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

Probability models

Stanley Gudder, Nino Zanghı̀

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento BFeb 1, 198413 citations

Optimal phase change for a generalized Grover's algorithm

Christopher Cardullo, Min Kang

We study the generalized Grover's algorithm with an arbitrary amplitude vector to find the optimal phase change for maximizing the gain in probability for the target of each iteration. In the classic setting of Grover's algorithm with a real initial amplitude vector, we find that a phase change of $π$ stays optimal until the probability of observing the target is quite close to 1. We provide a formula for identifying this cut-off point based on the size of the data set. When the amplitude is truly complex, we find that the optimal phase change depends non-trivially on the complexity of the amplitude vector. We provide an optimization formula to identify the required optimal phase change.We study the generalized Grover's algorithm with an arbitrary amplitude vector to find the optimal phase change for maximizing the gain in probability for the target of each iteration. In the classic setting of Grover's algorithm with a real initial amplitude vector, we find that a phase change of $π$ stays optimal until the probability of observing the target is quite close to 1. We provide a formula for identifying this cut-off point based on the size of the data set. When the amplitude is truly complex, we find that the optimal phase change depends non-trivially on the complexity of the amplitude vector. We provide an optimization formula to identify the required optimal phase change.

arXiv:2509.20610Sep 24, 20250 citations

Finite-size scaling of correlation ratio and generalized scheme for the probability-changing cluster algorithm

Yusuke Tomita, Yutaka Okabe

We study the finite-size scaling (FSS) property of the correlation ratio, the ratio of the correlation functions with different distances. It is shown that the correlation ratio is a good estimator to determine the critical point of the second-order transition using the FSS analysis. The correlation ratio is especially useful for the analysis of the Kosterlitz-Thouless (KT) transition. We also present a generalized scheme of the probability-changing cluster algorithm, which has been recently developed by the present authors, based on the FSS property of the correlation ratio. We investigate the two-dimensional spin- $\frac{1}{2}$ quantum $\mathrm{XY}$ model of with this generalized scheme, obtaining the precise estimate of the KT transition temperature with less numerical effort.We study the finite-size scaling (FSS) property of the correlation ratio, the ratio of the correlation functions with different distances. It is shown that the correlation ratio is a good estimator to determine the critical point of the second-order transition using the FSS analysis. The correlation ratio is especially useful for the analysis of the Kosterlitz-Thouless (KT) transition. We also present a generalized scheme of the probability-changing cluster algorithm, which has been recently developed by the present authors, based on the FSS property of the correlation ratio. We investigate the two-dimensional spin-$\frac{1}{2}$ quantum $\mathrm{XY}$ model of with this generalized scheme, obtaining the precise estimate of the KT transition temperature with less numerical effort.

arXiv:cond-mat/0210074Physical review. B, Condensed matterNov 7, 200251 citations

Generalized Random Events

Václav Skřivánek, Roman Frič

International Journal of Theoretical PhysicsMar 16, 201517 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.

π

stays optimal until the probability of observing the target is quite close to 1. We provide a formula for identifying this cut-off point based on the size of the data set. When the amplitude is truly complex, we find that the optimal phase change depends non-trivially on the complexity of the amplitude vector. We provide an optimization formula to identify the required optimal phase change.

\frac{1}{2}

quantum

\mathrm{XY}

model of with this generalized scheme, obtaining the precise estimate of the KT transition temperature with less numerical effort.