<p>In this paper, a dynamic quantum duopoly game model with bounded rationality and asymmetric information is established using the Frackiewicz quantum scheme. The analysis of this study reveals that the Nash equilibrium still exists even when introducing the quantum scheme. This paper rigorously proves the local stability of the Nash equilibrium points and investigates the complex behavior of the system through numerical simulations. Compared with the bounded rationality duopoly game model with asymmetric information that does not use the quantum scheme, we find that the dynamical system of the bounded rationality quantum duopoly game with asymmetric information under the Frackiewicz quantum scheme exhibits greater stability under the same strategy adjustment speed. Additionally, the study results show that increasing quantum entanglement delays the onset of chaos in the system. In other words, when firms compete, adopting quantum strategy schemes can delay the occurrence of market chaos. This has significant practical implications in reality.</p>