Distributional Property Testing in a Quantum World

Authors András Gilyén, Tongyang Li

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Author Details

András Gilyén
  • QuSoft, CWI and University of Amsterdam, The Netherlands
Tongyang Li
  • Department of Computer Science, Institute for Advanced Computer Studies, University of Maryland, USA
  • Joint Center for Quantum Information and Computer Science, University of Maryland, USA


A.G. thanks Ronald de Wolf, Ignacio Cirac and Yimin Ge for useful discussion. T.L. thanks Xiaodi Wu and Nengkun Yu for useful discussion.

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András Gilyén and Tongyang Li. Distributional Property Testing in a Quantum World. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 25:1-25:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


A fundamental problem in statistics and learning theory is to test properties of distributions. We show that quantum computers can solve such problems with significant speed-ups. We also introduce a novel access model for quantum distributions, enabling the coherent preparation of quantum samples, and propose a general framework that can naturally handle both classical and quantum distributions in a unified manner. Our framework generalizes and improves previous quantum algorithms for testing closeness between unknown distributions, testing independence between two distributions, and estimating the Shannon / von Neumann entropy of distributions. For classical distributions our algorithms significantly improve the precision dependence of some earlier results. We also show that in our framework procedures for classical distributions can be directly lifted to the more general case of quantum distributions, and thus obtain the first speed-ups for testing properties of density operators that can be accessed coherently rather than only via sampling.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Distribution functions
  • Theory of computation → Algorithm design techniques
  • Theory of computation → Quantum query complexity
  • distributional property testing
  • quantum algorithms
  • quantum query complexity


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