The three-level system represents the smallest quantum system capable of autonomous cycling in quantum heat engines. This study proposes a method to simulate the steady-state dynamics of a three-level quantum heat engine by designing and implementing superconducting quantum circuits. Following error mitigation, the outcomes from the quantum circuit model designed in this study, when executed on a real quantum device, closely align with theoretical predictions, thereby validating the effectiveness of the circuit model. This study offers a novel approach for investigating three-level quantum heat engines, enabling the verification of theoretical research findings while also reducing the complexity and cost of experimental procedures.