Many properties of Boolean functions can be tested far more efficiently than the function can be learned. However, this advantage often disappears when testers are limited to random samples--a natural setting for data science--rather than queries. In this work we investigate the quantum version of this scenario: quantum algorithms that test properties of a function $f$ solely from quantum data in the form of copies of the function state for $f$. For three well-established properties, we show that the speedup lost when restricting classical testers to samples can be recovered by testers that use quantum data. For monotonicity testing, we give a quantum algorithm that uses $\tilde{\mathcal{O}}(n^2)$ function state copies as compared to the $2^{\Omega(\sqrt{n})}$ samples required classically. We also present $\mathcal{O}(1)$-copy testers for symmetry and triangle-freeness, comparing favorably to classical lower bounds of $\Omega(n^{1/4})$ and $\Omega(n)$ samples respectively. These algorithms are time-efficient and necessarily include techniques beyond the Fourier sampling approaches applied to earlier testing problems. These results make the case for a general study of the advantages afforded by quantum data for testing. We contribute to this project by complementing our upper bounds with a lower bound of $\Omega(1/\varepsilon)$ for monotonicity testing from quantum data in the proximity regime $\varepsilon\leq\mathcal{O}(n^{-3/2})$. This implies a strict separation between testing monotonicity from quantum data and from quantum queries--where $\tilde{\mathcal{O}}(n)$ queries suffice when $\varepsilon=\Theta(n^{-3/2})$. We also exhibit a testing problem that can be solved from $\mathcal{O}(1)$ classical queries but requires $\Omega(2^{n/2})$ function state copies, complementing a separation of the same magnitude in the opposite direction derived from the Forrelation problem.

翻译：布尔函数的许多性质可以被检验的效率远高于学习该函数本身的效率。然而，当检验者被限制于使用随机样本（数据科学中的一种自然设定）而非查询时，这种优势常常会消失。在本工作中，我们研究了这一场景的量子版本：仅从量子数据（即函数$f$的函数态副本）出发检验函数$f$性质的量子算法。针对三个已确立的性质，我们证明了当经典检验者被限制于使用样本时所失去的速度优势，可以通过使用量子数据的检验者重新获得。对于单调性检验，我们给出了一种使用$\tilde{\mathcal{O}}(n^2)$个函数态副本的量子算法，而经典方法则需要$2^{\Omega(\sqrt{n})}$个样本。我们还提出了分别使用$\mathcal{O}(1)$个副本的对称性与无三角形性检验器，这优于经典方法所需的$\Omega(n^{1/4})$和$\Omega(n)$个样本的下界。这些算法是时间高效的，并且必然包含了超越早期检验问题中应用的傅里叶采样方法的技术。这些结果为系统研究量子数据在检验中所提供的优势奠定了基础。我们通过将上界与下界相结合来推进这一研究：在邻近区域$\varepsilon\leq\mathcal{O}(n^{-3/2})$内，基于量子数据的单调性检验存在$\Omega(1/\varepsilon)$的下界。这意味着基于量子数据的单调性检验与基于量子查询的检验之间存在严格分离——当$\varepsilon=\Theta(n^{-3/2})$时，$\tilde{\mathcal{O}}(n)$次查询便已足够。我们还展示了一个检验问题，它可以通过$\mathcal{O}(1)$次经典查询解决，但需要$\Omega(2^{n/2})$个函数态副本，这补充了从Forrelation问题导出的、在相反方向上具有相同量级的分离结果。