With the Lipkin-Meshkov-Glick (LMG) model as an illustration, we construct a thermodynamic cycle composed of two isothermal processes and two isomagnetic field processes, and we study the thermodynamic performance of this cycle accompanied by the quantum phase transition (QPT). We find that for a finite particle system working below the critical temperature, the efficiency of the cycle is capable of approaching the Carnot limit when the external magnetic field λ_{1} corresponding to one of the isomagnetic processes reaches the cross point of the ground states' energy level, which can become the critical point of the QPT in the large-N limit. Our analysis proves that the system's energy level crossings at low-temperature limits can lead to a significant improvement in the efficiency of the quantum heat engine. In the case of the thermodynamics limit (N→∞), the analytical partition function is obtained to study the efficiency of the cycle at high- and low-temperature limits. At low temperatures, when the magnetic fields of the isothermal processes are located on both sides of the critical point of the QPT, the cycle reaches maximum efficiency, and the Carnot efficiency can be achieved. This observation demonstrates that the QPT of the LMG model below critical temperature is beneficial to the thermodynamic cycle's operation.