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((log2 m)/є2) samples of a d-dimensional state ρ. That is, given observables 0 ≤ A1, A2, …, Am ≤ 1 such that (ρ Ai) ≥ 1/2 for at least one i, the algorithm finds j with (ρ Aj) ≥ 1/2−є. As a consequence, we obtain a Shadow Tomography algorithm requiring only O((log2 m)(logd)/є4) samples, which simultaneously achieves the best known dependence on each parameter m, d, є. This yields the same sample complexity for quantum Hypothesis Selection among m states; we also give an alternative Hypothesis Selection method using O((log3 m)/є2) samples.

We consider the problem of quantum state certification, where one is given n copies of an unknown d-dimensional quantum mixed state ρ, and one wants to test whether ρ is equal to some known mixed state σ or else is є-far from σ. The goal is to use notably fewer copies than the Ω(d2) needed for full tomography on ρ (i.e., density estimation). We give two robust state certification algorithms: one with respect to fidelity using n = O(d/є) copies, and one with respect to trace distance using n = O(d/є2) copies. The latter algorithm also applies when σ is unknown as well. These copy complexities are optimal up to constant factors.