Improved quantum data analysis
Costin Bădescu, Ryan O'Donnell
TL;DR
The paper targets sample-efficient quantum data analysis by introducing a Threshold Search framework, a chi-square-stable threshold reporting mechanism, and two strong applications in Shadow Tomography and Hypothesis Selection. It achieves near-optimal copy complexities across multiple parameters: threshold testing uses $n_{TS}(m,\epsilon,\delta)=\frac{\log^2 m+\textsc{l}}{\epsilon^2}\cdot O(\textsc{l})$ with $\textsc{l}=\log(1/\delta)$, Shadow Tomography attains $n=\tilde{O}((\log^2 m)(\log d)/\epsilon^4)$, and Hypothesis Selection can be performed with $n=\tilde{O}((\log^3 m)/\epsilon^2)$ in one route (or $\tilde{O}((\log^2 m)/\epsilon^4)$ in the Shadow Tomography route). The core ideas connect adaptive data analysis with quantum data via classical-quantum compatibility results, enabling online and reusable testing without excessive sample waste. These advances bridge theory and practice in quantum state learning, offering scalable tools for quantum tomography and state discrimination under adaptivity constraints. The results hold potential for advancing quantum information tasks where rapid, reliable property testing of states is essential, including efficient state identification and hypothesis testing in high-dimensional quantum systems.
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 $O((\log^2 m)/ε^2)$ samples of a $d$-dimensional state $ρ$. That is, given observables $0 \le A_1, A_2, ..., A_m \le 1$ such that $\mathrm{tr}(ρA_i) \ge 1/2$ for at least one $i$, the algorithm finds $j$ with $\mathrm{tr}(ρA_j) \ge 1/2-ε$. As a consequence, we obtain a Shadow Tomography algorithm requiring only $\tilde{O}((\log^2 m)(\log d)/ε^4)$ samples, which simultaneously achieves the best known dependence on each parameter $m$, $d$, $ε$. This yields the same sample complexity for quantum Hypothesis Selection among $m$ states; we also give an alternative Hypothesis Selection method using $\tilde{O}((\log^3 m)/ε^2)$ samples.