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

Article Type

Research Article

Publication Title

Quantum Information Processing


In this paper, we study the separation between the deterministic (classical) query complexity (D) and the exact quantum query complexity (QE) 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) = QE(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, QE(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 n2≤QE(f)≤⌈3n4⌉. 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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