A quantum algorithm for approximating the influences of Boolean functions

We investigate the influences of variables on a Boolean function f based on the quantum BernsteinVazirani algorithm. A previous paper has proved that if a n-variable Boolean function f(x1, · · · , xn) does not depend on an input variable xi, using the Bernstein-Vazirani circuit to f will always obtain an output y that has a 0 in the ith position. We generalize this result and show that after several times running the algorithm, the number of ones in each position i is relevant to the dependence degree of f on the variable xi, i.e. the influence of xi on f , and we give the relational expression between them. 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.