We employ here a quantum computational method to simulate the magnetic textures in a two-dimensional (2D) antiferromagnet with square structure where the antiferromagnetic (AF) Heisenberg exchange and anisotropic Dzyaloshinskii–Moriya (DM) interactions coexist. We find from our calculated results that well symmetric and periodic AF antiskyrmionic and antivortical crystals can be stabilized by external magnetic fields applied perpendicularly to the 2D monolayer in a broad area of the T−H phase diagram; each of these chiral AF crystals can be decomposed into two almost identical ferromagnetic (FM) sublattices that are FM antiskyrmionic or antivortical crystals; the topological charge density of each FM sublattice also forms 2D crystal that coincides with the corresponding FM spin texture; the averaged topological charge per antiskyrmion of the FM sublattice is found to be one if Berg's formulas [Berg and Lüscher (1981)] are used; and the spin texture factors of the AF ASL and AVL suggest that they are the superpositions of AF helicoids in the xy-plane.