  Determining the energy levels of a quantum system is a significant task, for
instance, to analyze reaction rates in drug discovery and catalysis or
characterize the compatibility of materials. In this paper we exploit quantum
metrology, the research field focusing on the estimation of unknown parameters
exploiting quantum resources, to address this problem for a three-level system
interacting with laser fields. The performance of simultaneous estimation of
the levels compared to independent one is also investigated in various
scenarios. Moreover, we introduce, the Hilbert-Schmidt speed (HSS), a special
type of quantum statistical speed, as a powerful figure of merit for enhancing
estimation of energy spectrum. This measure is easily computable, because it
does not require diagonalization of the system state, verifying its efficiency
in high-dimensional systems.
