A quantum algorithm for approximating the influences of Boolean functions and its applications

We investigate the influences of variables on a Boolean function $f$ based on the quantum Bernstein-Vazirani algorithm. A previous paper (Floess et al. in Math. Struct. in Comp. Science 23: 386, 2013) has proved that if a $n$-variable Boolean function $f(x_1,\ldots,x_n)$ does not depend on an input variable $x_i$, using the Bernstein-Vazirani circuit to $f$ will always obtain an output $y$ that has a $0$ in the $i$th position. We generalize this result and show that after one time running the algorithm, the probability of getting a 1 in each position $i$ is equal to the dependence degree of $f$ on the variable $x_i$, i.e. the influence of $x_i$ on $f$. On this foundation, we give an approximation algorithm to evaluate the influence of any variable on a Boolean function. Next, as an application, we use it to study the Boolean functions with juntas, and construct probabilistic quantum algorithms to learn certain Boolean functions. Compared with the deterministic algorithms given by Floess et al., our probabilistic algorithms are faster.