  Electronic states play a crucial role in many quantum systems of moire
superlattices, quasicrystals, and fractals. As recently reported in
\textit{Sierpi\'{n}ski} lattices [Phys. Rev. B 107, 115424 (2023)], the
critical states are revealed by the energy level-correlation spectra, which are
caused by the interplay between aperiodicity and determined self-similarity
characters. In the case of the \textit{Sierpi\'{n}ski Carpet}, our results
further demonstrate that there is some degree of spatial overlap between these
electronic states. These states could be strongly affected by its `seed
lattice' of the $generator$, and slightly modulated by the dilation pattern and
the geometrical self-similarity level. These electronic states are multifractal
by scaling the $q$-order inverse participation ratio or fractal dimension,
which correlates with the subdiffusion behavior. In the $gene$ pattern, the
averaged state-based multifractal dimension of second-order would increase as
its \textit{Hausdoff dimension} increases. Our findings could potentially
contribute to understanding quantum transports and single-particle quantum
dynamics in fractals.
