  In this work, we find that the complexity of quantum many-body states,
defined as a spread in the Krylov basis, may serve as a new probe that
distinguishes topological phases of matter. We illustrate this analytically in
one of the representative examples, the Su-Schrieffer-Heeger model, finding
that spread complexity becomes constant in the topological phase. Moreover, in
the same setup, we analyze exactly solvable quench protocols where the
evolution of the spread complexity shows distinct dynamical features depending
on the topological vs non-topological phase of the initial state as well as the
quench Hamiltonian.
