Improved Quantum data analysis

Abstract

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)/?2) samples of a d-dimensional state ?. That is, given observables 0 ? A1, A2, ..., Am ? 1 such that (? Ai) ? 1/2 for at least one i, the algorithm finds j with (? Aj) ? 1/2-?. As a consequence, we obtain a Shadow Tomography algorithm requiring only O((log2 m)(logd)/?4) samples, which simultaneously achieves the best known dependence on each parameter m, d, ?. 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)/?2) samples.