We study how efficiently a k-element set S ? [n] can be learned from a uniform superposition |Si of its elements. One can think of |Si = Pi?S |ii/p|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 |Si suffice, in contrast to the T(k log k) random samples needed by a classical coupon collector. On the other hand, if n - k = ?(k), then ?(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 |Si. 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.

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15th Conference on the Theory of Quantum Computation, Communication and Cryptography, TQC 2020
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

Arunachalam, S, Belovs, A, Childs, A.M, Kothari, R, Rosmanis, A, & de Wolf, R.M. (2020). Quantum coupon collector. In Leibniz International Proceedings in Informatics, LIPIcs. doi:10.4230/LIPIcs.TQC.2020.10