In this paper, we study the rate of convergence of a sequential quadratic programming (SQP) method for nonlinear semidefinite programming (SDP) problems. Since the linear SDP constraints does not contribute to the Hessian of the Lagrangian, we propose a reduced SQP-type method, which solves an equivalent and reduced type of the nonlinear SDP problem near the optimal point. For the reduced SDP problem, the well-known and often mentioned "$σ$-term" in the second order sufficient condition vanishes. We analyze the rate of local convergence of the reduced SQP-type method and give a sufficient and necessary condition for its superlinear convergence. Furthermore, we give a sufficient and necessary condition for superlinear convergence of the SQP-type method under the nondegeneracy condition, the second-order sufficient condition with $σ$-term and the strict complementarity condition.