We study the complexity of two closely related learning problems, one quantum and one classical. In the quantum setting, we consider agnostic tomography for the natural class of product mixed states. Given $N$ copies of an $n$-qubit state $ρ$, the goal is to output a nearly optimal product mixed state approximation in trace distance. While recent work has focused on pure-state ansatz (e.g., product or stabilizer states), no polynomial-time guarantees were previously known for mixed-state ansatz. In the classical setting, we study robust learning of binary product distributions: given samples from an unknown distribution on ${0,1}^n$, the goal is to output a nearly optimal product approximation. Our main contributions are as follows. (1) We give a semi-agnostic tomography algorithm for product mixed states with polynomial sample and computational complexity achieving error $O(\mathrm{opt}\log(1/\mathrm{opt}))$, where $\mathrm{opt}$ is the trace distance to the best product approximation. This is the first efficient algorithm with any nontrivial agnostic guarantee for mixed-state ansatz, using only single-qubit, single-copy measurements. We also prove a Quantum Statistical Query lower bound showing near-optimality, and an unconditional lower bound demonstrating that adaptivity is necessary under single-qubit measurements. (2) We give a semi-agnostic algorithm for robustly learning binary product distributions with matching guarantees and establish a Statistical Query lower bound, essentially resolving the efficient robust learnability of this class and improving on prior work since Diakonikolas et al. (2016).

翻译：我们研究两个密切相关学习问题的复杂性，一个为量子领域，另一个为经典领域。在量子设定中，我们考虑自然类乘积混合态的非预言层析成像。给定$N$份$n$量子比特态$ρ$的副本，目标是以迹距离输出一个近乎最优的乘积混合态近似。尽管近期工作聚焦于纯态假设（如乘积态或稳定子态），但此前对混合态假设尚无多项式时间保证。在经典设定中，我们研究二元乘积分布的鲁棒学习：给定$\{0,1\}^n$上未知分布的样本，目标输出一个近乎最优的乘积近似。我们的主要贡献如下：(1) 我们提出一种半非预言乘积混合态层析成像算法，其样本和计算复杂度为多项式，可实现误差$O(\mathrm{opt}\log(1/\mathrm{opt}))$，其中$\mathrm{opt}$为至最优乘积近似的迹距离。这是首个对混合态假设提供非平凡非预言保证的高效算法，且仅使用单量子比特、单副本测量。我们还证明了量子统计查询下界，表明其近乎最优性，以及无条件下界，说明在单量子比特测量下自适应性的必要性。(2) 我们提出一种半非预言算法，用于鲁棒学习二元乘积分布，具有匹配的保证，并建立了统计查询下界，本质上解决了该类的有效鲁棒可学习性问题，改进了自Diakonikolas等人(2016)以来的先前工作。