One of the most fundamental problems in distribution testing is the identity testing problem: given samples $x_1,\ldots,x_s$, the goal is to determine whether the samples are drawn from a target distribution $\mathcal{D}$. When $\mathcal{D}$ is a distribution over $\bit^n$, the optimal sample complexity of identity testing is known to be $Ω(\sqrt{2^n})$. Furthermore, most existing results assume that the samples $x_1,\ldots,x_s$ are generated independently from an unknown distribution. In this work, we overcome both of these limitations by initiating study of distribution testing in a more realistic setting. In our model, the unknown distribution is promised to be efficiently samplable, while allowing the observed samples $x_1,\ldots,x_s$ to be adversarially generated and arbitrarily correlated. Under this model, we show that polynomially many samples suffice to verify distributions. We further characterize the computational complexity of verifying classically- and quantumly-samplable distributions. Our techniques also extend to verifications of quantum states. In establishing some of our results, we employ Kolmogorov complexity techniques in a novel manner. We also present multiple applications of Kolmogorov complexity that are of independent interest. In particular, we show that certified randomness with a classical efficient prover can be achieved without computational assumptions when inefficient verification is allowed. Furthermore, we also show that a natural quantum extension of a well-studied Kolmogorov complexity measure provides a good benchmark for certifying sampling-based quantum advantage.

翻译：分布检验中最基本的问题之一是身份检验问题：给定样本$x_1,\ldots,x_s$，目标是判断这些样本是否来自目标分布$\mathcal{D}$。当$\mathcal{D}$是$\bit^n$上的分布时，身份检验的最优样本复杂度已知为$Ω(\sqrt{2^n})$。此外，大多数现有结果假设样本$x_1,\ldots,x_s$是独立地从未知分布中生成的。本文通过在一个更现实的场景下启动分布检验研究，克服了这两个限制。在我们的模型中，未知分布被承诺为可高效采样的，同时允许观测到的样本$x_1,\ldots,x_s$由对抗方式生成并具有任意相关性。在该模型下，我们证明多项式数量的样本足以验证分布。我们进一步刻画了验证经典可采样分布与量子可采样分布的计算复杂度。我们的技术还扩展到量子态的验证。在确立部分结果的过程中，我们以新颖的方式运用了柯尔莫哥洛夫复杂度技术。我们还展示了柯尔莫哥洛夫复杂度的多重应用，这些应用具有独立意义。特别地，我们证明：在允许非高效验证的条件下，无需计算假设即可实现具有经典高效证明者的认证随机性。此外，我们还表明，一个被广泛研究的柯尔莫哥洛夫复杂度度量的自然量子扩展，为认证基于采样的量子优势提供了一个良好的基准。