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Parity decision tree in classical–quantum separations for certain classes of Boolean functions

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Abstract

In this paper, we study the separation between the deterministic (classical) query complexity (D) and the exact quantum query complexity (\(Q_E\)) of several Boolean function classes using the parity decision tree method. We first define the query friendly (QF) functions on n variables as the ones with minimum deterministic query complexity D(f). We observe that for each n, there exists a non-separable class of QF functions such that \(D(f)=Q_E(f)\). Further, we show that for some values of n, all the QF functions are non-separable. Then, we present QF functions for certain other values of n where separation can be demonstrated, in particular, \(Q_E(f)=D(f)-1\). In a related effort, we also study the Maiorana–McFarland (MM)-type Bent functions. We show that while for any MM Bent function f on n variables \(D(f) = n\), separation can be achieved as \(\frac{n}{2} \le Q_E(f) \le \lceil \frac{3n}{4} \rceil \). Our results highlight how different classes of Boolean functions can be analyzed for classical–quantum separation exploiting the parity decision tree method.

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Acknowledgements

The authors would like to thank the anonymous reviewers for detailed comments that improved editorial, as well as technical quality of this article. The authors also acknowledge the project (20162021) “Cryptography & Cryptanalysis: How Far Can We Bridge the Gap Between Classical and Quantum Paradigm,” awarded by the Scientific Research Council of the Department of Atomic Energy, the Board of Research in Nuclear Sciences.

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  1. Indian Statistical Institute, Kolkata, India

    Chandra Sekhar Mukherjee & Subhamoy Maitra

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  1. Chandra Sekhar Mukherjee
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  2. Subhamoy Maitra
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Correspondence to Subhamoy Maitra.

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Mukherjee, C.S., Maitra, S. Parity decision tree in classical–quantum separations for certain classes of Boolean functions. Quantum Inf Process 20, 218 (2021). https://doi.org/10.1007/s11128-021-03158-1

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