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A new linear and conservative finite difference scheme which preserves discrete mass and energy is developed for the two-dimensional Gross–Pitaevskii equation with angular momentum rotation. In addition to the energy estimate method and mathematical induction, we use the lifting technique as well as some well-known inequalities to establish the optimal 
$H^{1}$-error estimate for the proposed scheme with no restrictions on the grid ratio. Unlike the existing numerical solutions which are of second-order accuracy at the most, the convergence rate of the numerical solution is proved to be of order 
$O(h^{4}+\unicode[STIX]{x1D70F}^{2})$ with time step 
$\unicode[STIX]{x1D70F}$ and mesh size 
$h$. Numerical experiments have been carried out to show the efficiency and accuracy of our new method.
