In this talk, I will present the first mathematical proof of the quantum mirror symmetry conjecture proposed by Aganagic-Cheng-Dijkgraaf-Krefl-Vafa about the NS-limit of closed refined topological string theory in the case of local P^2. The NS-limit of the refined topological string A-model is in fact the same as the higher genus logarithmic Gromov-Witten invariants of (P^2,E) by Bousseau’s work, while the mirror B-model is predicted to be the quantum curve and its WKB approximation, which will give the quantum correction of the classical period integrals, i.e. the quantum periods. The two key ingredients of the proof are the intrinsic definition of the quantum periods given by the deformation quantization theory and the Gross-Siebert construction of the proper Landau-Ginzburg model together with its deformation quantization by Bousseau. Due to the generality of those theories, our method could be easily generalized to all local (toric) del Pezzo surfaces. If time permits, I will make some remarks on how to use the Gross-Siebert program to explore the open string case. This talk is based on a joint work with Pierrick Bousseau and Bernd Siebert.