  Recently, Chapman et al. argued that holographic complexities for defects
distinguish action from volume. Motivated by their work, we study complexity of
quantum states in conformal field theory with boundary. In generic
two-dimensional BCFT, we work on the path-integral optimization which gives one
of field-theoretic definitions for the complexity. We also perform holographic
computations of the complexity in Takayanagi's AdS/BCFT model following by the
"complexity $=$ volume" conjecture and "complexity $=$ action" conjecture. We
find that increments of the complexity due to the boundary show the same
divergent structures in these models except for the CA complexity in the
AdS$_3$/BCFT$_2$ model as the argument by Chapman et al. Thus, we conclude that
boundary does not distinguish the complexities in general.
