  Elementary point charge excitations in 3+1D topological phases can condense
along a line and form a descendant excitation called the Cheshire string.
Unlike the elementary flux loop excitations in the system, Cheshire strings do
not have to appear as the boundary of a 2d disc and can exist on open line
segments. On the other hand, Cheshire strings are different from trivial
excitations that can be created with local unitaries in 0d and finite depth
quantum circuits in 1d and higher. In this paper, we show that to create a
Cheshire string, one needs a linear depth circuit that acts sequentially along
the length of the string. Once a Cheshire string is created, its deformation,
movement and fusion can be realized by finite depths circuits. This circuit
depth requirement applies to all nontrivial descendant excitations including
symmetry-protected topological chains and the Majorana chain.
