On the Quantum versus Classical Learnability of Discrete Distributions

Ryan Sweke1, Jean-Pierre Seifert2,3, Dominik Hangleiter1, and Jens Eisert1,4,5

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

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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.

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[2] Hsin-Yuan Huang, Michael Broughton, Masoud Mohseni, Ryan Babbush, Sergio Boixo, Hartmut Neven, and Jarrod R. McClean, "Power of data in quantum machine learning", arXiv:2011.01938.

[3] Hsin-Yuan Huang, Richard Kueng, and John Preskill, "Information-theoretic bounds on quantum advantage in machine learning", arXiv:2101.02464.

[4] Bob Coecke, Giovanni de Felice, Konstantinos Meichanetzidis, and Alexis Toumi, "Foundations for Near-Term Quantum Natural Language Processing", arXiv:2012.03755.

[5] Manuel S. Rudolph, Ntwali Bashige Toussaint, Amara Katabarwa, Sonika Johri, Borja Peropadre, and Alejandro Perdomo-Ortiz, "Generation of High-Resolution Handwritten Digits with an Ion-Trap Quantum Computer", arXiv:2012.03924.

[6] Daniel Herr, Benjamin Obert, and Matthias Rosenkranz, "Anomaly detection with variational quantum generative adversarial networks", arXiv:2010.10492.

[7] Brian Coyle, Maxwell Henderson, Justin Chan Jin Le, Niraj Kumar, Marco Paini, and Elham Kashefi, "Quantum versus Classical Generative Modelling in Finance", arXiv:2008.00691.

[8] Murphy Yuezhen Niu, Andrew M. Dai, Li Li, Augustus Odena, Zhengli Zhao, Vadim Smelyanskyi, Hartmut Neven, and Sergio Boixo, "Learnability and Complexity of Quantum Samples", arXiv:2010.11983.

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[10] Ieva Čepaitė, Brian Coyle, and Elham Kashefi, "A Continuous Variable Born Machine", arXiv:2011.00904.

[11] Michael L. Wall and Giuseppe D'Aguanno, "Tree tensor network classifiers for machine learning: from quantum-inspired to quantum-assisted", arXiv:2104.02249.

[12] Marcello Benedetti, Brian Coyle, Mattia Fiorentini, Michael Lubasch, and Matthias Rosenkranz, "Variational inference with a quantum computer", arXiv:2103.06720.

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