This work establishes a framework for solving inverse boundary problems with the geodesic-based quadratic Wasserstein distance ( W 2 W_{2} ). A general form of the Fréchet gradient is systematically derived from the optimal transportation (OT) theory. In addition, a fast algorithm based on the new formulation of OT on S 1 \mathbb {S}^{1} is developed to solve the corresponding optimal transport problem. The computational complexity of the algorithm is reduced to O ( N ) O(N) from O ( N 3 ) O(N^{3}) of the traditional method. Combining with the adjoint-state method, this framework provides a new computational approach for solving the challenging electrical impedance tomography problem. Numerical examples are presented to illustrate the effectiveness of our method.