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

Chandra Sekhar Mukherjee, Subhamoy Maitra

Abstract: In this paper, we study the separation between the deterministic (classical) query complexity (D) and the exact quantum query complexity ( $$Q_E$$ 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 $$D(f)=Q_E(f)$$ 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, $$Q_E(f)=D(f)-1$$ 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 $$D(f) = n$$ D ( f ) = n , separation can be achieved as $$\frac{n}{2} \le Q_E(f) \le \lceil \frac{3n}{4} \rceil $$ n 2 ≤ Q E ( f ) ≤ ⌈ 3 n 4 ⌉ . Our results highlight how different classes of Boolean functions can be analyzed for classical–quantum separation exploiting the parity decision tree method.