Stability for a magnetic Schr\"odinger operator on a Riemann surface with boundary

We consider a magnetic Schr"odinger operator $({\mathrm{\nabla }}^X{)}^{\ast }{\mathrm{\nabla }}^X+q$ on a compact Riemann surface with boundary and prove a $\log \log$-type stability estimate in terms of Cauchy data for the electric potential and magnetic field under the assumption that they satisfy appropriate a priori bounds. We also give a similar stability result for the holonomy of the connection 1-form $X$.