A Quantum Algorithm for Boolean Functions Processing

Abstract

Detecting junta variables is a critical issue in Boolean function analysis, circuit design optimization, and machine learning feature selection. In this paper, we investigate a novel quantum computation algorithm based on the Mz operator. The algorithm takes in an unknown oracle concealing a Boolean function with n variables and an unknown input state, which can be quantum or classical. The input state can be complete or incomplete quantum basis states, and it can be a weighted or uniform superposition of basis states. The proposed approach determines whether a given variable is a junta with a time cost of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(2/\epsilon ^{2})$ </tex-math></inline-formula> and a memory cost of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2n+6$ </tex-math></inline-formula>. The algorithm is analyzed and experimentally implemented using the Qiskit simulator and IBM’s real quantum computer. Experimental results show that the proposed approach achieved a quantum supremacy ratio 6300% higher than that of the classical method when verifying junta variables for Boolean functions with 12 variables. The results suggest that the proposed quantum method can verify junta variables in scenarios beyond the capabilities of current classical or quantum methods.