We study the fully generalized Grover's algorithm to find the optimal phase changes for each step of the iteration to maximize gain in probability of observation of the target, and when phase matching is required. We find that classical Grover's algorithm and phase matching remains to be optimal till the target probability gets close 1. However, as the probability of observation approaches 1, the optimal phase changes differ from $π$ and no longer observe phase matching. We provide the optimization statement to find the optimal phase changes given the current amplitude vector and the size of the set. To analyze this formula, we approach it from a numerical and analytical perspective, with the analytical perspective focusing on special cases that simplify the optimization and allow for general statements about its behavior. Finally, we provide an example of a 5 qubit system and show that for the final iteration the optimal phase changes differ from traditional Grover's algorithm and do not observe phase matching, but lead to an increase in the probability of the target.We study the fully generalized Grover's algorithm to find the optimal phase changes for each step of the iteration to maximize gain in probability of observation of the target, and when phase matching is required. We find that classical Grover's algorithm and phase matching remains to be optimal till the target probability gets close 1. However, as the probability of observation approaches 1, the optimal phase changes differ from $π$ and no longer observe phase matching. We provide the optimization statement to find the optimal phase changes given the current amplitude vector and the size of the set. To analyze this formula, we approach it from a numerical and analytical perspective, with the analytical perspective focusing on special cases that simplify the optimization and allow for general statements about its behavior. Finally, we provide an example of a 5 qubit system and show that for the final iteration the optimal phase changes differ from traditional Grover's algorithm and do not observe phase matching, but lead to an increase in the probability of the target.

quant-pharXiv:2605.13758May 14, 20260 citations

Phase Matching in Quantum Searching and the Improved Grover Algorithm

Yang Sun

We briefly introduced some of our recent work related to the phase matching condition in quantum searching algorithms and the improved Grover algorithm. When one replaces the two phase inversions in the Grover algorithm with arbitrary phase rotations, the modified algorithm usually fails in searching the marked state unless a phase matching condition is satisfied between the two phases. The Grover algorithm is not 100% in success rate, an improved Grover algorithm with zero-failure rate is given by replacing the phase inversions with angles that depends on the size of the database. Other aspects of the Grover algorithm such as the SO(3) picture of quantum searching, the dominant gate imperfections in the Grover algorithm are also mentioned.We briefly introduced some of our recent work related to the phase matching condition in quantum searching algorithms and the improved Grover algorithm. When one replaces the two phase inversions in the Grover algorithm with arbitrary phase rotations, the modified algorithm usually fails in searching the marked state unless a phase matching condition is satisfied between the two phases. The Grover algorithm is not 100% in success rate, an improved Grover algorithm with zero-failure rate is given by replacing the phase inversions with angles that depends on the size of the database. Other aspects of the Grover algorithm such as the SO(3) picture of quantum searching, the dominant gate imperfections in the Grover algorithm are also mentioned.

Yuanzihe wuli pinglunJan 1, 20047 citations

Phase matching in grover's algorithm

Xin Li, Kaoping Song, Ning Sun +1 more

When the Grover's algorithm is applied to search an unordered database, the probability of getting correct results usually decreases with the increase of marked items. The reason for this phenomenon is analyzed in this Letter, the Grover iteration is studied, and a new phase matching is proposed. With application of the new phase matching, when the fraction of marked items is greater than 1/3, the probability of getting correct results is greater than 25/27 with only one Grover iteration. The validity of the new phase matching is verified by a search example.When the Grover's algorithm is applied to search an unordered database, the probability of getting correct results usually decreases with the increase of marked items. The reason for this phenomenon is analyzed in this Letter, the Grover iteration is studied, and a new phase matching is proposed. With application of the new phase matching, when the fraction of marked items is greater than 1/3, the probability of getting correct results is greater than 25/27 with only one Grover iteration. The validity of the new phase matching is verified by a search example.

Chinese Control Conference (CCC)Jul 26, 20131 citation

A Comparative Analysis of Grover's Algorithm with Classical Algorithms

Issac Diana Jeba Jingle, Sharlene Anna Pereira

Lecture notes in networks and systemsJan 1, 20260 citations

π

and no longer observe phase matching. We provide the optimization statement to find the optimal phase changes given the current amplitude vector and the size of the set. To analyze this formula, we approach it from a numerical and analytical perspective, with the analytical perspective focusing on special cases that simplify the optimization and allow for general statements about its behavior. Finally, we provide an example of a 5 qubit system and show that for the final iteration the optimal phase changes differ from traditional Grover's algorithm and do not observe phase matching, but lead to an increase in the probability of the target.