  Based on large-scale density matrix renormalization group techniques, we
investigate the critical behaviors of quantum three-state Potts chains with
long-range interactions. Using fidelity susceptibility as an indicator, we
obtain a complete phase diagram of the system. The results show that as the
long-range interaction power $\alpha$ increases, the critical points
$f_{c}^{*}$ shift towards lower values. In addition, the critical threshold
$\alpha_{c}(\approx 1.43$) of the long-range interaction power is obtained for
the first time by a non-perturbative numerical method. This indicates that the
critical behavior of the system can be naturally divided into two distinct
universality classes, namely the long-range ($\alpha \textless \alpha_c$) and
short-range ($\alpha \textgreater \alpha_c$) universality classes,
qualitatively consistent with the classical $\phi^{3}$ effective field theory.
This work provides a useful reference for further research on phase transitions
in quantum spin chains with long-range interaction.
