  We investigate multiple-$Q$ instability on a square lattice at particular
ordering wave vectors. We find that a superposition of quadruple-$Q$ spin
density waves, which are connected by fourfold rotational and mirror
symmetries, gives rise to a checkerboard bubble lattice with a collinear spin
texture as a result of the geometry among the constituent ordering wave vectors
in the Brillouin zone. By performing the simulated annealing for a fundamental
spin model, we show that such a checkerboard bubble lattice is stabilized under
an infinitesimally small easy-axis two-spin anisotropic interaction and
biquadratic interaction at zero field, while it is degenerate with an
anisotropic double-$Q$ state in the absence of the biquadratic interaction. The
obtained multiple-$Q$ structures have no intensities at high-harmonic wave
vectors in contrast to other multiple-$Q$ states, such as a magnetic skyrmion
lattice. We also show that the checkerboard bubble lattice accompanies the
charge density wave and exhibits a nearly flat band dispersion in the
electronic structure. Our results provide another route to realize exotic
multiple-$Q$ spin textures by focusing on the geometry and symmetry in terms of
the wave vectors in momentum space.
