| Summary: | In this article, we propose an efficient quantum k-collision search algorithm with low quantum memory O(n). The previous quantum k-collision algorithms can not be converted into a low quantum memory k-collision algorithm directly, because the time complexity of the converted algorithm is larger than the basic k-collision algorithm. To solve this problem, we shall not only divide our low memory quantum k-collision algorithm into several subroutines, but also need to achieve some balances between these subrou tines. The time complexity of our k-collision search algorithm is O(2(2<sup>k</sup>-2)n/2<sup>k+1</sup>-3 ), and the classical memory and quantum memory complexities are O(2(2<sup>k</sup>-2)n/2<sup>k+1</sup>-3 ) and O(n) respectively. In addition, we propose an efficient k-claw search algorithm, which can output a k-claw with O(n) qubits. Given 2<sup>s</sup> quantum processors, we can construct our quantum k-collision and k-claw parallel algorithm with the time of O(2(2<sup>k</sup>-2)n-(2<sup>k+1</sup>-2<sup>k-1</sup>-3)s/2<sup>k+1</sup>-3 )while the classical memory and quantum memory complexities are O(2(2<sup>k-1</sup>-1)n+s-2<sup>k-2</sup>/2<sup>k+1</sup>-3 ) and O(n), respectively.
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