An Exact Quantum Polynomial-Time Algorithm for Solving k-Junta Problem with One Uncomplemented Product

This study modifies Chen’s algorithm, the first exact quantum algorithm for testing 2-junta, and proposes an exact quantum learning algorithm for finding four dependent variables of the Boolean function f : {0, 1} n → {0, 1} with one uncomplemented product of four variables. Typically, the dependent variables are obtained by evaluating the function 2 n times in the worst-case. However, our proposed quantum algorithm only requires 8 log 2 n function operations in the worst-case. Additionally, we analyze the average-case of our algorithms. Our algorithm requires on the average 10.16 function operations at the most. Furthermore, we propose an exact quantum learning algorithm for finding k dependent variables of the Boolean function with one uncomplemented product of k variables, where k > 4. Based on our analysis, the proposed quantum algorithm only requires 4 klog 2 n function operations in the worst-case, provided that k is given. Additionally, in the average-case, the proposed algorithm requires 16 5 k function operations at the most to find k dependent variables.