We provide more sample-efficient versions of some basic routines in quantum data analysis, along with simpler proofs. Particularly, we give a quantum "Threshold Search" algorithm that requires only $O((\log^2 m)/\epsilon^2)$ samples of a $d$-dimensional state $\rho$. That is, given observables $0 \le A_1, A_2, ..., A_m \le 1$ such that $\mathrm{tr}(\rho A_i) \ge 1/2$ for at least one $i$, the algorithm finds $j$ with $\mathrm{tr}(\rho A_j) \ge 1/2-\epsilon$. As a consequence, we obtain a Shadow Tomography algorithm requiring only $\tilde{O}((\log^2 m)(\log d)/\epsilon^4)$ samples, which simultaneously achieves the best known dependence on each parameter $m$, $d$, $\epsilon$. This yields the same sample complexity for quantum Hypothesis Selection among $m$ states; we also give an alternative Hypothesis Selection method using $\tilde{O}((\log^3 m)/\epsilon^2)$ samples.

翻译：我们提供了量子数据分析中一些基本例程的更高效样本版本，并给出了更简洁的证明。特别地，我们提出了一种量子“阈值搜索”算法，该算法仅需 $O((\log^2 m)/\epsilon^2)$ 个 $d$ 维态 $\rho$ 的样本。即，给定可观测量 $0 \le A_1, A_2, ..., A_m \le 1$，且至少存在一个 $i$ 使得 $\mathrm{tr}(\rho A_i) \ge 1/2$，该算法能够找到满足 $\mathrm{tr}(\rho A_j) \ge 1/2-\epsilon$ 的 $j$。作为结果，我们获得了一种仅需 $\tilde{O}((\log^2 m)(\log d)/\epsilon^4)$ 个样本的影子断层扫描算法，该算法同时达到了每个参数 $m$、$d$、$\epsilon$ 上的最佳已知依赖关系。这为从 $m$ 个态中进行量子假设选择提供了相同的样本复杂度；我们还给出了一种替代的假设选择方法，仅需 $\tilde{O}((\log^3 m)/\epsilon^2)$ 个样本。