Stabilizing Grover Search via Density-Aware Phase Control

Abstract

Grover’s algorithm provides a quadratic speedup for unstructured search and represents one of the most fundamental primitives of quantum computation. However, its practical performance can be sensitive to the precise choice of the iteration count, which depends on the number of marked elements. This sensitivity becomes more pronounced when the density of solutions increases, where even small inaccuracies in estimating the number of marked states may lead to overshooting of the target subspace and a corresponding drop in success probability. In this work we introduce a density-aware parametrization of Grover search in which the phase rotations of the Grover operator are computed explicitly as a function of the solution density. The resulting construction can be interpreted as a density-dependent instantiation of Høyer’s arbitrary-phase amplitude amplification framework. The resulting operator performs a controlled rotation in the two–dimensional search subspace and naturally interpolates between sparse and dense regimes of the search space. We provide a theoretical analysis of the resulting dynamics and show that the proposed parametrization stabilizes the amplification process across a wide range of solution densities. Numerical simulations demonstrate that the method maintains near–optimal success probability for densities where the standard Grover algorithm becomes unstable, while also exhibiting improved robustness with respect to errors in estimating the number of marked states. These results suggest that density-aware phase parametrizations offer a simple yet effective strategy for extending the practical applicability of Grover-style quantum search procedures.