  We ask whether Krylov complexity is mutually compatible with the circuit and
Nielsen definitions of complexity. We show that the Krylov complexities between
three states fail to satisfy the triangle inequality and so cannot be a measure
of distance: there is no possible metric for which Krylov complexity is the
length of the shortest path to the target state or operator. We show this
explicitly in the simplest example, a single qubit, and in general.
