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Two New Results About Quantum Exact Learning

Authors Srinivasan Arunachalam, Sourav Chakraborty, Troy Lee, Manaswi Paraashar, Ronald de Wolf

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Author Details

Srinivasan Arunachalam
  • Center for Theoretical Physics, MIT, Cambridge, MA, USA
Sourav Chakraborty
  • Indian Statistical Institute, Kolkata, India
Troy Lee
  • Centre for Quantum Software and Information, School of Software, Faculty of Engineering and Information Technology, University of Technology Sydney, Australia
Manaswi Paraashar
  • Indian Statistical Institute, Kolkata, India
Ronald de Wolf
  • QuSoft, CWI and University of Amsterdam, The Netherlands

Funding

  • Arunachalam, Srinivasan: Work done when at QuSoft, CWI, Amsterdam, the Netherlands. Supported by ERC Consolidator Grant 615307 QPROGRESS and MIT-IBM Watson AI Lab under the project Machine Learning in Hilbert space.
  • Chakraborty, Sourav: Work done while on sabbatical at CWI, supported by ERC QPROGRESS.
  • Lee, Troy: Part of this work was done while at the School for Physical and Mathematical Sciences, Nanyang Technological University and the Centre for Quantum Technologies, Singapore, supported by the Singapore National Research Foundation under NRF RF Award No. NRF-NRFF2013-13.
  • de Wolf, Ronald: Supported by ERC QPROGRESS, and QuantERA project QuantAlgo 680-91-034, and NWO Gravitation project QSC.

Cite AsGet BibTex

Srinivasan Arunachalam, Sourav Chakraborty, Troy Lee, Manaswi Paraashar, and Ronald de Wolf. Two New Results About Quantum Exact Learning. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.16

Abstract

We present two new results about exact learning by quantum computers. First, we show how to exactly learn a k-Fourier-sparse n-bit Boolean function from O(k^{1.5}(log k)^2) uniform quantum examples for that function. This improves over the bound of Theta~(kn) uniformly random classical examples (Haviv and Regev, CCC'15). Our main tool is an improvement of Chang’s lemma for sparse Boolean functions. Second, we show that if a concept class {C} can be exactly learned using Q quantum membership queries, then it can also be learned using O ({Q^2}/{log Q} * log|C|) classical membership queries. This improves the previous-best simulation result (Servedio-Gortler, SICOMP'04) by a log Q-factor.

Subject Classification

ACM Subject Classification
  • Hardware → Quantum computation
  • Theory of computation → Sample complexity and generalization bounds
  • Theory of computation → Boolean function learning
Keywords
  • quantum computing
  • exact learning
  • analysis of Boolean functions
  • Fourier sparse Boolean functions

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