Ferromagnetic (FM) skyrmions observed in ferromagnet/heavy-metal multilayers are considered as promising candidates for future spintronic racetrack memory. Unfortunately, the Magnus force acting on the skyrmions seriously limits this application. In contrast, that force experienced by the two sublattices of an antiferromagnetic (AFM) skyrmion can be cancelled completely, so that they are expected to move straightly along the applied electric current with greatly enhanced velocity. However, Monte Carlo simulations done by previous authors show that in the presence of Dzyaloshinsky-Moriya (DM) interaction the AFM skyrmion crystals (SLs) can only be induced in two-dimensional (2D) triangle lattice near zero temperature, or in a finite square lattice at elevated temperatures. For the sake, in this work we perform simulations for an infinite 2D AFM square lattice by means of a quantum computational method which we develop in recent years. We find from our the simulated results that the AFM-SLs can also be induced by a strong external magnetic field applied in the perpendicular direction; each of these AFM-SLs can be decomposed into two identical ferromagnetic sublattices which form a dual pair; in every sublattice, an FM skyrmion is always surrounded by four shallow vortices curling in the opposite direction; the skyrmions and vortices are all left-handed. Our findings explain the reasons why the AFM-SLs are so hard to be observed in experiments, and also suggests that quantum theory is really indispensable in order to accurately describe the magnetic systems when the AFM Heisenberg exchange (HE), DM and Zemann interactions are all involved.