String operators for Cheshire strings in topological phases

Elementary point charge excitations in three-plus-one-dimensional (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 disk 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 zero dimensions and finite depth quantum circuits in one dimension 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.