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Quantum learning of concentrated Boolean functions

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Abstract

In this paper, we present a series of new results about learning of concentrated Boolean functions in the quantum computing model. Given a Boolean function f on n variables, its concentration refers to the dominant terms in its Fourier–Walsh spectrum. We show that a quantum probabilistically approximately correct learning model to learn a Boolean function characterized by its concentration yields improvements over the best-known classical method. All of our results are presented within the framework of query complexity, and therefore, our advantage represents asymptotic improvements in the number of queries using a quantum approach over its classical counterpart. Next, we prove a lower bound in the number of quantum queries needed to learn the function in the distribution-independent settings. Further, we examine the case of exact learning which is the learning variant without error. Here, we show that the query complexity grows as \(2^{{\beta }n}\) for some \(0< \beta < 1\) and therefore remains intractable even when quantum approaches are considered. This proof is based on the quantum information theoretic approach developed by researchers for the restricted case of k-sparse functions.

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Funding

This material is based upon work supported by Defense Advanced Research Projects Agency under the Grant No. FA8750-16-2-0004.

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Authors and Affiliations

  1. Department of Computer Science, Rice University, Houston, USA

    Krishna Palem & Duc Hung Pham

  2. Department of Computer Science and Engineering, IIT Hyderabad, Hyderabad, India

    M. V. Panduranga Rao

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  1. Krishna Palem
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  3. M. V. Panduranga Rao
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Correspondence to Duc Hung Pham.

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Palem, K., Pham, D.H. & Rao, M.V.P. Quantum learning of concentrated Boolean functions. Quantum Inf Process 21, 256 (2022). https://doi.org/10.1007/s11128-022-03607-5

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