  According to the Heisenberg uncertainty principle between time and energy
fluctuation, a concept of the quantum speed limit (QSL) has been established to
determine the minimum evolutionary time between quantum states. Considerable
theoretical and experimental efforts are invested in obtaining the QSL time
bounds in various scenarios. However, it remains a long-standing goal to derive
a meaningful QSL bound for a general quantum problem. Here, we propose a
geometrical approach to derive a QSL bound for closed and open quantum systems.
By solving a quantum brachistochrone problem in the framework of the Riemannian
metric, we show that the QSL between a given initial state to a final state is
determined not only by the entire dynamics of the system but also by the
individual dynamics of a critical parameter. We exemplify the utility of the
new bound in three representative scenarios, demonstrating a pronounced
advantage in finding a tight and meaningful QSL bound of a general quantum
evolution problem.
