License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.TQC.2020.10
URN: urn:nbn:de:0030-drops-120692
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12069/
Go to the corresponding LIPIcs Volume Portal

Arunachalam, Srinivasan ; Belovs, Aleksandrs ; Childs, Andrew M. ; Kothari, Robin ; Rosmanis, Ansis ; de Wolf, Ronald

Quantum Coupon Collector

pdf-format:
LIPIcs-TQC-2020-10.pdf (0.5 MB)

Abstract

We study how efficiently a k-element set S⊆[n] can be learned from a uniform superposition |S> of its elements. One can think of |S>=∑_{i∈S}|i>/√|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 Θ(k log k) random samples needed by a classical coupon collector. On the other hand, if n-k=Ω(k), then Ω(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.

BibTeX - Entry

@InProceedings{arunachalam_et_al:LIPIcs:2020:12069,
  author =	{Srinivasan Arunachalam and Aleksandrs Belovs and Andrew M. Childs and Robin Kothari and Ansis Rosmanis and Ronald de Wolf},
  title =	{{Quantum Coupon Collector}},
  booktitle =	{15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)},
  pages =	{10:1--10:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-146-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{158},
  editor =	{Steven T. Flammia},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12069},
  URN =		{urn:nbn:de:0030-drops-120692},
  doi =		{10.4230/LIPIcs.TQC.2020.10},
  annote =	{Keywords: Quantum algorithms, Adversary method, Coupon collector, Quantum learning theory}
}

Keywords: Quantum algorithms, Adversary method, Coupon collector, Quantum learning theory
Collection: 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)
Issue Date: 2020
Date of publication: 08.06.2020

DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI