Grover algorithm for multi-objective searching with iteration auto-controlling

Abstract

This paper presents an improved Grover searching algorithm [1] which can auto-control the iterative processing when the number of target states is unknown. The final amplitude of the target states will be decreased if this number is unknown. So the question is how to perform Grover searching algorithm without the number of target states? As for this question, there are two conventional solutions. One solution tries to find the number of target states before performing the original algorithm. The other solution guesses a random k as the number of target states before performing the original algorithm. Both the two solutions need O(√N) additional times Oracle calls than original algorithm and the answer of the first solution is non-deterministic while the second solution needs to check the correctness of the result. Assuming an operator which can judge the sign of the phases of superposition state, based on this technical, this paper shows a novel solution, which can perform Grover searching algorithm even if the number of target states is unknown. This solution only needs adding one gate, which can judge the sign of phase, and one more time Oracle call than the original algorithm.