Quantum Information Processing

, Volume 4, Issue 5, pp 355–386 | Cite as

Improved Bounds on Quantum Learning Algorithms

  • Alp Atici
  • Rocco A. Servedio


In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples.

  • Hunziker et al.[Quantum Information Processing, to appear] conjectured that for any class C of Boolean functions, the number of quantum black-box queries which are required to exactly identify an unknown function from C is \(O(\frac{\log |C|}{\sqrt{{\hat{\gamma}}^{C}}})\), where \(\hat{\gamma}^{C}\) is a combinatorial parameter of the class C. We essentially resolve this conjecture in the affirmative by giving a quantum algorithm that, for any class C, identifies any unknown function from C using \(O(\frac{\log |C| \log \log |C|}{\sqrt{{\hat{\gamma}}^{C}}})\) quantum black-box queries.

  • We consider a range of natural problems intermediate between the exact learning problem (in which the learner must obtain all bits of information about the black-box function) and the usual problem of computing a predicate (in which the learner must obtain only one bit of information about the black-box function). We give positive and negative results on when the quantum and classical query complexities of these intermediate problems are polynomially related to each other.

  • Finally, we improve the known lower bounds on the number of quantum examples (as opposed to quantum black-box queries) required for ɛ, Δ-PAC learning any concept class of Vapnik-Chervonenkis dimension d over the domain \(\{0,1\}^n\) from \(\Omega({\frac d n})\) to \(\Omega(\frac{1}{\epsilon}\log \frac{1}{\delta}+d+\frac{\sqrt{d}}{\epsilon})\). This new lower bound comes closer to matching known upper bounds for classical PAC learning.


Quantum query algorithms quantum computation computational learning theory PAC learning 


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Copyright information

© Springer Science+Business Media 2005

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Department of Computer ScienceColumbia UniversityNew YorkUSA

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