Efficient quantum algorithms for set operations

Abstract

Abstract Analyzing the relations between Boolean functions has many applications in many fields, such as database systems, cryptography, and collision problems. This paper proposes four quantum algorithms that use amplitude amplification techniques to perform set operations, including Intersection, Difference, and Union, on two Boolean functions in $$\mathcal {O}(\sqrt{\mathcal {N}} )$$ O ( N ) time complexity. The proposed algorithms employ two quantum amplitude amplification techniques divided into two stages. The first stage uses the Younes et al. algorithm for quantum searching via entanglement and partial diffusion to prepare incomplete superpositions of the truth set of the first Boolean function. In the second stage, a modified version of Arima’s algorithm, along with an oracle that represent the second Boolean function, is employed to handle the set operations. The proposed algorithms have a higher probability of success in more general and comprehensive applications when compared with relevant techniques in literature.

Keywords

• Set operations
• Quantum search algorithm
• Union
• Intersection
• Difference
• Amplitude amplification