Classical-Quantum Separations in Minimal Query Complexity of Boolean Functions

Query complexity is a model of computation in which functions are evaluated by making queries to the variables. In a very recent paper [Physical Review A 101, 022325 (2020)], Chen, Ye and Li provided a characterization of exact one-query quantum algorithms. We first note that this result is an immediate corollary of what was proved five years back by Montanaro, Jozsa and Mitchison [Algorithmica 71, pp. 775-796 (2015)]. Further, in this model, we concentrate on understanding the differences between classical and quantum computation in terms of number of influencing variables of a function given lowest query complexity in the classical domain. We explore the parity decision tree model to portray possible separations between classical and quantum computations in specific cases. Our results classify different classes of Boolean functions on $n$-variables ($n$ being a function of $k$), that can be evaluated (i) with exactly $k$ queries in both classical and exact quantum model or (ii) with $k$ queries in classical model and $(k-1)$ queries in exact quantum model.