Geometry and quantum brachistochrone analysis of multiple entangled spin-1/2 particles under all-range Ising interaction

We present a unified geometric and dynamical framework for a physical system consisting of n spin-1/2 particles with all-range Ising interaction. Using the Fubini-Study formalism, we derive the metric tensor of the associated quantum state manifold and compute the corresponding Riemann curvature. Our analysis reveals that the system evolves over a smooth, compact, two-dimensional manifold with spherical topology and a dumbbell-like structure shaped by collective spin interactions. We further investigate the influence of the geometry and topology of the resulting state space on the behavior of geometric and topological phases acquired by the system. We explore how this curvature constrains the system's dynamical behavior, including its evolution speed and Fubini-Study distance between the quantum states. Within this geometric framework, we address the quantum brachistochrone problem and derive the minimal time required for optimal evolution, a result useful for time-efficient quantum circuit design. Subsequently, we explore the role of entanglement in shaping the state space geometry, modulating geometric phase, and controlling evolution speed and brachistochrone time. Our results reveal that entanglement enhances dynamics up to a critical threshold, beyond which geometric constraints begin to hinder evolution. Moreover, entanglement induces critical shifts in the geometric phase, making it a sensitive indicator of entanglement levels and a practical tool for steering quantum evolution. This approach offers valuable guidance for developing quantum technologies that require time-efficient control strategies rooted in the geometry of quantum state space.