文章基本信息

标题：Quantum Coupon Collector
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作者：Srinivasan Arunachalam ; Aleksandrs Belovs ; Andrew M. Childs 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：158
页码：10:1-10:17
DOI：10.4230/LIPIcs.TQC.2020.10
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We study how efficiently a k-element set SâS[n] can be learned from a uniform superposition S> of its elements. One can think of S>=â^'_{iâ^^S} i>/â^S S as the quantum version of a uniformly random sample over S, as in the classical analysis of the "coupon collector problem." We show that if k is close to n, then we can learn S using asymptotically fewer quantum samples than random samples. In particular, if there are n-k=O(1) missing elements then O(k) copies of S> suffice, in contrast to the Î~(k log k) random samples needed by a classical coupon collector. On the other hand, if n-k=Î©(k), then Î©(k log k) quantum samples are necessary. More generally, we give tight bounds on the number of quantum samples needed for every k and n, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through S>. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case.
关键词：Quantum algorithms; Adversary method; Coupon collector; Quantum learning theory