Exact solution of the $q$-deformed $D_3^{(1)}$ vertex model with open boundaries

In this paper, we study the exact solution of the $q$-deformed $D_3^{(1)}$ quantum lattice model with non-diagonal open boundary condition. We demonstrate the crossing symmetry of the transfer matrix and obtain the quantum determinant. We construct the independent transfer matrix fusion identities and show that the fusion processes can be closed. Based on the fusion hierarchies and polynomial analysis, we obtain the inhomogeneous $T-Q$ relations, exact energy spectrum and Bethe ansatz equations of the system.