We consider the problem of finding the minimum element in a list of length $N$ using a noisy comparator. The noise is modelled as follows: given two elements to compare, if the values of the elements differ by at least $α$ by some metric defined on the elements, then the comparison will be made correctly; if the values of the elements are closer than $α$, the outcome of the comparison is not subject to any guarantees. We demonstrate a quantum algorithm for noisy quantum minimum-finding that preserves the quadratic speedup of the noiseless case: our algorithm runs in time $\tilde O(\sqrt{N (1+Δ)})$, where $Δ$ is an upper-bound on the number of elements within the interval $α$, and outputs a good approximation of the true minimum with high probability. Our noisy comparator model is motivated by the problem of hypothesis selection, where given a set of $N$ known candidate probability distributions and samples from an unknown target distribution, one seeks to output some candidate distribution $O(\varepsilon)$-close to the unknown target. Much work on the classical front has been devoted to speeding up the run time of classical hypothesis selection from $O(N^2)$ to $O(N)$, in part by using statistical primitives such as the Scheffé test. Assuming a quantum oracle generalization of the classical data access and applying our noisy quantum minimum-finding algorithm, we take this run time into the sublinear regime. The final expected run time is $\tilde O( \sqrt{N(1+Δ)})$, with the same $O(\log N)$ sample complexity from the unknown distribution as the classical algorithm. We expect robust quantum minimum-finding to be a useful building block for algorithms in situations where the comparator (which may be another quantum or classical algorithm) is resolution-limited or subject to some uncertainty.
Simons Institute for the Theory of Computing（西蒙斯计算理论研究所）；Boston University（波士顿大学）；University of Waterloo（滑铁卢大学）；Microsoft Research（微软研究院）；IBM Research（IBM研究院）；UC Berkeley（加州大学伯克利分校）；Google（谷歌）；University of Minnesota, Twin Cities（明尼苏达大学双城分校）；Carnegie Mellon University（卡内基梅隆大学）
Minimum Description Feature Selection for Complexity Reduction in Machine Learning-based Wireless Positioning
Elmore Family School of Electrical and Computer Engineering, Purdue University（埃尔莫家族电气与计算机工程学院，普渡大学）；School of Electrical, Computer and Energy Engineering, Arizona State University（电气、计算机与能源工程学院，亚利桑那州立大学）
