Sensitivity of symmetric Boolean functions

Abstract

In quantum computing theory, the well-known Deutsch’s problem and Deutsch–Jozsa problem can be equivalent to symmetric Boolean functions. Meanwhile, sensitivity of Boolean functions is a quite important complexity measure in the query model. So far, whether symmetry means high-sensitivity problems is still considered as a challenge. In symmetric setting, based on whether all inputs in \(\{0,1\}^{n}\) are defined, this paper investigates sensitivity of total and partial Boolean functions, respectively. Firstly, we point out that the computation of sensitivity requires at most \(n+1\) classical queries or n quantum queries. Secondly, we show that the lower bound of sensitivity is not less than \(\frac{n}{2}\) except for the sensitivity 0. Finally, we discover and prove some non-trivial bounds on the number of symmetric (total and partial) Boolean functions with each possible sensitivity.