  Quantum speed limits (QSLs) impose fundamental constraints on the evolution
speed of quantum systems. Traditionally, the Mandelstam-Tamm (MT) and
Margolus-Levitin (ML) bounds have been widely employed, relying on the standard
deviation and mean of energy distribution to define the QSLs. However, these
universal bounds only offer loose restrictions on the quantum evolution. Here
we introduce the generalized ML bounds, which prove to be more stringent in
constraining dynamic evolution, by utilizing moments of energy spectra of
arbitrary orders, even noninteger orders. To validate our findings, we conduct
experiments in a superconducting circuit, where we have the capability to
prepare a wide range of quantum photonic states and rigorously test these
bounds by measuring the evolution of the system and its photon statistics using
quantum state tomography. While, in general, the MT bound is effective for
short-time evolution, we identify specific parameter regimes where either the
MT or the generalized ML bounds suffice to constrain the entire evolution. Our
findings not only establish new criteria for estimating QSLs but also
substantially enhance our comprehension of the dynamic evolution of quantum
systems.
