $\#P$-hard. Proving this conjecture has since become the central question in the theory of quantum advantage. In this work we make progress by proving that $e^{-n\log n -n - O(n^\delta)}$ additive error estimates to output probabilities of most random BosonSampling experiments are $\#P$-hard, for any $\delta>0$. In the process, we circumvent all known barrier results for proving the hardness of BosonSampling experiments. This is nearly the robustness needed to prove hardness of BosonSampling -- the remaining hurdle is now "merely" to show that the $n^\delta$ in the exponent can be improved to $O(\log n).$ We also obtain an analogous result for Random Circuit Sampling. Our result allows us to show, for the first time, a hardness of classical sampling result for random BosonSampling experiments, under an anticoncentration conjecture. Specifically, we prove the impossibility of multiplicative-error sampling from random BosonSampling experiments with probability $1-e^{-O(n)}$, unless the Polynomial Hierarchy collapses.

翻译：玻色采样是实现近期量子优势的主流候选方案，目前已在多个实验平台上实现。Aaronson与Arkhipov于2011年提出的原始理论证明，玻色采样的经典计算难度可由"高斯积和式估计"猜想的证明推导得出。该猜想指出：对绝大多数随机玻色采样实验的输出概率进行$e^{-n\log{n}-n-O(\log n)}$加性误差估计属于$\#P$难问题。证明该猜想此后成为量子优势理论的核心课题。本研究取得突破性进展：对于任意$\delta>0$，证明对绝大多数随机玻色采样实验输出概率进行$e^{-n\log n -n - O(n^\delta)}$加性误差估计属于$\#P$难问题。在证明过程中，我们规避了现有所有关于玻色采样实验难度证明的障碍性结论。这已接近证明玻色采样计算难度所需的鲁棒性要求——当前仅需将指数项中的$n^\delta$改进为$O(\log n)$即可完成最终证明。我们同时获得了随机电路采样的类似结论。基于反集中性猜想，本研究首次证明了随机玻色采样实验在经典采样任务中的计算难度：除非多项式层级结构坍缩，否则以$1-e^{-O(n)}$概率实现随机玻色采样实验的乘性误差采样是不可能的。