In this work we make progress in understanding the relationship between learning models with access to entangled, separable and statistical measurements in the quantum statistical query (QSQ) model. To this end, we show the following results. $\textbf{Entangled versus separable measurements.}$ The goal here is to learn an unknown $f$ from the concept class $C\subseteq \{f:\{0,1\}^n\rightarrow [k]\}$ given copies of $\frac{1}{\sqrt{2^n}}\sum_x \vert x,f(x)\rangle$. We show that, if $T$ copies suffice to learn $f$ using entangled measurements, then $O(nT^2)$ copies suffice to learn $f$ using just separable measurements. $\textbf{Entangled versus statistical measurements}$ The goal here is to learn a function $f \in C$ given access to separable measurements and statistical measurements. We exhibit a class $C$ that gives an exponential separation between QSQ learning and quantum learning with entangled measurements (even in the presence of noise). This proves the "quantum analogue" of the seminal result of Blum et al. [BKW'03]. that separates classical SQ and PAC learning with classification noise. $\textbf{QSQ lower bounds for learning states.}$ We introduce a quantum statistical query dimension (QSD), which we use to give lower bounds on the QSQ learning. With this we prove superpolynomial QSQ lower bounds for testing purity, shadow tomography, Abelian hidden subgroup problem, degree-$2$ functions, planted bi-clique states and output states of Clifford circuits of depth $\textsf{polylog}(n)$. $\textbf{Further applications.}$ We give and $\textit{unconditional}$ separation between weak and strong error mitigation and prove lower bounds for learning distributions in the QSQ model. Prior works by Quek et al. [QFK+'22], Hinsche et al. [HIN+'22], and Nietner et al. [NIS+'23] proved the analogous results $\textit{assuming}$ diagonal measurements and our work removes this assumption.

翻译：在这项工作中，我们在理解量子统计查询（QSQ）模型中基于纠缠测量、可分测量和统计测量的学习关系方面取得了进展。为此，我们展示了以下结果。$\textbf{纠缠测量与可分测量。}$ 目标是学习概念类 $C\subseteq \{f:\{0,1\}^n\rightarrow [k]\}$ 中的未知函数 $f$，给定 $\frac{1}{\sqrt{2^n}}\sum_x \vert x,f(x)\rangle$ 的副本。我们证明，如果使用纠缠测量学习 $f$ 需要 $T$ 个副本，那么使用可分测量学习 $f$ 仅需 $O(nT^2)$ 个副本。$\textbf{纠缠测量与统计测量。}$ 目标是学习函数 $f \in C$，给定可分测量和统计测量的访问权限。我们展示了一个类 $C$，它在 QSQ 学习与使用纠缠测量的量子学习（即使在噪声存在下）之间实现了指数级分离。这证明了 Blum 等人 [BKW'03] 开创性结果的“量子类比”，该结果将经典 SQ 学习与带有分类噪声的 PAC 学习区分开来。$\textbf{QSQ 学习状态的下界。}$ 我们引入了量子统计查询维度（QSD），用于给出 QSQ 学习的下界。据此，我们证明了纯度测试、阴影层析成像、阿贝尔隐子群问题、二次函数、种植双团态以及深度 $\textsf{polylog}(n)$ 的 Clifford 电路输出态的 QSQ 超多项式下界。$\textbf{进一步应用。}$ 我们给出了弱错误缓解与强错误缓解之间的$\textit{无条件}$ 分离，并证明了 QSQ 模型中分布学习的下界。先前 Quek 等人 [QFK+'22]、Hinsche 等人 [HIN+'22] 和 Nietner 等人 [NIS+'23] 的工作$\textit{假设}$ 了对角测量，而我们的工作移除了这一假设。