We employ a new quantum simulation approach to study the magnetism of nanodisks with Dzyaloshinsky-Moriya (DM) interaction. We find that a weak external magnetic field normal to the disk plane cannot obviously affect the single AFM skyrmion structure on small disk; but if it is sufficiently strong, it can destroy the AFM skyrmion completely. By increasing DM interaction, more self-organized magnetic domains appear, the average distance of the neighboring domains agrees well with a grid theory. In this case, a magnetic anisotropy normal to the disk plane can re-construct AFM skyrmion, providing a way for experimentalists to create AMF skyrmions.