Quantum Coupon Collector

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

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Author Details

Srinivasan Arunachalam
  • IBM Research, Yorktown Heights, NY, USA
Aleksandrs Belovs
  • Faculty of Computing, University of Latvia, Riga, Latvia
Andrew M. Childs
  • Department of Computer Science, Institute for Advanced Computer Studies, University of Maryland, College Park, MD, USA
  • Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, USA
Robin Kothari
  • Microsoft Quantum, Redmond, WA, USA
  • Microsoft Research, Redmond, WA, USA
Ansis Rosmanis
  • Graduate School of Mathematics, Nagoya University, Japan
Ronald de Wolf
  • QuSoft, Amsterdam, The Netherlands
  • CWI, Amsterdam, The Netherlands
  • University of Amsterdam, The Netherlands

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Srinivasan Arunachalam, Aleksandrs Belovs, Andrew M. Childs, Robin Kothari, Ansis Rosmanis, and Ronald de Wolf. Quantum Coupon Collector. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 158, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Quantum algorithms
  • Adversary method
  • Coupon collector
  • Quantum learning theory


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