<p>Based on the Li–Du–Massar quantization scheme, this paper constructs a quantum mixed duopoly game model with quadratic costs, involving one state-owned enterprise aiming to optimize domestic social surplus and one private enterprise focused on maximizing its own profit. Considering the bounded rational expectations of the state-owned enterprise and the naïve expectations of the private enterprise, the nonlinear dynamic behavior of the system is investigated. The existence and stability of the quantum Nash equilibrium are analyzed. Complex dynamical behaviors are comprehensively examined using bifurcation diagrams, the maximum Lyapunov exponents, strange attractors, and sensitive dependence on initial conditions, confirming the emergence of chaos via period-doubling bifurcations. Furthermore, an effective method is proposed to control and restore the system from chaos back to a stable state.</p>