An exact quantum algorithm for testing 3-junta in Boolean functions with one uncomplemented product

Abstract

This paper modifies Chen’s algorithm, which is the first exact quantum algorithm for testing 2-junta, and proposes an exact quantum learning algorithm for finding dependent variables of the Boolean function $$ f: \left\{ {0, 1} \right\}^{n} \to \left\{ {0, 1} \right\} $$ with one uncomplemented product of three variables. Typically, the dependent variables are obtained by evaluating the function $$ 2n $$ times in the worst case. However, our proposed quantum algorithm only requires $$ O\left( {\log_{2} n} \right) $$ function operations in the worst case. In addition, the average number to perform the function is evaluated. Our algorithm requires an average of $$ 7.23 $$ function operations at the most when $$ n \ge 16 $$ . We also show that our algorithm cannot solve $$ k $$ -junta problem with one uncomplemented product if $$ 4 \le k < n/2 $$ .