  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.

Data Structures and Algorithms;Quantum Physics