An efficient quantum multi-collision search algorithm

Abstract

In this article, we propose an efficient quantum κ-collision search algorithm with low quantum memory O(n). The previous quantum κ-collision algorithms can not be converted into a low quantum memory κ-collision algorithm directly, because the time complexity of the converted algorithm is larger than the basic κ-collision algorithm. To solve this problem, we shall not only divide our low memory quantum κ-collision algorithm into several subroutines, but also need to achieve some balances between these subroutines. The time complexity of our κ-collision search algorithm is (equation presented) and the classical memory and quantum memory complexities are (equation presented) and O(n) respectively. In addition, we propose an efficient κ-claw search algorithm, which can output a κ-claw with O(n) qubits. Given 2s quantum processors, we can construct our quantum κ-collision and κ-claw parallel algorithm with the time of (equation presented) while the classical memory and quantum memory complexities are (equation presented) andO(n), respectively.