  We use Nielsen's geometric approach to quantify the circuit complexity in a
one-dimensional Kitaev chain across a topological phase transition. We find
that the circuit complexities of both the ground states and non-equilibrium
steady states of the Kitaev model exhibit non-analytical behaviors at the
critical points, and thus can be used to detect both {\it equilibrium} and {\it
dynamical} topological phase transitions. Moreover, we show that the locality
property of the real-space optimal Hamiltonian connecting two different ground
states depends crucially on whether the two states belong to the same or
different phases. This provides a concrete example of classifying different
gapped phases using Nielsen's circuit complexity. We further generalize our
results to a Kitaev chain with long-range pairing, and discuss generalizations
to higher dimensions. Our result opens up a new avenue for using circuit
complexity as a novel tool to understand quantum many-body systems.
