We investigate the influences of variables on a Boolean function f\documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document} based on the quantum Bernstein–Vazirani algorithm. A previous paper (Floess et al. in Math Struct Comput Sci 23:386, 2013) has proved that if an n\documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}-variable Boolean function f(x1,…,xn)\documentclass[12pt]{minimal}
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\begin{document}$$f(x_1,\ldots ,x_n)$$\end{document} does not depend on an input variable xi\documentclass[12pt]{minimal}
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\begin{document}$$x_i$$\end{document}, using the Bernstein–Vazirani circuit for f\documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document} will always output y\documentclass[12pt]{minimal}
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\begin{document}$$y$$\end{document} that has a 0 in the i\documentclass[12pt]{minimal}
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\begin{document}$$i$$\end{document}th position. We generalize this result and show that, after running this algorithm once, the probability of getting a 1 in each position i\documentclass[12pt]{minimal}
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\begin{document}$$i$$\end{document} is equal to the dependence degree of f\documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document} on the variable xi\documentclass[12pt]{minimal}
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\begin{document}$$x_i$$\end{document}, i.e., the influence of xi\documentclass[12pt]{minimal}
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\begin{document}$$x_i$$\end{document} on f\documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document}. Based on this, 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.
