On the Quantum versus Classical Learnability of Discrete Distributions

Ryan Sweke; Jean-Pierre Seifert; Dominik Hangleiter; Jens Eisert

doi:10.22331/q-2021-03-23-417

On the Quantum versus Classical Learnability of Discrete Distributions

Ryan Sweke¹, Jean-Pierre Seifert^2,3, Dominik Hangleiter¹, and Jens Eisert^1,4,5

¹Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, D-14195 Berlin, Germany
²Department of Electrical Engineering and Computer Science, TU Berlin, D-10587 Berlin, Germany
³FhG SIT, D-64295 Darmstadt, Germany
⁴Helmholtz Center Berlin, D-14109 Berlin, Germany
⁵Department of Mathematics and Computer Science, Freie Universität Berlin, D-14195 Berlin, Germany

Published:	2021-03-23, volume 5, page 417
Eprint:	arXiv:2007.14451v2
Doi:	https://doi.org/10.22331/q-2021-03-23-417
Citation:	Quantum 5, 417 (2021).

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Abstract

Here we study the comparative power of classical and quantum learners for generative modelling within the Probably Approximately Correct (PAC) framework. More specifically we consider the following task: Given samples from some unknown discrete probability distribution, output with high probability an efficient algorithm for generating new samples from a good approximation of the original distribution. Our primary result is the explicit construction of a class of discrete probability distributions which, under the decisional Diffie-Hellman assumption, is provably not efficiently PAC learnable by a classical generative modelling algorithm, but for which we construct an efficient quantum learner. This class of distributions therefore provides a concrete example of a generative modelling problem for which quantum learners exhibit a provable advantage over classical learning algorithms. In addition, we discuss techniques for proving classical generative modelling hardness results, as well as the relationship between the PAC learnability of Boolean functions and the PAC learnability of discrete probability distributions.

► BibTeX data

@article{Sweke2021quantumversus,
  doi = {10.22331/q-2021-03-23-417},
  url = {https://doi.org/10.22331/q-2021-03-23-417},
  title = {On the {Q}uantum versus {C}lassical {L}earnability of {D}iscrete {D}istributions},
  author = {Sweke, Ryan and Seifert, Jean-Pierre and Hangleiter, Dominik and Eisert, Jens},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {417},
  month = mar,
  year = {2021}
}

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