Cryptographic Conditions for Efficient Testing of Distributions and Quantum States

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 $\Omega(\sqrt{2^n})$. 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.