Abstract: Least square Toeplitz solutions and Hermitian Toeplitz solutions of the quaternion matrix equation $\sum_{i=1}^{k} \boldsymbol{A}_{i} \boldsymbol{X}_{i} \boldsymbol{B}_{i}=\boldsymbol{C}$ are studied. Utilizing real vector representation of the quaternion matrix and semi-sensor product theory, the quaternion matrix equation is transformed into its equivalent real matrix equation. Considering the structural characteristics of the Toeplitz matrix and Hermitian Toeplitz matrix, independent elements of the solution matrix are extracted to reconstruct a new solution vector, thus the computational complexity of the problem is reduced. The existing conditions of Toeplitz solutions and Hermitian Toeplitz solutions of the equation are obtained, and the general solutions of the equation are given. Finally, a numerical example is given to demonstrate the precision degree and effectiveness of the algorithm.

Key words: quaternion matrix equation, semi-tensor product of matrices, least square Toeplitz solution, least square Hermitian Toeplitz solution