  Topological quantum states cannot be created from product states with local
quantum circuits of constant depth and are in this sense more entangled than
topologically trivial states, but how entangled are they? Here we quantify the
entanglement in one-dimensional topological states by showing that local
quantum circuits of linear depth are necessary to generate them from product
states. We establish this linear lower bound for both bosonic and fermionic
one-dimensional topological phases and use symmetric circuits for phases with
symmetry. We also show that the linear lower bound can be saturated by
explicitly constructing circuits generating these topological states. The same
results hold for local quantum circuits connecting topological states in
different phases.
